Planet Musings

The notorious Jacobian conjecture can be formulated concretely over the complex numbers as follows.

Conjecture 1 (Jacobian Conjecture) Let be a polynomial map in complex variables, whose Jacobian is a non-zero constant. Then is invertible (with polynomial inverse).

The condition that the Jacobian is non-zero is equivalent to being locally invertible. (The implication of local invertibility from non-vanishing Jacobian follows from the inverse function theorem; the converse implication can be derived from the Weierstrass preparation theorem, but is omitted here.) Also, from the fundamental theorem of algebra, once the Jacobian polynomial is non-zero, it must be constant. So the hypothesis “Jacobian is a non-zero constant” can be replaced with “ is locally invertible”. So the Jacobian conjecture can be viewed as an assertion that local invertibility implies global invertibility. The complex numbers can be easily replaced with other fields of characteristic zero by the Lefschetz principle, but I prefer to work in the concrete setting of the complex numbers.

It was recently shown (using the Fable AI) that the conjecture is false in three dimensions (and thus in higher dimensions as well):

Theorem 2 (Counterexample to conjecture) There exists a polynomial which has non-zero constant Jacobian, but is not invertible.

The example can be stated completely explicitly: one can take

The example has since been retroactively explained in more geometric terms. As a “digestion” exercise to myself, I sought to write this explanation with relatively little use of algebraic geometry, in a manner that minimizes the amount of “miracles” required, although there are still a few places where some remarkable phenomena occur.

It is convenient to use the local injectivity formulation, and to generalize the domain to an equivalent affine variety. Namely, we will show

Theorem 3 (Counterexample, reformulated) There exists an affine variety that is isomorphic to by polynomial changes of variable, and a polynomial map which is locally injective, but not globally injective.

Clearly one can get from Theorem 3 to Theorem 2 by composing with the isomorphism and using the previously mentioned fact that local injectivity implies non-zero constant Jacobian. Our objective is now to find data , that obeys three separate properties:

• (a) is locally injective on .
• (b) is not globally injective on .
• (c) is isomorphic to by polynomial changes of variable.

(A pedantic remark: strictly speaking, in the arguments below, we not only replace the domain of by an equivalent variety , but also replace the range of by an equivalent variety . But the equivalence between and is a boring linear isomorphism ( will just be a hyperplane in a four-dimensional vector space ), so we do not highlight this aspect of the construction.)

It turns out that and can be built out of the operation of multiplication of low degree polynomials. Namely, consider the following three simple affine spaces:

• The space of linear homogeneous polynomials of two complex variables .
• The space of quadratic homogeneous polynomials of two complex variables .
• The space of cubic homogeneous polynomials of two complex variables .

The map , essentially a map from to , is clearly polynomial; it is given explicitly in coordinates as

• If one applies a scaling for some non-zero complex numbers , then the product is scaled by : .
• If one applies a change of variables for some invertible linear transformation , then the product is transformed by : .

The five-dimensional domain is of course larger than the four-dimensional range , so the map clearly cannot be injective. This can already be seen from the scaling symmetry, as the specific scalings

It will be convenient to “spend” the scaling symmetry to obtain a useful normalization. If is a linear polynomial and is a quadratic polynomial, the resultant can be defined by the determinant

We now have a restricted multiplication map (which by abuse of notation we will continue to call ) from the four-dimensional variety

But we now also have property (a)! Suppose we want to show the local injectivity of in the neighborhood of a pair with . As the resultant is non-vanishing, the root of (which exists in the Riemann sphere, or projective line if you prefer) is distinct from the two roots of (though the latter two roots could be equal to each other). Applying the action (which performs Möbius transforms on the roots), one can assume without loss of generality that is the point at infinity (or equivalently ), thus for some complex number and for some complex numbers , with the resultant condition (7) simplifies to (so in particular are also non-zero). It is then clear that if one perturbs and by a small amount (say, modifying each coefficient by ), then the root of will perturb to something large (), while the roots of stay bounded. Thus, just from knowledge of the product , one can reconstruct which of the three roots of this cubic polynomial will be the perturbed root of , and which two will be the perturbed roots of ; from this and (6), (7) we can also reconstruct the leading coefficient of , and this completely determines both and . This establishes the local injectivity property (a). (In fact it is étale, but we will not need the machinery of étale maps here.)

Unfortunately, (the four-dimensional analogue of) condition (c) fails: the quadric hypersurface (8) is not isomorphic to the affine space . But we can try to get around this by passing to a three-dimensional slice. Let be some three-dimensional affine plane of (which we will take to avoid the origin for technical reasons), then we can restrict as a map from the set

Let’s see how. The affine hyperplanes in avoiding the origin are parameterized by the dual space of avoiding the origin, which one can think of as the non-zero third order homogeneous differential operators in two variables. Indeed, every such operator generates an affine hyperplane that avoids the origin, and conversely by duality every affine hyperplane avoiding the origin arises in this form uniquely. Just as the cubic polynomials in can be factored into three linear polynomials, the differential operators in the dual space can also be factored into three linear differential operators, e.g.,

• Operators where the three roots are all distinct, thus for independent first-order operators .
• Operators where two roots coincide and one is distinct, thus for independent first-order operators .
• Operators where all three roots coincide, thus for some first-order operator .

It turns out that the affine miracle for (9) occurs precisely in the second case, when has two identical roots. I do not have a completely satisfactory geometric explanation for this miracle, but one can verify it by the following coordinate computation.

By applying the action, we can normalize so that , thus is now the affine hyperplane of cubic polynomials with . Using (2) and (5), the variety (9) can now be described explicitly in coordinates as

So we just need to glue back in the fiber. Indeed, from (10) we see that the fiber at is just

The standard way to proceed here is to manipulate various tangent spaces using the modern machinery of algebraic geometry and commutative algebra, but given my own background, I prefer to adopt the language of analysis, and in particular big-O notation (in place of the ideals used in algebraic geometry), in order to investigate the limit by hand. On the variety (10), let us use to denote any multiple of by a polynomial expression in . Thus, for instance, the equation implies that

We can get some more precise asymptotics by also taking advantage of (15). Substituting into (15), we obtain after some algebra

Expanding the error term in (16) as , and doing a little more algebra, we thus have a polynomial change of variables

The previous computations, when expanded out, also gives polynomial inverse maps:

AI disclosure: I used an AI chatbot to discuss various aspects of this problem and to confirm several of the calculations made here.

As many of you have probably heard already, yesterday morning, Levent Alpöge tweeted that Fable had found a counterexample to the Jacobian Conjecture. Specifically, let

Then the Jacobian of is easily checked to be . However, the map is generically three to one, not bijective.

I’m sure many of you are playing with these polynomials to see what you can figure out about them. This is a place for us to share our observations. I’ll post a few minor observations of my own soon.

Correct outcome, but imagine if Argentina had won this on penalties without taking a single shot on goal in 120 minutes of actual play! That could easily have happened, and what an embarrassment.

My proposed rule change: the number of penalty kicks a team gets is either five or the number of shots taken on the field of play, whichever is fewer. Couldn’t mount a serious attack the whole game and only made two shots on goal? Too bad, so sad.

Maxwell’s equations are invariant under Lorentz transformations. The usual equations of fluid flow are not! Like the rest of Newtonian mechanics, they’re invariant under Galilean transformations like

So, if we simply slap these two theories together, we get a mess! How can we study electrically conductive fluids—like plasma—without bringing special relativity into the game?

We can use a limiting case of Maxwell’s equations where we ignore terms that become tiny when all the particles are moving much slower than light.

There seem to be at least two ways to do this: there’s an ‘electric limit’ of Maxwell’s equations and a ‘magnetic limit’. Both are invariant under Galilean transformations. The original derivation of these limits by Le Bellac and Lévy-Leblond in 1973 used the version of Maxwell’s equations including the electric permittivity and magnetic permeability of the vacuum, whose product is . This is convenient but not necessary, as explained here:

• Jose A. Heras, The Galilean limits of Maxwell’s equations.

In the magnetic limit of Maxwell’s equations, we throw out effects due to time-varying electric fields:

People often use the magnetic limit when studying nonrelativistic electrically conductive fluids. In this situation they often consider a version of the magnetic limit where the charge density is zero, since this is typically close to true in a plasma. However Heras does not do this, nor does the original paper:

• Le Bellac and Levy-Leblond, Galilean electromagnetism.

In the electric limit of Maxwell’s equations, we throw out effects due to time-varying magnetic fields:

It’s fun to compare the magnetic and electric limits.

The magnetic limit has been called ‘pre-Maxwellian’, because it’s like electromagnetism before Maxwell added the extra term that makes a changing electric field create a curl in the magnetic field. Without this term there is no light!

In the electric limit you also can’t have light, because it’s missing the term that makes a changing magnetic field create a curl in the electric field.

In the magnetic limit you can’t have capacitors, because those store energy in the electric field, and in the magnetic limit the energy density is just .

Similarly, in the electric limit you can’t have inductors, because inductors store energy in the magnetic field, and in this limit the energy density is just .

It’s all nicely symmetrical! But still somewhat mysterious to me. All the derivations of these limits that I’ve seen involve too many parameters for my taste, and too much talk. But that’s how I often feel when I’m just starting to study a piece of physics.

Besides the two papers mentioned in my last post, I’ve been looking at this:

• Giovanni Manfredi, Non-relativistic limits of Maxwell’s equations.

There’s a lot I haven’t explained here. I haven’t even said how the electric or magnetic fields transform under Galilean boosts in these limiting theories! I find this subject fairly confusing, and I’d probably have to redo all the calculations to really understand them. As Feynman said, “what I cannot create I do not understand”.

Someday I should dig deeper into this subject and explain how the two limits work in a way I find satisfying. I should also draw the connections to this earlier article of mine:

• Magnetohydrodynamics.

I don’t usually talk about particle physics here. I have a whole series of articles about octonions and the Standard Model on my other blog. But I’m kind of excited about this new paper, so I’ll talk about it here too:

• John Baez, Endre Bokor and Latham Boyle, Jordan pair quantum theory and the Standard Model.

Jordan algebras were introduced by Jordan, von Neumann and Wigner in 1934 in an attempt to formalize algebras of observables in quantum theory. They come in 4 infinite series—but there’s one more, the ‘exceptional Jordan algebra’, consisting of 3 × 3 self-adjoint matrices of octonions. For years physicists sought to find some use for it.

In 2018, Todorov and Dubois–Violette noticed that the symmetries of the exceptional Jordan include the Standard Model gauge group in a nice way. But it was unclear how to bring in the fermions—the quarks and leptons. That’s what our new paper does.

To do this, we need to go beyond Jordan algebras. Jordan pairs and Jordan triples are two closely linked formalisms that generalize Jordan algebras. Our paper explains them in detail—and how they’re connected to geometry and quantum mechanics. But here I will mostly skip that wonderful story, so I can quickly explain the connection to the Standard Model.

Here’s how the Standard Model gauge group, together with its representation on one generation of fermions, drops out of a Jordan triple.

The bi-Cayley triple

Let

be the bioctonions: octonions with complex coefficients. Write for the space of column vectors with two bioctonion entries.

has a certain triple product

which obey the axioms of a gadget called a ‘positive hermitian Jordan triple’. It’s called the bi-Cayley triple.

Now, every positive hermitian Jordan triple gives rise to a -graded real Lie algebra

Not a Lie superalgebra: a plain old-fashioned Lie algebra with a -grading!

How does this work? We take the hermitian Jordan triple itself to be The Lie algebra consists of all linear maps from to itself that are of this form:

for some These maps are called real inner derivations. They form a Lie algebra since the commutator of two such maps is another such map. With a bit more work we can define other operations making all of into a -graded Lie algebra.

So, we get a big Lie algebra and a Lie subalgebra sitting inside it. From this we get two Lie groups: a big one whose Lie algebra is and a subgroup whose Lie algebra is

The quotient is is a nice kind of manifold called a hermitian symmetric space. Conversely, any compact hermitian symmetric space give rise to a positive hermitian Jordan triple!

This geometric picture is revealing. The group acts transitively as symmetries of our hermitian symmetric space, while the stabilizer of any point is isomorphic to Our original Jordan triple, is then the tangent space of that point! So, acts on this Jordan triple. This action preserves the triple product, and we call the real inner automorphism group of our Jordan triple.

Here’s another great thing about the geometric picture: hermitian symmetric spaces were classified by Eli Cartan (who seems to have spent his life classifying things). As a result we also know the classification of positive hermitian Jordan triples. They come in four infinite series together with two exceptions. One is the bi-Cayley triple, and other is the Albert triple, which is the complexification of the exceptional Jordan algebra. The bi-Cayley triple is a subtriple of the Albert triple. It’s these two exceptions that are connected to the Standard Model. But we’ll start with the bi-Cayley triple.

The 3-graded Lie algebra coming from the bi-Cayley triple is the compact real form of :

The even part of this Lie algebra is in brackets. The corresponding hermitian symmetric space is called the bioctonionic plane The even part of our 3-graded Lie algebra, generates the stabilizer of a point in the bioctonionic plane. The odd part, our friend is the tangent space of that point.

Here’s the first big surprise. The even part transforms as the adjoint representation of while the odd part itself transforms as the 16-dimensional complex spinor representation of Ignoring the extra for a moment, this is exactly what we see in a grand unified theory: gauge bosons in the adjoint representation, and one generation of fermions in the 16-dimensional spinor representation.

So before we do anything, the bi-Cayley triple already smells like it contains the ingredients of an grand unified theory.

Tripotents

In a Jordan algebra the important elements are the idempotents, In a Jordan triple their role is played by tripotents: elements with

A tripotent always lets us split into three parts via something called its Peirce decomposition. The operator has eigenvalues 0, 1/2, and 1, so splits into the corresponding eigenspaces

which are called the Peirce 0-space, Peirce 1/2-space and Peirce 1-space of A tripotent is called minimal when its Peirce 1-space is one-dimensional. Two tripotents are called colinear when each lies in the other’s Peirce 1/2-space.

I can’t resist explaining some of the quantum physics here. In a hermitian Jordan triple, the triple product is linear in the first and last slot, but conjugate-linear in the middle slot. So, if you multiply a tripotent by a phase you get a new tripotent:

This should remind you of how when you multiply a unit vector in a Hilbert space by a phase, you get a new unit vector. In Jordan triple quantum mechanics, minimal tripotents take the place of these unit vectors. The hermitian symmetric space that I was talking about earlier is the same as the space of minimal tripotents mod phase! So, it generalizes the familiar space of ‘pure states’ in quantum mechanics: unit vectors mod phase.

But let’s get back to the Standard Model.

A chain of Jordan triples

From here on, the single fact driving everything is this: in any hermitian Jordan triple, any minimal tripotent’s Peirce 1/2-space is itself a hermitian Jordan triple!

If we run this starting from the bi-Cayley triple, we get this chain of hermitian Jordan triples, where each row’s 1/2-space is the next row’s triple:

Jordan triple	Lie algebra (even part in brackets)

Here is the Jordan triple of antisymmetric complex matrices, is the Jordan triple of complex matrices, and

is the true Standard Model gauge group.

The gauge group from two tripotents

Start with the bi-Cayley triple. Choose two colinear minimal tripotents Descend the table twice:

• Start with which has real inner automorphism group

• Fix Its Peirce latex W’ = \mathfrak{a}_5(\mathbb{C}),$ with real inner automorphism group

• Fix (colinear with so living in ). Its Peirce latex W’$ is with real inner automorphism group exactly

In other words, the subspace of the bi-Cayley triple colinear with both and is a Jordan triple whose real inner automorphism group is the Standard Model gauge group.

The choice of and also pins down how sits inside the original group At each we step take the subgroup that acts with determinant and preserves the chosen tripotent up to a phase; this gives a chain of subgroups whose members are and so we get the embeddings

In particle physics, this is the classic chain taking us from the so-called grand unified theory down to the grand unified theory down to the Standard Model. And it’s well known that restricting the 16-dimensional complex spinor representation of along this chain gives precisely the Standard Model representation on one generation of fermions! So we get one generation of Standard Model fermions this way.

The six particles types as Peirce spaces

We have gotten the representation of the Standard Model gauge group on one generation of fermions without any fuss. But it’s also fun to peer into the details, and see how the different kinds of fermions emerge.

For any tripotent we have projections and onto its three eigenspaces: its so-called Peirce projectors. Since we get the Standard Model structure using two minimal tripotents and in the bi-Cayley triple , there are nine composites of two Peirce projectors we can apply to this triple. This is how we pick out the different kinds of fermions!

As a representation of the Standard Model Lie algebra

any generation of Standard Model fermions transforms as the direct sum of six irreducible representations:

These correspond to the six types of left-handed fermion: Six irreducible pieces, six particle types.

It turns out these are exactly the six nonzero components of the Peirce decomposition of with respect to both and Those six match up one-to-one with the particle types:

Peirce projector	representation of	particle type
	(3, 2, +1/6)
	( 1, +1/3)
	( 1, −2/3)
	(1, 2, −1/2)
	(1, 1, +1)
	(1, 1, 0)

The remaining three combinations— and —all vanish, which is why we land on six pieces and not nine.

So the whole package—the gauge group the embedding the representation and even the split of one generation into its six particle multiplets as distinct Peirce components—all comes out of the single object once you choose two colinear minimal tripotents.

And if you prefer to start one level up, with the Albert triple you get the same result by choosing three mutually colinear tripotents instead of two—but for that, read our paper!

Last time I described a way to get the Standard Model gauge group from the exceptional Jordan algebra. But that approach gave no obvious nice way to put quarks and leptons into the picture. This new paper tackles that problem:

• John Baez, Endre Bokor and Latham Boyle, Jordan pair quantum theory and the Standard Model.

Jordan pairs and Jordan triples are two closely linked formalisms that generalize Jordan algebras. Our paper explains them in detail — and how they’re connected to geometry and quantum mechanics. Here I will mostly skip that wonderful story, so I can quickly explain the connection to the Standard Model.

Here’s how the Standard Model gauge group, together with its representation on one generation of fermions, drops out of a Jordan triple.

The bi-Cayley triple

Let

$$\mathbb{O}_\mathbb{C} = \mathbb{C} \textstyle{\otimes}_\mathbb{R} \mathbb{O}$$

be the bioctonions: octonions with complex coefficients. Write $\mathbb{O}_\mathbb{C}^2$ for the space of column vectors with two bioctonion entries.

$\mathbb{O}_\mathbb{C}^2$ has a certain triple product

$$[x,y,z]=\frac{1}{2}(x(y^{\dagger}z)+z(y^{\dagger}x))$$

which obey the axioms of a gadget called a ‘positive hermitian Jordan triple’. It’s called the bi-Cayley triple.

Now, every positive hermitian Jordan triple gives rise to a $\mathbb{Z}_2$-graded real Lie algebra

$$\mathbf{k} = \mathbf{k}_0 \textstyle{\oplus} \mathbf{k}_1$$

Not a Lie superalgebra: a plain old-fashioned Lie algebra with a $\mathbb{Z}_2$-grading!

How does this work? We take the hermitian Jordan triple itself to be $\mathbf{k}_1$. The Lie algebra $\mathbf{k}_0$ consists of all linear maps from $\mathbf{k}_1$ to itself that are of this form:

$$x \mapsto [a,b,x] - [b,a,x]$$

for some $a,b \in \mathbf{k}_1$. These maps are called real inner derivations. They form a Lie algebra since the commutator of two such maps is another such map. With a bit more work we can define other operations making all of $\mathbf{k}$ into a $\mathbb{Z}_2$-graded Lie algebra.

So, we get a big Lie algebra $\mathbf{k}$, and a Lie subalgebra $\mathbf{k}_0$ sitting inside it. From this we get two Lie groups: a big one $K$ whose Lie algebra is $\mathbf{k}$, and a subgroup $K_0$, whose Lie algebra is $\mathbf{k}_0$.

The quotient is $K/K_0$ is a nice kind of manifold called a hermitian symmetric space. Conversely, any compact hermitian symmetric space give rise to a positive hermitian Jordan triple!

This geometric picture is revealing. The group $K$ acts transitively as symmetries of our hermitian symmetric space, while the stabilizer of any point is isomorphic to $K_0$. Our original Jordan triple, $\mathbf{k}_1$, is then the tangent space of that point. So, $K_0$ acts on the Jordan triple. This action preserves the triple product, and we call $K_0$ the real inner automorphism group of our Jordan triple.

Here’s another great thing about the geometric picture: hermitian symmetric spaces were classified by Eli Cartan (who seems to have spent his life classifying things). As a result we also know the classification of positive hermitian Jordan triples. They come in four infinite series together with two exceptions. One is the bi-Cayley triple, and other is the Albert triple, which is the complexification of the exceptional Jordan algebra. The bi-Cayley triple is a subtriple of the Albert triple. It’s these two exceptions that are connected to the Standard Model. But we’ll start with the bi-Cayley triple.

The 3-graded Lie algebra coming from the bi-Cayley triple is the compact real form of $\mathfrak{e}_6$:

$$\mathfrak{e}_6 = \big[\mathfrak{so}(10) \textstyle{\oplus} \mathfrak{u}(1)\big] \textstyle{\oplus} \mathbb{O}_\mathbb{C}^2.$$

The even part of this Lie algebra is in brackets. The corresponding hermitian symmetric space is called the bioctonionic plane $(\mathbb{C}\otimes\mathbb{O})P^2$. I explained it in Part 12. The even part of our 3-graded Lie algebra, $\mathfrak{so}(10)\oplus \mathfrak{u}(1)$, generates the stabilizer of a point in the bioctonionic plane. The odd part, our friend $\mathbb{O}_\mathbb{C}^2$, is the tangent space of that point.

Here’s the first big surprise. The even part transforms as the adjoint representation of $\mathrm{Spin}(10)$, while the odd part itself transforms as the 16-dimensional complex spinor representation of $\mathrm{Spin}(10)$. Ignoring the extra $\mathrm{U}(1)$ for a moment, this is exactly what we see in a $\mathrm{SO}(10)$ grand unified theory: gauge bosons in the adjoint representation, and one generation of fermions in the 16-dimensional spinor representation.

So before we do anything, the bi-Cayley triple already smells like it contains the ingredients of an $\mathrm{SO}(10)$ grand unified theory.

Tripotents

In a Jordan algebra the important elements are the idempotents, $e^2 = e$. In a Jordan triple $W$ their role is played by tripotents: elements $e$ with

$$[e,e,e] = e.$$

A tripotent always lets us split $W$ into three parts via something called its Peirce decomposition. The operator $w \mapsto [e,e,w]$ has eigenvalues $0, \tfrac{1}{2}, 1$, and $W$ splits into the corresponding eigenspaces

$$W = W_0(e) \textstyle{\oplus} W_{1/2}(e) \textstyle{\oplus} W_1(e),$$

which are called the Peirce 0-space, Peirce $\tfrac{1}{2}$-space and Peirce 1-space of $e$. A tripotent is called minimal when its Peirce $1$-space is one-dimensional: minimal tripotents are the analogues of unit vectors in ordinary quantum theory. Two tripotents $e_1, e_2$ are called colinear when each lies in the other’s Peirce $\tfrac{1}{2}$-space.

I can’t resist explaining some of the quantum physics here. I said I wouldn’t, but I can’t help it. In a hermitian Jordan triple, the triple product $[-,-,-]$ is linear in the first and last slot, but conjugate-linear in the middle slot. So, if you multiply a tripotent by a phase $\alpha$, you get a new tripotent:

$$[\alpha e, \alpha e, \alpha e] = \alpha \overline{\alpha} \alpha e = \alpha e$$

This should remind you of how when you multiply a unit vector in a Hilbert space by a phase, you get a new unit vector. In Jordan triple quantum mechanics, minimal tripotents take the place of these unit vectors. And guess what: the hermitian symmetric space $K/K_0$ that I was talking about earlier is also the space of minimal tripotents mod phase! So, it generalizes the familiar space of ‘pure states’ in quantum mechanics, which are unit vectors mod phase.

But let’s get back to the Standard Model.

A chain of Jordan triples

From here on, the single fact driving everything is this: in any hermitian Jordan triple, any minimal tripotent’s Peirce $\tfrac{1}{2}$-space is itself a hermitian Jordan triple!

If we run this starting from the bi-Cayley triple, we get this chain of hermitian Jordan triples, where each row’s $\tfrac{1}{2}$-space is the next row’s triple:

Here $\mathfrak{a}_5(\mathbb{C})$ is the Jordan triple of antisymmetric $5\times 5$ complex matrices, $\mathrm{M}_{3,2}(\mathbb{C})$ is the Jordan triple of $3\times 2$ complex matrices, $\mathfrak{g}_{\mathrm{SM}} = \mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus \mathfrak{u}(1)$, and

$$G_{\mathrm{SM}} = \mathrm{S}(\mathrm{U}(2) \times \mathrm{U}(3)) \cong (\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1))/\mathbb{Z}_6$$

is the true Standard Model gauge group.

The gauge group from two tripotents

Now pick two colinear minimal tripotents $e_1, e_2 \in W$. Descend the table twice:

• Start with $W = \mathbb{O}_\mathbb{C}^2$, which has real inner automorphism group $(\mathrm{Spin}(10)\times\mathrm{U}(1))/\mathbb{Z}_4$.
• Fix $e_1$. Its Peirce $\tfrac{1}{2}$-space is $W' = \mathfrak{a}_5(\mathbb{C})$, with real inner automorphism group $\mathrm{SU}(5)\times\mathrm{U}(1)$.
• Fix $e_2$ (colinear with $e_1$, so living in $W'$). Its Peirce $\tfrac{1}{2}$-space in $W'$ is $W'' = \mathrm{M}_{3,2}(\mathbb{C})$, with real inner automorphism group exactly $G_{\mathrm{SM}}$.

In other words, the subspace of the bi-Cayley triple colinear with both $e_1$ and $e_2$ is a Jordan triple whose real inner automorphism group is the Standard Model gauge group.

The choice of $e_1$ and $e_2$ also pins down how $G_{\mathrm{SM}}$ sits inside the original group $\mathrm{E}_6$. At each we step take the subgroup that acts with determinant $1$ and preserves the chosen tripotent up to a phase; this gives a chain of subgroups whose members are $\mathrm{Spin}(10)$, $\mathrm{U}(5)$, and $G_{\mathrm{SM}}$, so we get the embedding

$$G_{\mathrm{SM}} \subset \mathrm{SU}(5) \subset \mathrm{Spin}(10).$$

In particle physics, this is the classic chain taking us from the so-called $\mathrm{SO}(10)$ grand unified theory down to the $\mathrm{SU}(5)$ grand unified theory down to the Standard Model. And it’s well known that restricting the 16-dimensional complex spinor representation of $\mathrm{Spin}(10)$ along this chain gives precisely the Standard Model representation $\rho_{\mathrm{SM}}$ on one generation of fermions! So we get one generation of Standard Model fermions this way.

The six particles types as Peirce spaces

We have gotten the representation of the Standard Model gauge group on one generation of fermions without any fuss. But it’s also fun to peer into the details, and see how the different kinds of fermions emerge. We can get them using the fact that for any tripotent $e$, we have projections $P_0(e), P_{1/2}(e)$ and $P_1(e)$ onto its three eigenspaces: its so-called Peirce projectors.

Since we get the Standard Model gauge group and its representation on fermions from two minimal tripotents $e_1$ and $e_2$, we have nine Peirce projectors we can apply to our Jordan triple $\mathbb{O}_{\mathbb{C}}^2$. Let’s use these to pick out various kinds of particles!

As a representation of the Standard Model Lie algebra

$$\mathfrak{g}_{\mathrm{SM}} = \mathfrak{su}(3) \textstyle{\oplus} \mathfrak{su}(2) \textstyle{\oplus} \mathfrak{u}(1) ,$$

any generation of Standard Model fermions transforms as the direct sum of six irreducible representations:

$$\rho_{\mathrm{SM}} = (3,2,\tfrac{1}{6}) \textstyle{\oplus} (\bar 3,1,\tfrac{1}{3}) \textstyle{\oplus} (\bar 3,1,-\tfrac{2}{3}) \textstyle{\oplus} (1,2,-\tfrac{1}{2}) \textstyle{\oplus} (1,1,1) \textstyle{\oplus} (1,1,0),$$

These correspond to the six types of left-handed fermion: $q_L, \overline{d_R}, \overline{u_R}, \ell_L, \overline{e_R}, \overline{\nu_R}$. Six irreducible pieces, six particle types.

It turns out these are exactly the six nonzero components of the Peirce decomposition of $\mathbb{O}_\mathbb{C}^2$ with respect to both $e_1$ and $e_2$. Those six match up one-to-one with the particle types:

The remaining three combinations — $P_1(e_2)P_1(e_1)$, $P_1(e_2)P_0(e_1)$, and $P_0(e_2)P_1(e_1)$ — all vanish, which is why we land on six pieces and not nine.

So the whole package — the gauge group $G_{\mathrm{SM}}$, the embedding $G_{\mathrm{SM}} \subset \mathrm{Spin}(10)$, the representation $\rho_{\mathrm{SM}}$, and even the split of one generation into its six particle multiplets as distinct Peirce components — all comes out of the single object $\mathbb{O}_\mathbb{C}^2$ once you choose two colinear minimal tripotents.

And if you prefer to start one level up, with the Albert triple $\mathfrak{h}_3(\mathbb{O}) \otimes \mathbb{C}$, you get the same result by choosing three mutually colinear tripotents instead of two — but for that, read our paper!

A week ago, a philosopher named Amit Hagar put out a preprint entitled The NISQ Trap: Eight Years of Demonstrations the Hardware was Built to Lose. Here’s the abstract:

With a single clear exception, every NISQ-era flagship demonstration of ‘quantum advantage’ has, within eighteen months of its announcement, been classically reproduced, shown to rest on classically tractable structure, or closed by a simulability theorem. Six theoretical results from 2024 through April 2026 explain the pattern: the regions of circuit-space NISQ hardware can run with sufficient fidelity coincide with the regions classical algorithms compress efficiently, because the features that admit one (low effective depth, strong algebraic structure, geometric locality) are the features that admit the other. This reading dates the NISQ programme from its 2018 articulation as an interim retreat from the unmet conditions of the 1996 threshold theorems, characterises the eight years that followed as a closed loop in which the demonstrations the hardware could run were drawn from regions classical methods could already reach, and locates the exit from the loop where the threshold theorems originally located it: in fault tolerance. The empirical pattern could in principle break with a demonstration that escapes the current simulability results. After eight years and more than thirty advantage-class announcements, the burden of producing such a demonstration falls to the defenders of NISQ.

You can also read some debates about the paper on SciRate here. I think it’s fair to say that the paper is purely polemical, without new ideas, and Pangram agrees with my suspicion (and that of a SciRate commenter) that significant portions of it are AI-generated.

Nevertheless, the basic thesis—that quantum supremacy in the NISQ (Noisy Intermediate Scale Quantum computing) era has been a failure, or even an example of pathological science—seems surprisingly widely shared, along with the opposite thesis that quantum computing already gives oodles of useful advantages for optimization and finance.

So it seems worth stating for the record that I have an extremely different view. I would say:

1. Sampling-based quantum supremacy experiments, including those based on Random Circuit Sampling and BosonSampling, passed the point about two years ago where, absent a breakthrough in classical algorithms, they quite clearly are beating what can easily be simulated on any existing classical computer. Hagar seems to claim that these experiments have been killed by the October 2025 paper Classical simulation of noisy random circuits from exponential decay of correlation, but he ignores that the algorithm from that paper still needs time that’s exponential in the circuit depth (see Theorem 2).
2. Indeed, simulating deep ~100-qubit random circuits, like those that Google and Quantinuum have now demonstrated experimentally, still seems pretty hopeless with any current classical method. This is particularly true for Quantinuum’s experiments, which had high enough gate fidelity to maintain a Linear Cross-Entropy score of order 1 (i.e., they’re no longer all that “noisy”). The central drawback of these experiments is no longer lack of confidence about quantum advantage; rather, it’s just that we only get samples as output, and directly verifying the quality of the samples seems just as intractable for a classical computer as spoofing the samples.
3. As of this past year, however, we have some strong candidates for verifiable quantum advantage. One is the Google OTOC experiment, as even Hagar himself acknowledges (that’s his “single clear exception”). A second is the simulations of the 2D Fermi-Hubbard model on Quantinuum and Google machines, like this one. The 1D Fermi-Hubbard model can be classically simulated pretty easily (see here for example), but the 2D one still presents challenges, meaning that in some regimes, the best available estimates of certain observables apparently now come from quantum computers. I wish I could write about other examples that will be public shortly.
4. Yes, the “real” goal remains, as it’s been since the 1990s, to build a scalable fault-tolerant quantum computer—and I’m glad that Hagar (unlike, say, Gil Kalai) never suggests that we’ve learned anything to rule that goal out. In the meantime, an intermediate goal would be to use NISQ devices to do physics and chemistry simulations that are commercially useful, or that help solve important scientific problems. The point of quantum supremacy experiments, you might say, is that by demonstrating the reality of quantum speedup about as clearly as it can be demonstrated with current hardware, they let us cleanly turn our attention to those more ambitious goals.

Anyway, my son and I need to catch a plane to Utah now, for the next iteration of the wonderful Epsilon Camp, where I’ll again be teaching theoretical computer science to 11- and 12-year-olds. But feel free to discuss in the comments! Nothing about world affairs in this thread please, just quantum supremacy.

Update (July 19): Not unrelated to the subject of this post, here’s a podcast I did with Gill Eapen of “Scientific Sense” about the current situation in quantum computing including recent experimental victories.

Mark Wise, the John A. McCone Professor of High Energy Physics at Caltech, passed away on July 10 at age 72. At a recent memorial service, John Preskill made these remarks.

I’m John Preskill, Mark’s colleague on the Caltech physics faculty for more than four decades. Our friendship goes back even farther. My wife Roberta and I met Mark and Jackie not long after they arrived at Harvard in 1980. We’ve been friends since then. We attended the bris for both Barry and Jonathan during those Harvard days. Mark and Jackie have two boys and we have two girls who are a few years younger, who were thrilled to connect with Barry and Jonathan when the families would get together for occasions like Passover or Hanukkah or Thanksgiving. When the kids were little, Mark and I would sometimes muse about the potential for forging even closer family ties if those relationships blossomed.

That didn’t happen. But Mark would preside at each Seder with a light hand, sprinkling the occasion with corny jokes as was his style, and Jackie would be determined to make it to the end of that customized family-friendly Haggadah she had meticulously prepared. The children, meanwhile, would be wondering when they’d be able to continue their game of sock baseball.

Many of you know that Mark was deeply dedicated to his family and friends. I’ll make some brief remarks about three facets of Mark I know especially well: Mark the scientist, Mark the teacher and mentor, and Mark the colleague and friend.

Because of his self-deprecating manner, those of you who are not scientists may not appreciate Mark’s stature as a physicist. He was one of the most influential figures in theoretical particle physics of his generation. It was not obvious things would turn out that way. Growing up in Toronto, Mark was an indifferent student, and his poor grades reflected that lack of interest. As a 9^th grader, though, it struck him that he better change his ways and figure out how to make something of his life. He liked sports — the possibility of being a professional athlete was briefly considered, but discarded. Somehow he decided that science would be a better fit. I’m not sure why — he had recently failed math. But he worked hard and had inspiring teachers, so by the time he finished high school Mark was an excellent student, and he sailed into the University of Toronto well prepared to major in physics,

At U of T, Mark came under the influence of a young professor, Nathan Isgur, who would later become his close research collaborator. Under Nathan’s guidance, Mark sought admission to the PhD programs of the most prestigious US research universities, intent on a career devoted to deep exploration of the fundamental laws of physics. He was rejected everywhere he applied. He should have been discouraged. But he wasn’t. Mark shrugged and said: “It’s okay. I’ll stay another year in Toronto, I’ll get a master’s degree, I’ll apply again and I’ll get in somewhere.” And that’s what happened. He went to Stanford, where, under the kind tutelage of Fred Gilman, Mark took off like a rocket. Hired to the Caltech faculty in 1982, he was a tenured full professor three years later at the age of 31, and appointed as the John A. McCone Professor of High Energy Physics while still in his 30s.

Mark liked action movies, such as those starring Arnold Schwarzenegger or Clint Eastwood. In serious moments, we would sometimes ponder together why we’re successful at what we do, and Mark would always quote Clint Eastwood as Harry Callahan in Magnum Force: “A man’s got to know his limitations.” We would both laugh, but those were words of wisdom. Mark understood what he did well as a research scientist and what he was less good at. Finding problems he could solve that would have interesting consequences for experiments that had been done or could be done was where he excelled – he did it again and again. Mark never lost his zest for calculating things, often by hand with pen and paper, his head resting on one arm with his glasses pushed up onto his forehead as he scribbled. Getting to an answer that was experimentally relevant never stopped giving him a thrill.

Mark also never lost his sense of appreciation for the teachers and mentors who had inspired and helped him. Perhaps that’s why he became such a dedicated teacher and mentor himself. It’s hard to impress Caltech students, but Mark’s lectures where extremely popular, not just for their pedagogical value but also for the humanity and humor he displayed. Students had to pay attention because otherwise one might miss the jokes, which inevitably became known as “Wisecracks.” There is even an account on X with the handle @MarkWiseSays, curated by students who want to preserve Mark’s pithy lessons in physics and in life.

For example, Mark might say: “If you really get depressed, I recommend diagonalizing a 2×2 matrix.” For physics students, this is both funny and sage advice. Or he might say. “This calculation will knock your socks off.” A cliché you might hear from anyone. But who besides Mark would then proceed to remove his shoes, rip off his socks, hurl them at the blackboard, put his shoes back on and resume lecturing?

Most famously, Mark would come to class with an ample supply of coins. He would ask the class questions, sometimes about physics and sometimes random trivia, rewarding a student who gave an answer Mark approved of by tossing a coin. At first the coins were quarters. But Mark, who had a scholarly interest in finance as well as physics, eventually felt that due to inflation he needed to upgrade to dollar coins. These are harder to come by, so it took frequent visits to the bank to make sure he wouldn’t run out. His antics in class made Mark human and approachable, and students responded. Mark felt that many Caltech students don’t fully realize how smart they are. He saw part of his job as building their self-confidence and relieving their stress.

As a colleague and mentor to graduate students and postdoctoral scholars, Mark was highly collaborative. He believed that interactions with others sparked his creativity. He was never at all pompous. I know this started early. When we were in the Harvard Society of Fellows we were obligated to have dinner with the Senior Fellows on Monday nights. It was a rather stuffy occasion. And, though I don’t think they do this anymore, after a sumptuous meal we would literally retire for brandy and cigars. Once, while puffing on his cigar after dinner, Mark had an inspiration. He gathered up a few junior fellows and led them to a theater for a movie he thought everyone should see right away. The movie was Conan the Barbarian. And everyone had a blast. That was a perfect Mark moment.

As the news about Mark has spread, accolades have poured in from physicists all over the world. He was admired not just for his scientific brilliance, but almost as much for his quirky sense of humor and his kindness. Mark was a wonderful friend to many of us. When you were with him, you were sure to laugh and feel good. He touched the lives of countless colleagues, students and friends. We miss him terribly but there are so many memories that we’ll cherish. We are all so very fortunate to have known and loved Professor Mark Wise.

I don’t think people who argue about reductionism are really arguing about reductionism.

Reductionism is the idea that the behavior of big, complicated things (people, economies, ecosystems) boils down to the behavior of their smallest constituents (molecules, atoms, subatomic particles). It’s often contrasted with emergence, the idea that new rules emerge in those big, complicated systems that are more than just the rules that govern the smallest scales.

Emergence can be divided into two kinds: weak and strong. In strong emergence, the big, complicated things have their own causal powers that aren’t due to smaller things at all. This tends to get mystical, with ideas like “lifeforce” and “consciousness”. Weak emergence is much milder, and while the big complicated things are best described by their own laws, in weak emergence they are in principle still caused by laws on smaller scales.

In practice, basically no-one believes in strong emergence: it seems way too much like magic for most scientists. And basically everyone believes in weak emergence: it would be nuts to insist that economists and biologists aren’t discovering important rules that would be almost impossible to find for someone who just had physics and chemistry to work with.

So if everyone agrees, what do people argue about?

Arguments about reductionism are really arguments about attitudes. If you position yourself as a reductionist, or an emergentist, you’re defending a particular way of thinking about the world, one that privileges one scale over another. When people argue for emergence, what they really seem to be doing is opposing a kind of “reductionist chauvinism”, where people like physicists insist that their perspective is the most valuable one.

And that’s understandable, because physicists can definitely be jerks sometimes. Let it be known I am no fan of jerks.

But I think it’s worth defending reductionism, not as a philosophy, but as an attitude or perspective. Worth arguing not about whether things in practice reduce or not, but about whether reduction is a good goal, about whether science that reduces more successfully is healthier science.

Because I think it is. And the reason boils down to agreement.

Simpler systems are less controversial systems. When we write down the laws that govern subatomic particles, they’re more definite: less heuristic, more precisely specified, with fewer exceptions. That isn’t to say there’s zero controversy in these subjects, there’s even controversy between mathematicians. But the more thoroughly you can boil something down to simple rules, the easier time you have of convincing others you’re right.

In contrast, the laws of the largest scales, like psychology and biology, are deeply heuristic. They thrive on exceptions and guidelines, general tendencies without clearly defined limits. And the problem with such laws is that they can lead to intractable arguments. Different schools of thought in psychology may simply never be able to convince each other, and may just have to wait for one to die off, the aesthetic feel of one set of ideas falling out of fashion as people become preoccupied with different sorts of problems.

Every time you can reduce, you avoid those insoluble disagreements. The simpler a system you can invoke, the more you can cooperate and build off each other’s work, the less time you have to waste disagreeing, the more you can accept people with different aesthetic and philosophical preferences as just different perspectives on what are ultimately the same facts. Reductionism is a technology for peace, and one of the most powerful we have.

So yeah, I’m a bit of a chauvinist about reductionism. That’s typical of an ex-physicist, sure. But it comes from a place of concern. I want a peaceful world, where we can learn from each other and find ways to agree. And reductionism is how I get there.

Dream. I’m giving some kind of lecture to grad students with not fully prepared slides. There are slides that I made some time ago and have forgotten whe I meant by them. But what feels important is an image that’s just the number “7”, in black, sans serif font, against a plain white background; this is meant to suggest an unequivocal decision, something that just is a certain way, without ambiguity.

When I woke up this morning I found this dream and its central image puzzling. But it’s actually very transparent. I had to decide when to have surgery to repair my torn rotator cuff. Oh did I forget to mention that when I knocked my shoulder out of the socket I also tore my rotator cuff? So it has transpired. Apparently this happens a lot.

I worried a lot over the optimal date to schedule the surgery. Eventually I decided on August 7. The other option was today.

I believe that the creation of visualization apps to illustrate mathematical or scientific concepts is a particularly favorable use case for modern coding agents, as many of the downside risks attached to other LLM use cases are limited:

1. Not mission-critical. As such apps are not authorative sources of truth and only used for secondary purposes, a small positive error rate in the output can be acceptable.
2. Stand-alone. As the applets are not destined to be incorporated into a larger codebase or literature, the technical debt incurred by delegating all the coding to an LLM agent is bounded.
3. End product is deterministic (and sandboxed). As the applets run on a deterministic language (Javascript), are sandboxed against file or internet access, and do not make any LLM calls at run-time, security and privacy concerns are minimal, and the applet can be maintained without continued premium LLM access or resource-intensive compute.
4. Not replacing primary skills. While deskilling is the tradeoff one accepts when relying on these tools to accelerate output, I am perfectly willing to forego the opportunity to keep my Javascript skills at a high level, as this is a tertiary skill for me at best in my chosen profession. (I continue to manually program in Lean and in Python to keep in practice with programming in general.)
5. Not competing with humans. To my knowledge, there is no existing human effort that is being duplicated by these applets (the activity in this direction appears to have peaked two decades ago).

I would however caution against unrestricted LLM use when one or more of the above five favorable situations is not in effect.

With these points in mind, I have used such an agent to create two further apps. The first app illustrates the “zeta process” that was introduced in my recent paper with Alexeev, Barreto, Li, Lichtman, Price, Shah, and Tang, though it was first discovered by an AI. For each , the zeta distribution is a random natural number with distribution

It has long been known that this distribution has good number-theoretic properties: for instance, the number of times a given prime divides has a geometric distribution of mean . However, the new observation is that these random variables can be chained together into a single stochastic process, which we call the “zeta process”, which is an infinite divisibility chain. I used an agent to create an app to visualize this process:

The underlying process is generated by several exponential random variables at each prime: in the above instantiation of the process, two such variables are visible at the prime , and one variable at the primes . At a given choice of , is formed by collecting all the variables below this threshold (and for which all predecessors also lie below the threshold); in the above illustration, this amounts to one variable at each of the primes , leading to in this case. Additional visualizations in the app display the distribution of each , as well as the distribution of the hitting probability , which among other things can be used to give a quick solution to Erdős problem #1196.

The second app is rather different in nature, and is a somewhat whimsical attempt to display the motion of the heavens, both at “human” scales of space and time, and at more “astronomical” scales (in which the motion of the planets in particular are more apparent). It is very loosely inspired by the game “Katamari Damacy“, in which one absorbs both terrestrial and celestial objects of many different scales. Here is how the app typically looks at a human scale:

And here is how it looks when one’s perspective leaves the Earth’s atmosphere:

(As I did not want to render an entire explorable world in this app, the observer in the app is only limited to changing his or her size, from a human to a creature of comparable size to the Earth itself; they cannot move horizontally on the planet.) At the largest scales of space and time, the classic orrery diagram appears:

After lengthy conversations with the agent, I was able to implement many astronomical phenomena, including phases of the Moon, the effect of Earth’s rotation against the fixed stars (though one can also stabilize one’s view against those stars to see the Earth’s rotation more directly), and so forth.

I have trouble sleeping — trouble staying asleep, not trouble falling asleep. I have found that the best way to get back to sleep at three or four in the morning is to listen to a podcast on a 10-minute fadeout timer. But the podcast has to land in a very specific zone: interesting enough that I don’t actively mind listening to it, but not so interesting that my brain maintains awakeness in order to catch what’s next. After a long process of trial and error I have found that millennial dating advice podcasts are perfect. Getting these little glimpses into the lives of the young and confused is never dull, but at the same time, anything you miss is going to happen again to some other hapless millennial single in a week or two, so it’s OK to drift off.

Anyway: Jared Freid, one of the hosts of my go-to millennial dating podcast, U Up, has a book out, called Walking Red Flag. It’s really good, even while being of no practical relevance to me! I like it, Dr. Mrs. Q likes it, AB likes the parts I’ve read to her. Presumably actual lonely millennials like it too. Freid is a stand-up comic, and as such has a deep feel for words and sounds. Pardon this coarse language, but I will never forget him describing his small sad NYC apartment with the sentence “I fuck next to the sink.” That is writing! That’s what you learn in poetry class! If it’s a good poetry class.

And not to be corny, but as the father of a son, I appreciate that Freid can be out there being a straight man who both wants to date women and also clearly actually likes women and is relentlessly normal about it. Bizarrely little of this in the male advice space!

Anyway, this is probably the only comic millennial dating advice book I am ever going to recommend on this blog, but it really is — there’s no other way to put this — a hoot.

One byproduct of learning how to use coding agents to create visualization apps is that it now becomes straightforward to convert any figure in one’s papers that had already been generated by code (e.g., in Python) into a more interactive, animated applet.

I can illustrate this with Figure 1 from my recent paper on the Gilbreath conjecture with Chase and Hunter, reproduced below:

This plot displays both exact and numerically simulated values of a certain poorly understood sequence relating to the Gilbreath conjecture, which I will call the “Gilbreath expectation sequence” here for lack of a better name. The definition of the sequence is as follows. Consider a “Gilbreath array” which is an inverted pyramid, where the top entries are independent exponential random variables of mean 1, and all the other entries are the absolute values of the differences of the two entries immediately above it. Thanks to the visualizer app, I can quickly give an example (with ):

The left diagonal entries are then random variables; the sequence are defined to be the expectation of these values. (The process is stationary, so in fact any entry on the row will have expectation .)

If one starts with the first normalized prime gaps (which have expectation about , and are conjecturally distributed asymptotically according to a geometric distribution), then standard conjectures (e.g., the prime tuples conjecture) predict that the row entries should decay like , at least for small . So the Gilbreath conjecture appears to be tied to how fast the sequence decays with .

One can in principle work out each value of as an explicit rational number by performing a certain complicated multivariate integral, but in the paper we only did this for (the orange line in the above figure); for the remaining we performed a Monte Carlo simulation with Gilbreath arrays to obtain a numerical approximation (in blue), which (as per the law of large numbers) agreed well with the theoretical values. A later calculation of Michael Ross extended the theoretical values to , maintaining the good fit:

The asymptotic behavior of the sequence remains mysterious. Clearly, it is not monotonic; in fact we cannot even prove it is bounded. The best we could do in our paper was establish an inequality which, roughly speaking, showed that cannot decay faster than .

In a recent preprint of Ross, these numerics were extended, and a rough empirical prediction

was proposed for some constants and (empirically ), where is the number of 1’s in the binary expansion of ; in particular, it is the fluctuation in this quantity that is intended to explain much of the non-monotonic behavior of . These are now all displayed in the following companion applet, which was a routine matter to generate in about an hour by the coding agent (which by this point has extensive experience with creating such apps, encoded via a “skill” markdown file that it maintains):

The appearance of the quantity may initially appear mysterious, but it is related to Lucas’s theorem, Kummer’s theorem, and the Sierpinski gasket. Consider for instance a Gilbreath array where all the entries are zero except for a single “spike”. Then the following Sierpinski pattern emerges:

Here is what an version of this picture looks like (with the spike positioned at the 32th entry):

The number of 1s in the row is then (if we index the rows starting from zero), which is at least of the same shape as the empirical prediction, albeit with different constants. (This sequence is also known as Gould’s sequence.)

Numerically, we seem to observe fragments of Sierpinski gaskets being generated before decaying (often due to “collisions” with other gaskets):

However, it is not clear to me at all what the asymptotic probabilistic model should be, even heuristically; it does not resemble any random shape model that I am familiar with. But perhaps there are readers more expert in probability theory or statistical physics who may be able to suggest such an asymptotic limit?

(I am writing here in my capacity as Director of Special Projects at IPAM.)

IPAM seeks program proposals from the mathematical, statistical, and scientific communities for long programs, workshops, and summer schools.  Most program proposals are reviewed at IPAM’s Science Advisory Board meeting, held in November each year.  Programs are selected on the basis of their scientific impact and contribution to IPAM’s goals. IPAM is committed to supporting a community where people of all backgrounds and points of view can engage, learn, and thrive.  If you would like to discuss your program ideas and prepare a proposal for IPAM’s consideration, you are encouraged to contact the IPAM Director. For more information visit: https://www.ipam.ucla.edu/propose-a-program/long-programs-2/

When I wrote this post last month, I really did not want it to be first in a series.

The inciting incident to write that post was a news article in Science reporting that there have been draconian ~ 30% cuts in present fiscal year budgets within the NSF.  Program officers were instructed to keep this confidential and not talk about it with PIs.  The rumor was that this funding would go to support the TIP directorate and its activities, particularly the "X-Labs".  

Now Dan Garisto has broken this story in Nature. For some, this is behind a paywall, so let me hit the highlights.

• "NSF staff members — who asked to remain anonymous out of fear of retaliation — and an internal NSF ledger seen by Nature suggest that the NSF plans to claw back around US$500 million that has already been distributed to grant-making divisions. "
• "Several NSF staff members told Nature that at least some of the funds will be funnelled to another project, a brainchild of the White House OSTP."
• "Even though programme officers have less to spend than they had expected, those who spoke to Nature are unsure how many new grants will make it out the door. “They could make it through if the process is allowed to work without interference or interruption,” one staff member says. But most of “the process is now a black box and unpredictable”. "
• "NSF staff members estimate that if the withdrawals are finalized, hundreds more proposals that have been recommended for funding across the engineering, computer and information science and engineering, and maths and physical science directorates would need to be sent back to programme officers for revision. Some proposals would be held until a later date when funds are available. Others would have their budgets reduced, and some would simply be declined."
• "Staff members say that they are frustrated by the planned diversion of funds and the lack of communication about the agency’s spending. “We don’t know where the money’s going or what’s going on,” says one staff member. Programme officers are not allowed to pass on what they know to researchers. “We cannot communicate to the community at all. We’re forbidden.”"

I am finding the newly revealed capability to code old applet ideas into reality to be very tempting to sink more time into, though I am certainly encountering the common “vibe coding” experience that the process can produce something that superficially resembles a finished product well before a satisfactory level of testing and review has been completed; indeed, it is the review process which is now the most time-consuming, to the point where I think any further advances in coding agent capability will have little impact on the new bottlenecks in the design process.

In any event, I spent a few hours working to realize a proposal I had made back in 2023 to automatically create diagrams to visually illustrate the logical flow of a given mathematical paper. At the time, Freddie Manners, extrapolating from the half-decent capability of the then-newly released ChatGPT 3.5 at this task, presciently predicted that “by the time a dedicated tool had been completed, the next general purpose engine would be better than it”.

With that in mind, I decided to focus not on the generation of the diagram – which now can be done at various levels of quality by any number of large language models – but on its presentation. The result is the following app, which can take a certain formatted JSON file of dependencies between theorem objects and produce an interactive graph which can be explored, edited and also exported (somewhat lossily) into other standard formats such as SVG, TikZ, quiver, or Mermaid. Here is a screenshot of a diagramming of the celebrated proof by Wang and Zahl of the three-dimensional Kakeya conjecture:

Using an LLM, I generated diagrams for eight papers for demonstration purposes, including for instance a diagram for Wiles’s proof of Fermat’s last theorem, or of Szemeredi’s proof of his famous theorem on arithmetic progressions (which sports a notoriously convoluted such diagram in the original paper), as well as a few papers of my own. If there are other requests to diagram particular papers, I can try to use an LLM to generate more examples; but my intention is for users of the app to create their own such diagrams, either by manually constructing them, or by directing their own AI tools to build the diagram in the required format (which is a JSON, with the precise specification given here).

I mentioned in the previous post that for these sorts of visualization apps, which work deterministically for a given set of inputs, the downside risk of LLM use to build the app is acceptably low. For this particular app, there is a complicating factor, which is that while the app does remain deterministic, the data I used to populate the app – namely, the above diagrams – are also LLM-generated. I have done spot checks comparing the diagrams against the source papers, and did not find any errors; however, they are not guaranteed to be 100% accurate, and should only be used as approximations to the logical structure of these papers rather than completely exact representations. (The latter might become deterministically extractable should the results of these papers become formalized in a proof assistant language, but this is not currently the case.) Still, I hope these sorts of diagrams can serve as a helpful initial guide when first trying to read and understand a complex paper.

Here at the National Academy of Sciences, it seems that my first job is to serve on the selection committee for the prestigious Michael and Sheila Held Prize in combinatorial and discrete optimization and related areas. The committee chair, my former MIT colleague Madhu Sudan (now at Harvard), invited me to share the following message here on Shtetl-Optimized. (I’d add: put in the effort to nominate someone, and you can actually influence how things go!)

Dear Colleagues

I am writing to seek nominations for the 2027 Michael and Sheila Held Prize. The scope of the prize and nomination needs are described below. If you intend to nominate someone I would appreciate a heads up by email to madhu@cs.harvard.edu one month before the deadline (so email by Sept 8, 2026) to let me know your nomination is coming. (We may also reach out to you in response to coordinate multiple/overlapping nominations.)

The Held prize honors outstanding, innovative, creative, and influential research in the areas of combinatorial and discrete optimization, or related parts of computer science, such as the design and analysis of algorithms and complexity theory. This $100,000 prize is intended to recognize recent work (defined as published within the last eight years, i.e., on or after October 6, 2018).

All nominations must be submitted online by Monday, October 5, 2026 and include:

1. Nomination letter describing the candidate’s work and why he or she should be selected for the award. No more than three (3) pages.

2. Curriculum vitae. No more than two (2) pages.

3. Bibliography listing no more than twelve (12) of the nominee’s most significant publications.

4. Suggested citation. A 50-word summary stating why the nominee should be considered for this award.

5. Two letters of support. No more than one letter of support can be written by someone of the same primary work institution as the nominee.

The Held Prize is given to a person or a set of persons, as supported by a paper or a body of work. Unless otherwise stated, preference will be given to scientists who may be earlier in their careers or those whose work has not been recognized by other prizes or awards. Nomination restrictions can be found here. Joint nominations will only be considered when nominees have collaborated closely on the paper to be recognized by the award. If nominating multiple individuals for a paper with additional authors, please clearly explain the reason for nominating those chosen, as well as the reason for excluding other collaborators, if applicable. 

Please feel free to circulate this call further within your department

Best
Madhu Sudan, on behalf of The Michael and Sheila Held Prize Selection Committee

And while I have your attention, a second CS theory announcement: David Soloveichik, my wonderful friend and colleague in UT Austin’s Electrical and Computer Engineering Department, has funding for a postdoc for 1-2 years, to work on the thermodynamics of computation here at UT. This is a topic that I’ve been trying to learn more about as well, so I might get involved too! David writes, “the big picture is to think of thermodynamics (energy dissipation / entropy production) as CS complexity measures like time and space usage.” If you’re on the postdoc market and this sounds potentially up your alley, email David to learn more.

When Lee and Yang suggested that the laws of physics might not be invariant under spatial reflection — that there’s a fundamental difference between left and right — Pauli was skeptical. In a letter to Victor Weisskopf in January 1957, he wrote:

“Ich glaube aber nicht, daß der Herrgott ein schwacher Linkshänder ist.”

(I do not believe that the Lord is a weak left-hander.)

But just two days after Pauli wrote this letter, Chien-Shiung Wu’s experiment confirmed that Lee and Yang were correct. There’s an inherent asymmetry in nature.

We can trace this back to how the ‘left-handed’ fermions and antifermions live in a different representation of the Standard Model gauge group than the right-handed ones. And when we try to build grand unified theories that take this into account, we run into the fact that while we can fit the Standard Model gauge group into $\text{Spin}(10)$ in various ways, not all these ways produce the required asymmetry. There’s a way where it fits into $\text{Spin}(9)$, which is too symmetrical to work… and alas, this one has a nice octonionic description!

To keep things simple I’ll explain this by focusing, not on the whole Standard Model gauge group, but its subgroup $\text{SU}(2) \times \text{SU}(3)$. Here is a theorem proved by Will Sawin in response to a question of mine on MathOverflow:

Theorem 10. There are exactly two conjugacy classes of subgroups of $\text{Spin}(10)$ that are isomorphic to $\text{SU}(2) \times \text{SU}(3)$. One of them has a representative that is a subgroup of $\text{Spin}(9) \subset \text{Spin}(10)$, while the other does not.

I’ll describe representatives of these two subgroups; then I’ll say a bit about how they show up in physics, and then I’ll show you Sawin’s proof.

We can get both subgroups in a unified way! There’s always an inclusion

$$\text{SO}(m) \times \text{SO}(n) \to \text{SO}(m+n)$$

and taking double covers of each group we get a 2-1 homomorphism

$$\text{Spin}(m) \times \text{Spin}(n) \to \text{Spin}(m+n)$$

In particular we have

$$\text{Spin}(4) \times \text{Spin}(6) \to \text{Spin}(10)$$

so composing with the exceptional isomorphisms:

$$\text{Spin}(4) \cong \text{SU}(2) \times \text{SU}(2), \qquad \text{Spin}(6) \cong \text{SU}(4)$$

we get a 2-1 homomorphism

$$k \colon \text{SU}(2) \times \text{SU}(2) \times \text{SU}(4) \to \text{Spin}(10)$$

Now, there are three obvious ways to include $\text{SU}(2) \times \text{SU}(3)$ in $\text{SU}(2) \times \text{SU}(2) \times \text{SU}(4)$. There is an obvious inclusion

$$j \colon \text{SU}(3) \hookrightarrow \text{SU}(4)$$

but there are three obvious inclusions

$$\ell, r, \delta \colon \text{SU}(2) \hookrightarrow \text{SU}(2) \times \text{SU}(2)$$

namely the left one:

$$\begin{array}{ccc} \ell \colon \text{SU}(2) &\to& \text{SU}(2) \times \text{SU}(2) \\ g & \mapsto & (g,1) \end{array}$$

the right one:

$$\begin{array}{ccc} r \colon \text{SU}(2) &\to& \text{SU}(2) \times \text{SU}(2) \\ g & \mapsto & (1,g) \end{array}$$

and the diagonal one:

$$\begin{array}{ccc} \delta \colon \text{SU}(2) &\to& \text{SU}(2) \times \text{SU}(2) \\ g & \mapsto & (g,g) \end{array}$$

Combining these with our earlier maps, we actually get a one-to-one map from $\text{SU}(2) \times \text{SU}(3)$ to $\text{Spin}(10)$. So we get three subgroups of $\text{Spin}(10)$, all isomorphic to $\text{SU}(2) \times \text{SU}(3)$:

• There’s the left subgroup $G_\ell$, which is the image of this composite homomorphism:

$$\text{SU}(2) \times \text{SU}(3) \stackrel{\ell \times j}{\longrightarrow} \text{SU}(2) \times \text{SU}(2) \times \text{SU}(4) \cong \text{Spin}(4) \times \text{Spin}(6) \stackrel{k}{\longrightarrow} \text{Spin}(10)$$

• There’s the diagonal subgroup $G_\delta$, which is the image of this:

$$\text{SU}(2) \times \text{SU}(3) \stackrel{\delta \times j}{\longrightarrow} \text{SU}(2) \times \text{SU}(2) \times \text{SU}(4) \cong \text{Spin}(4) \times \text{Spin}(6) \stackrel{k}{\longrightarrow} \text{Spin}(10)$$

• And there’s the right subgroup $G_r$, which is the image of this:

$$\text{SU}(2) \times \text{SU}(3) \stackrel{r \times j}{\longrightarrow} \text{SU}(2) \times \text{SU}(2) \times \text{SU}(4) \cong \text{Spin}(4) \times \text{Spin}(6) \stackrel{k}{\longrightarrow} \text{Spin}(10)$$

The left and right subgroups are actually conjugate, but the diagonal one is truly different! We’ll prove this by taking a certain representation of $\text{Spin}(10)$, called the Weyl spinor representation, and restricting it to those two subgroups. We’ll get inequivalent representations of $\text{SU}(2) \times \text{SU}(3)$. This proves the two subgroups aren’t conjugate.

This argument is also interesting for physics. When restrict to the left subgroup, we get a representation of $\text{SU}(2) \times \text{SU}(3)$ that matches what we actually see for one generation of fermions! This is the basis of the so-called $\text{SO}(10)$ grand unified theory, which should really be called the $\text{Spin}(10)$ grand unified theory.

(In fact this works not only for $\text{SU}(2) \times \text{SU}(3)$ but for the whole Standard Model gauge group, which is larger. I’m focusing on $\text{SU}(2) \times \text{SU}(3)$ just because it makes the story simpler.)

When we restrict the Weyl spinor representation to the diagonal subgroup, we get a representation of $\text{SU}(2) \times \text{SU}(3)$ that is not physically correct. Unfortunately, it’s the diagonal subgroup that shows up in several papers connecting the Standard Model gauge group to the octonions. I plan to say a lot more about this later.

The left subgroup

Let’s look at the left subgroup $G_\ell$, the image of this composite:

$$\text{SU}(2) \times \text{SU}(3) \stackrel{\ell \times j}{\longrightarrow} \text{SU}(2) \times \text{SU}(2) \times \text{SU}(4) \cong \text{Spin}(4) \times \text{Spin}(6) \stackrel{k}{\longrightarrow} \text{Spin}(10)$$

$\text{Spin}(10)$ has a 32-dimensional unitary representation called the ‘Dirac spinor’ representation. This representation is really on the exterior algebra $\Lambda \mathbb{C}^5$. It’s the direct sum of two irreducible parts, the even grades and the odd grades:

$$\Lambda \mathbb{C}^5 \cong \Lambda^{\text{even}} \mathbb{C}^5 \oplus \Lambda^{\text{odd}} \mathbb{C}^5$$

Physicists call these two irreducible representations ‘right- and left-handed Weyl spinors’, and denote them as $\mathbf{16}$ and $\mathbf{16}\ast$ since they’re 16-dimensional and one is the dual of the other.

Let’s restrict the $\mathbf{16}$ to the left subgroup $G_\ell$ and see what we get.

To do this, first we can restrict the $\mathbf{16}$ along $k$ and get

$$\mathbf{2} \otimes \mathbf{1} \otimes \mathbf{4} \; \oplus \; \mathbf{1} \otimes \mathbf{2} \otimes \mathbf{4}\ast$$

Here $\mathbf{1}$ is the trivial representation of $\text{SU}(2)$, $\mathbf{2}$ is the tautologous representation of $\text{SU}(2)$, and $\mathbf{4}$ is the tautologous rep of $\text{SU}(4)$.

Then let’s finish the job by restricting this representation along $\ell \times j$. Restricting the $\mathbf{4}$ of $\text{SU}(4)$ to $\text{SU}(3)$ gives $\mathbf{3} \oplus \mathbf{1}$: the sum of the tautologous representation of $\text{SU}(3)$ and the trivial representation. Restricting $\mathbf{2} \otimes \mathbf{1}$ to the left copy of $\text{SU}(2)$ gives the tautologous representation $\mathbf{2}$, while restricting $\mathbf{1} \otimes \mathbf{2}$ to this left copy gives $\mathbf{1} \oplus \mathbf{1}$: the sum of two copies of the trivial representation. All in all, we get this representation of $\text{SU}(2) \times \text{SU}(3)$:

$$\mathbf{2} \otimes (\mathbf{3} \oplus \mathbf{1}) \; \oplus \; (\mathbf{1} \oplus \mathbf{1}) \otimes (\mathbf{3}\ast \oplus \mathbf{1})$$

This is what we actually see for one generation of left-handed fermions and antifermions in the Standard Model! The representation $\mathbf{3} \oplus \mathbf{1}$ describes how the left-handed fermions in one generation transform under $\text{SU}(3)$: 3 colors of quark and one ‘white’ lepton. The representation $\mathbf{3}\ast \oplus \mathbf{1}$ does the same for the left-handed antifermions. The left-handed fermions form an isospin doublet, giving us the $\mathbf{2}$, while the left-handed antifermions have no isospin, giving us the $\mathbf{1} \oplus \mathbf{1}$.

This strange lopsidedness is a fundamental feature of the Standard Model.

The right subgroup would work the same way, up to switching the words ‘left-handed’ and ‘right-handed’. And by Theorem 10, the left and right subgroups must be conjugate in $\text{Spin}(10)$, because now we’ll see one that’s not conjugate to either of these.

The diagonal subgroup

Consider the diagonal subgroup $G_\delta$, the image of this composite:

$$\text{SU}(2) \times \text{SU}(3) \stackrel{\delta \times j}{\longrightarrow} \text{SU}(2) \times \text{SU}(2) \times \text{SU}(4) \cong \text{Spin}(4) \times \text{Spin}(6) \stackrel{k}{\longrightarrow} \text{Spin}(10)$$

Let’s restrict the $\mathbf{16}$ to $G_\delta$.

To do this, first let’s restrict the $\mathbf{16}$ along $k \colon \text{SU}(2) \times \text{SU}(2) \times \text{SU}(4) \to \text{Spin}(10)$ and get

$$\mathbf{2} \otimes \mathbf{1} \otimes \mathbf{4} \; \oplus \; \mathbf{1} \otimes \mathbf{2} \otimes \mathbf{4}\ast$$

as before. Then let’s restrict this representation along $\delta \times j$. The $\SU(3)$ part works as before, but what happens when we restrict $\mathbf{2} \otimes \mathbf{1}$ or $\mathbf{1} \otimes \mathbf{2}$ along the diagonal map $\delta \colon \text{SU}(2) \to \text{SU}(2) \times \text{SU}(2)$? We get $\mathbf{2}$. So, this is the representation of $G_\delta$ that we get:

$$\mathbf{2} \otimes (\mathbf{3} \oplus \mathbf{1}) \; \oplus \; \mathbf{2} \otimes (\mathbf{3}\ast \oplus \mathbf{1})$$

This is not good for the Standard Model. It describes a more symmetrical universe than ours, where both left-handed fermions and antifermions transform as doublets under $\text{SU}(2)$.

The fact that we got a different answer this time proves that $G_\ell$ and $G_\delta$ are not conjugate in $\text{Spin}(10)$. So to complete the proof of Theorem 10, we only need to prove

1. Every subgroup of $\text{Spin}(10)$ isomorphic to $\text{SU}(2) \times \text{SU}(3)$ is conjugate to $G_\ell$ or $G_\delta$.

2. $G_\delta$ is conjugate to a subgroup of $\text{Spin}(9) \subset \text{Spin}(10)$, but $G_\ell$ is not.

I’ll prove 2, and then I’ll turn you over to Will Sawin to do the rest.

Why the diagonal subgroup fits in $\text{Spin}(9)$

Every rotation of $\mathbb{R}^n$ extends to a rotation of $\mathbb{R}^{n+1}$ that leaves the last coordinate fixed, so we get an inclusion $\text{SO}(n) \hookrightarrow \text{SO}(n+1)$, which lifts to an inclusion of the double covers, $\text{Spin}(n) \hookrightarrow \text{Spin}(n+1)$. Since we have exceptional isomorphisms

$$\text{Spin}(3) \cong \text{SU}(2), \qquad \text{Spin}(4) \cong \text{SU}(2) \times \text{SU}(2)$$

it’s natural to ask how the inclusion $\text{Spin}(3) \hookrightarrow \text{Spin}(4)$ looks in these terms. And the answer is: it’s the diagonal map! In other words, we have a commutative diagram

$$\begin{array}{ccc} \text{SU}(2) & \xrightarrow{\sim} & \text{Spin}(3) \\ \delta \downarrow & & \downarrow \\ \text{SU}(2) \times \text{SU}(2) & \xrightarrow{\sim} & \text{Spin}(4) \end{array}$$

Now, we can easily fit this into a larger commutative diagram involving some natural maps $\text{Spin}(m) \times \text{Spin}(n) \to \text{Spin}(m+n)$ and $\text{Spin}(n) \to \text{Spin}(n+1)$:

$$\begin{array}{ccccccc} \text{SU}(2) & \xrightarrow{\sim} & \text{Spin}(3) & \to & \text{Spin}(3) \times \text{Spin}(6) & \to & \text{Spin}(9) \\ \delta \downarrow & & \downarrow & & \downarrow & & \downarrow \\ \text{SU}(2) \times \text{SU}(2) & \xrightarrow{\sim} & \text{Spin}(4) & \to & \text{Spin}(4) \times \text{Spin}(6) & \to & \text{Spin}(10) \end{array}$$

We can simplify this diagram using the isomorphism $\text{Spin}(6) \cong \text{SU}(4)$:

$$\begin{array}{ccccccc} \text{SU}(2) \times \text{SU}(4) & \to & \text{Spin}(9) \\ \delta \times 1 \downarrow & & \downarrow \\ \text{SU}(2) \times \text{SU}(2) \times \text{SU}(4) & \to & \text{Spin}(10) \end{array}$$

and then we can use our friend the inclusion $j \colon \text{SU}(3) \to \text{SU}(4)$:

$$\begin{array}{ccccccc} \text{SU}(2) \times \text{SU}(3) & \xrightarrow{1 \times j} & \text{SU}(2) \times \text{SU}(4) & \to & \text{Spin}(9) \\ & & \delta \times 1 \downarrow & & \downarrow \\ & & \text{SU}(2) \times \text{SU}(2) \times \text{SU}(4) & \to & \text{Spin}(10) \end{array}$$

This shows that the diagonal subgroup $G_\delta$ of $\text{Spin}(10)$ is actually a subgroup of $\text{Spin}(9)$!

Why the left subgroup does not fit in $\text{Spin}(9)$

The three-fold way is a coarse classification of irreducible complex representations of compact Lie group. Every such representation is of one and only one of these three kinds:

1) not self-dual: not isomorphic to its dual,

2a) orthogonal: isomorphic to its dual via an invariant nondegenerate symmetric bilinear form, also called an orthogonal structure,

2b) symplectic: isomorphic to its dual via an invariant nondegenerate antisymmetric bilinear form, also called a symplectic structure.

I’ve written about how these three cases are related to the division algebras $\mathbb{C}, \mathbb{R}$ and $\mathbb{H}$, respectively:

• John Baez, Division algebras and quantum theory.

A complex representation is orthogonal iff it’s the complexification of a representation on a real vector space, and symplectic iff it’s the underlying complex representation of a representation on a quaternionic vector space.

But we don’t need most of this yet. For now we just need to know one fact: when $n$ is odd, every irreducible representation of $\text{Spin}(n)$, and thus every representation of this Lie group, is self-dual: that is, isomorphic to its dual. In particular this is true of $\text{Spin}(9)$.

Why does this matter? Assume the left subgroup $G_\ell \subset \text{Spin}(10)$ is a subgroup of $\text{Spin}(9)$. When we restrict the Weyl spinor representation of $\text{Spin}(10)$ to $\text{Spin}(9)$ it will be self-dual, like every representation of $\text{Spin}(9)$. Then when we restrict this representation further to $\text{SU}(2) \times \text{SU}(3)$ it must still be self-dual, since the restriction of a self-dual representation is clearly self-dual.

However, we know this representation is

$$\mathbf{2} \otimes (\mathbf{3} \oplus \mathbf{1}) \; \oplus \; (\mathbf{1} \oplus \mathbf{1}) \otimes (\mathbf{3}\ast \oplus \mathbf{1})$$

and this is not self-dual, since $\mathbf{1}\ast \cong \mathbf{1}$ and $\mathbf{2}\ast \cong \mathbf{2}$ but $\mathbf{3}\ast \ncong \mathbf{3}$.

So, it must be that $G_\ell$ is not a subgroup of $\text{Spin}(9)$.

Proof of Theorem 10

To complete the proof of Theorem 10 we just need to see why there are just two conjugacy classes of subgroups of $\text{Spin}(10)$ isomorphic to $\text{SU}(2) \times \text{SU}(3)$. But in fact Will Sawin proved a stronger result! He was answering this question of mine:

Define the Standard Model gauge group to be $\text{S}(\text{U}(2) \times \text{U}(3))$, the subgroup of $\text{SU}(5)$ consisting of block diagonal matrices with a $2 \times 2$ block and then a $3 \times 3$ block. (This is isomorphic to the quotient of $\text{U}(1) \times \text{SU}(2) \times \text{SU}(3)$ by the subgroup of elements $(\alpha, \alpha^{-3}, \alpha^2$) where $\alpha$ is a 6th root of unity.)

Up to conjugacy, how many subgroups isomorphic to the Standard Model gauge group does $\text{Spin}(10)$ have?

This question is relevant to grand unified theories of particle physics, as explained here:

• John C. Baez and John Huerta, The algebra of grand unified theories, Bull. Amer. Math. Soc. 47 (2010), 483-552.

This paper focuses on one particular copy of $\text{S}(\text{U}(2) \times \text{U}(3))$ in $\text{Spin}(10)$, given as follows. By definition we have an inclusion $\text{S}(\text{U}(2) \times \text{U}(3)) \hookrightarrow \text{SU}(5)$, and we also have an inclusion $\text{SU}(5) \hookrightarrow \text{Spin}(10)$ because for any $n$ we have an inclusion $\text{SU}(n) \hookrightarrow \text{SO}(2n)$, and $\text{SU}(n)$ is simply connected so this gives a homomorphism $\text{SU}(n) \hookrightarrow \text{Spin}(2n)$.

However I think there is also an inclusion $\text{S}(\text{U}(2) \times \text{U}(3)) \hookrightarrow \text{Spin}(9)$, studied by Krasnov:

• Kirill Krasnov, SO(9) characterisation of the Standard Model gauge group.

Composing this with $\text{Spin}(9) \hookrightarrow \text{Spin}(10)$, this should give another inclusion $\text{S}(\text{U}(2) \times \text{U}(3)) \hookrightarrow \text{Spin}(10)$, and I believe this one is ‘truly different from’ — i.e., not conjugate to — the first one I mentioned.

So I believe my current answer to my question is “at least two”. But that’s not good enough.

Sawin’s answer relies heavily on the 3-fold way — that’s why I told you that stuff about orthogonal and symplectic representations. When we embed the group $\text{SU}(2) \times \text{SU}(3)$ in $\text{Spin}(10)$, we are automatically giving this group an orthogonal 10-dimensional representation, thanks to the map $\text{Spin}(10) \to \text{SO}(10)$. We can classify the possibilities.

He writes:

There are infinitely many embeddings. However, all but one of them is “essentially the same as” the one you studied as they become equal to the one you studied on restriction to $\text{SU}(2)\times \text{SU}(3)$. The remaining one is the one studied by Krasnov.

I follow the strategy suggested by Kenta Suzuki.

$\text{SU}(3)$ has irreducible representations of dimensions $1,3,3,6,8,6, 10, 10$, and higher dimensions. The $10$-dimensional ones are dual to each other, as are the $6$-dimensional ones, so they can’t appear. The $3$-dimensional ones are dual to each other and can only appear together. So the only $10$-dimensional self-dual representations of $\text{SU}(3)$ decompose as irreducibles as $8+1+1$, $3+3+1+1+1+1$, or ten $1$s. All of these are orthogonal because the 8-dimensional representation is orthogonal. However, the ten $1$s cannot appear because then $\text{SU}(3)$ would act trivially.

A representation of $\text{SU}(3) \times \text{SU}(2)$ is a sum of tensor products of irreducible representations of $\text{SU}(3)$ and irreducible representations of $\text{SU}(2)$. Restricted to $\text{SU}(3)$, each tensor product splits into a sum of copies of the same irreducible representation. So $\text{SU}(2)$ can only act nontrivially when the same representation appears multiple times. Since the $3+3$ is two different $3$-dimensional representation, only the $1$-dimensional representation can occur twice. Thus, our 10-dimensional orthogonal representation of $\text{SU}(3) \times \text{SU}(2)$ necessarily splits as either the $8$-dimensional adjoint repsentation of $\text{SU}(3)$ plus a $2$-dimensional orthogonal representation of $\text{SU}(2)$ or the $6$-dimensional sum of standard and conjugate [i.e., dual] representations of $\text{SU}(3)$ plus a $4$-dimensional orthogonal representation of $\text{SU}(2)$. However, $\text{SU}(2)$ has a unique nontrivial representation of dimension $2$ and it isn’t orthgonal, so only the second case can appear. $\text{SU}(2)$ has representations of dimension $1,2,3,4$ of which the $2$ and $4$-dimensional ones are symplectic and so must appear with even multiplicity in any orthogonal representation, so the only nontrivial $4$-dimensional orthogonal ones are $2+2$ or $3+1$.

So there are two ten-dimensional orthogonal representations of $\text{SU}(2) \times \text{SU}(3)$ that are nontrivial on both factors, those being the sum of two different $3$-dimensional irreducible representations of $\text{SU}(3)$ with either two copies of the two-dimensional irreducible representation of $\text{SU}(2)$ or the three-dimensional and the one-dimensional irreducible representation of $\text{SU}(2)$. The orthogonal structure is unique up to isomorphisms, so these give two conjugacy classes of homomorphisms $\text{SU}(2) \times \text{SU}(3) \to SO(10)$ and thus two conjugacy classes of homomorphisms $\text{SU}(2) \times \text{SU}(3) \to \text{Spin}(10)$. The first one corrresponds to the embedding you studied while only the second one restricts to $\text{Spin}(9)$ so indeed these are different.

To understand how to extend these to $\text{S}(\text{U}(2) \times \text{U}(3))$, I consider the centralizer of the representation within $\text{Spin}(10)$. Since the group is connected, this is the same as the centralizer of its Lie algebra, which is therefore the inverse image of the centralizer in $\text{SO}(10)$. Now there is a distinction between the two examples because the example with irrep dimensions $3+3+2+2$ has centralizer with identity component $\text{U}(1) \times \text{SU}(2)$ while the example with irrep dimensions $3+3+3+1$ has centralizer with identity component $\text{U}(1)$. In the second case, the image of $\text{U}(2) \times \text{U}(3)$ must be the image of $\text{SU}(2) \times \text{SU}(3)$ times the centralizer of the image of $\text{SU}(2) \times \text{SU}(3)$, so this gives a unique example, which must be the one considered by Krasnov.

In the first case, we can restrict attention to a torus $\text{U}(1) \times \text{U}(1)$ in $\text{SU}(2) \times \text{SU}(2)$. The center of $\text{S}(\text{U}(2) \times \text{U}(3))$ maps to a one-dimensional subgroup of this torus, which can be described by a pair of integers. Explicitly, given a two-by-two-unitary matrix $A$ and a three-by-three unitary matrix $B$ with $\det(A) \det(B) =1$, we can map to $\text{U}(5)$ by sending $(A,B)$ to $A \gamma^a \oplus B \gamma^b$ where $\gamma = \det (A) = \det(B)^{-1}$, and then map from $\text{U}(5)$ to $SO(10)$. This lifts to the spin group if and only if the determinant in $\text{U}(5)$ is a perfect square. The determinant is $\gamma^{ 1 + 2a - 1 + 3b} = \gamma^{2a+3b}$ so a lift exists if and only if $b$ is even.

The only possible kernel of this embedding is the scalars. The scalar $A = \lambda^3 I_2, B = \lambda^{-2} I_3$ maps to $\lambda^{3+ 6a} I_2 \oplus \lambda^{-2 + 6b} I_3$ and so the kernel is trivial if and only if $\gcd(3+6a,-2 + 6b)=1$.

However, there are infinitely many integer solutions $a,b$ to $\gcd(3+6a,-2a+6b)=1$ with $b$ even (in fact, a random $a$ and even $b$ works with probability $9/\pi^2$), so this gives infinitely many examples.

• Part 1. How to define octonion multiplication using complex scalars and vectors, much as quaternion multiplication can be defined using real scalars and vectors. This description requires singling out a specific unit imaginary octonion, and it shows that octonion multiplication is invariant under $\mathrm{SU}(3)$.
• Part 2. A more polished way to think about octonion multiplication in terms of complex scalars and vectors, and a similar-looking way to describe it using the cross product in 7 dimensions.
• Part 3. How a lepton and a quark fit together into an octonion — at least if we only consider them as representations of $\mathrm{SU}(3)$, the gauge group of the strong force. Proof that the symmetries of the octonions fixing an imaginary octonion form precisely the group $\mathrm{SU}(3)$.
• Part 4. Introducing the exceptional Jordan algebra $\mathfrak{h}_3(\mathbb{O})$: the $3 \times 3$ self-adjoint octonionic matrices. A result of Dubois-Violette and Todorov: the symmetries of the exceptional Jordan algebra preserving their splitting into complex scalar and vector parts and preserving a copy of the $2 \times 2$ adjoint octonionic matrices form precisely the Standard Model gauge group.
• Part 5. How to think of $2 \times 2$ self-adjoint octonionic matrices as vectors in 10d Minkowski spacetime, and pairs of octonions as left- or right-handed spinors.
• Part 6. The linear transformations of the exceptional Jordan algebra that preserve the determinant form the exceptional Lie group $\mathrm{E}_6$. How to compute this determinant in terms of 10-dimensional spacetime geometry: that is, scalars, vectors and left-handed spinors in 10d Minkowski spacetime.
• Part 7. How to describe the Lie group $\mathrm{E}_6$ using 10-dimensional spacetime geometry. This group is built from the double cover of the Lorentz group, left-handed and right-handed spinors, and scalars in 10d Minkowski spacetime.
• Part 8. A geometrical way to see how $\mathrm{E}_6$ is connected to 10d spacetime, based on the octonionic projective plane.
• Part 9. Duality in projective plane geometry, and how it lets us break the Lie group $\mathrm{E}_6$ into the Lorentz group, left-handed and right-handed spinors, and scalars in 10d Minkowski spacetime.
• Part 10. Jordan algebras, their symmetry groups, their invariant structures — and how they connect quantum mechanics, special relativity and projective geometry.
• Part 11. Particle physics on the spacetime given by the exceptional Jordan algebra: a summary of work with Greg Egan and John Huerta.
• Part 12. The bioctonionic projective plane and its connections to algebra, geometry and physics.
• Part 13. Two ways to embed $\text{SU}(2) \times \text{SU}(3)$ in $\text{Spin}(10)$, and their consequences for particle physics.

As I’ve written before, these past couple years I’ve often felt like the last remaining person in either quantum computing or AI who lacked a stake in some startup company whose valuation is right now shooting into interstellar space. My academic colleagues, including the ones who seemed the most singleminded about quantum oracle separations and other gloriously useless pursuits? One by one, like in a zombie movie, I learn that they too have now launched startups, and invariably raised tens of millions of dollars, for the sorts of ideas we might’ve idly traded at coffee breaks back in the day, before getting back to our real work.

So why didn’t I join this rollicking party? Partly because of a lifelong fear that, the instant my self-worth became tied to how much money I made, I’d need to humble myself before people who bluster and bully and lie and hype and conceal … yet who nevertheless succeed at becoming orders of magnitude richer than me. I’ve been terrified of even starting down that road, of whether I’d still be myself at the end of it.

It’s also partly that I can’t stand failure, or regret, or being wrong. Of course, as an academic researcher I also fail, and regret things, and am wrong constantly—but there it feels tolerable, because normally I can tell myself that it’s all just down to my inborn limitations. After all, if I could’ve solved the major open problem that someone else solved, or written the brilliant book that someone else wrote, then presumably I would’ve done it!

Clearly, though, I could’ve mined bitcoin in 2010. I could’ve gotten an early stake in Amazon or Google. It’s not even like those ideas never crossed my mind. I just … didn’t act on them, for some reason. (But even if I had, I’d probably just be full of regret that I hadn’t done even more.) Thus, my only way to avoid paralyzing regrets, has been to tell myself constantly that I’m not in the forecasting or money-making businesseses in the first place.

It helped that, insofar as I’m shallow or covetous, insofar as I’ve desired things of this world rather than insight or eternal truth, it’s never really been money that I cared about, but just being respected and liked. Elon Musk is the richest man on earth, but also one of the most despised—which isn’t a bargain that I could imagine ever appealing to me.

Plus, when I actually meet billionaires, I don’t find myself envious of their mansions or cars or anything else that they have; I don’t feel like such things would make my life any happier. Maybe I slightly envy their ability to fund the causes they care about, or their professional staffs who relieve them of drudgery, but mostly I envy the way their wealth announces, to whatever extent it does: “I was right when others weren’t.” Again, though, I’ve never trusted the world to cause me to be right about the future valuations of companies or anything similar, so I’ve settled for having been right about PostBQP and algebrization and BosonSampling.

The bottom line is that I made a choice decades ago to forgo trying to get rich, no matter how many of my friends did the same, and to strive instead to discover and tell the truth—to be a professor, a blogger, a jokester, and an “objective” arbiter and commentator. “Then, surely, everyone will like me!” my internal monologue went. “Then, surely, they’ll be grateful for all the free service I’ve rendered them—for decades of blogging, without once so much as asking for a donation or running an ad!”

HAHAHAHAHAHA.

As any regular reader will know, my attempts to be loved as a blogger backfired pretty spectacularly. Or rather: they did lead to thousands of strangers liking me (and I’m grateful for every last one of you), but they also led to probably an order of magnitude more strangers hating me, and congregating on Reddit and Twitter and elsewhere to discuss how badly I suck. And of course, trying to shift that balance by writing what people want to hear, rather than what I actually believe, was never within my realistic option set.

In the startup context, it didn’t matter how carefully I avoided taking a direct stake for or against any of the companies I blogged about. People on Twitter simply assumed that I had a stake—for example, that I must’ve shorted D-Wave or IonQ, or invested in their competitors, or had equity in AI companies. For why else would anyone write what I wrote?

Amusingly, my attackers here typically did have precisely the conflicts-of-interest that they falsely accused me of having, but that was never at issue; only my imaginary conflicts-of-interest were. Even as the Scott-haters greedily filled their pockets (or tried to), I alone needed to keep turning my pockets out to prove that they were still empty.

So then, screw it! In partnership with my brother David Aaronson, who’s long done investing professionally, and on David’s guidance and encouragement, I’m hereby embarking on a new policy.

Namely: when I hear about a brand-new startup that sounds relevant to my interests—in quantum, AI, or anything else—and I like and trust the founders (ideally, because of their previous academic research work), David and I will often make a small seed investment if the founders are open to it. Or, of course, we might become advisors or get involved in some other way.

In fact, David and I are launching BQP Partners—the link goes to our AngelList, where you can read about how to invest with us if you’re interested. (See also whether you can spot any differences between David’s writing style and preoccupations and mine!)

So far, David and I are investing in:

• Oratomic (the new neutral-atom quantum computing startup out of Pasadena started by Dolev Bluvstein, John Preskill, and Hsin-Yuan Huang among others, which I previously blogged about here). See here for their Series A announcement this week.
• A non-quantum startup being cofounded by someone whose scientific work I’ve admired. I’ll write more about this one as soon as I’m able to!

I have little doubt that more potential investments will come our way very soon (some, probably, as a direct result of this post).

Crucially, I can handle my burden of regret—the “why didn’t I do this much earlier, if I was going to do it at all?” question—by telling myself that friends of mine were not founding companies left and right until very recently. I can also tell myself that I’m doing this less as a bet about the future (in which case … what if I’m wrong?), than simply as a way to support brilliant colleagues doing things that I genuinely admire.

When I blog about a company, I’ll always disclose if I have a financial position that presents a clear conflict of interest, so you can judge for yourself whether to listen to me. (Although, if that’s the sort of thing you’d demand, then you probably weren’t listening to me in the first place, were you?)

Having reflected on it a lot these past few months, I’m happy with my new policy and with my and David’s new venture, and I’m curious to see where it goes. I’m at peace with the possibility that we’ll lose our shirts, but I’m even at peace with a more disturbing possibility—that we’ll make millions and then people will scream at me online for being a sellout, a hack, and a shill. Those people, as I’ve learned, were going to scream at me anyway.

This is a continuation of my conference coverage from last week. The same warnings apply: this is much more technical than my usual posts, readers beware!

Last week I covered the talks from Monday through Wednesday, so I’ll jump right in here with Thursday morning, where the first speaker was Francesco Riva, who after a brief ad for his new board game Tutti Quantum gave a review of positivity constraints, first-principles restrictions on quantum field theories based on their behavior at high energies. After covering some of the research program’s successes like arguments against Galileons and massive gravity, he talked about how new methods allow one to take into account the possibility of loops of massless particles, essentially by invoking a formal version of the idea that real experiments have finite size. He was followed by Grant Remmen, who talked about his work deriving string theory-like amplitudes from increasingly minimal assumptions. While one can always quibble with the assumptions they impose, I do find it encouraging that they are now managing to do this game with both gravity and gauge theory, and with five-particle amplitudes, not just four.

Paul Heslop then covered applications of bootstrap techniques to non-supersymmetric theories, where analytic superspace still finds a way to be useful. Tomasz Taylor covered progress in calculating Yang-Mills amplitudes in de Sitter space. Andrea Puhm and Nima Arkani-Hamed don’t have slides online yet: Puhm’s title suggests her talk was part of the celestial holography field, while Nima’s was likely similar to his talk at Lancefest, where he talked about calculations of amplitudes in the limit of a very large number of loops (represented in the field with a capital L) or very large numbers of particles (represented in the field with a lower-case n). Given the different context, I’m guessing he left out the self-effacing jokes where he was “little n” and Lance was “big L”, though I’m hoping he at least mentioned his students had been checking their results against what they called the “lanswer”.

The evening ended with a gong show, which for the non-initiated is a series of short student talks with a strict time limit (hence the gong). I’m not quite so intrepid as to read all of the slides for these, commenters who attended are welcome to highlight special examples.

Friday began with a talk by Agnese Bissi, who reviewed the connection between holographic correlators and amplitudes in AdS space. I liked her emphasis on this as a lab to find nice new representations of amplitudes, and her summary at the end of the current frontiers. Axel Kleinschmidt followed with a talk on one-loop string amplitudes, where he and his collaborators have gotten gradually more proficient at manipulating the rich structure of elliptic functions that make an appearance. Piotr Tourkine talked about his work using the S-matrix bootstrap to find scattering amplitudes in higher dimensions, a context where accounting for thresholds presented a new challenge. This is a problem people are approaching with genuine supercomputers, he quoted one calculation at 100,000 CPU hours. Lauren Williams is one of a small community of mathematicians who have been intrigued by the amplituhedron, her talk was a walk through a series of conjectures, some made by physicists, some by mathematicians, most with counterexamples found in the last few years.

Finally, Zvi Bern closed the conference with a talk on the frontier of amplitudes-based gravitational wave calculations, referring to it, probably to David Kosower’s annoyance, as 5PM. Some parts of this frontier have been calculated, but a few have proved hard going, bottlenecked by immensely challenging integrals, which go beyond the capabilities of publicly available codes. The new integrals have a variety of strange functions, including the elliptics and Calabi-Yaus I spent time on in my own career, as well as Heun integrals, which I imagine I will have to learn more about from the slides of Elliptics & beyond ’26, as I don’t remember people talking about them when I was in the field. New integration strategies have led to improvements in the public codes and seem to be making good progress, but Zvi highlighted that ideally they want to not need supercomputers at all, as they’ll need to go higher in loops to see effects from, for example, the deformability of neutron stars.

Amplitudes 2027 will be in Munich, with the summer school in Mainz under Stefan Weinzierl’s capable hands. I’m looking forward to seeing what the state of the art looks like then!

An album of GoPro headcam footage climbing Zinalrothorn (AD, 4,221m) in Switzerland.

Full album (49 videos): https://youtube.com/playlist?list=PLFMVEM4j3NZ0

As a scientist in the later stages of my career, the managerial and mentorship load has increased, leaving less time for math and programming. These technical activities were also why I wanted to be a scientist in the first place, and I regretted losing the time for this hands-on research. Formerly the core of my scientific work, these activities require sustained attention, hours at a time, a resource now in short supply.

Also like many scientists, I’ve watched the coming of artificial intelligence over the last few years with growing interest. The technical predecessors and underlying substructure of these large language models (LLMs) are neural networks, a computer technology that has already begun to revolutionise many scientific fields, including cosmology and astrophysics. I wrote about LLMs in my recent book, The Random Universe, but hadn’t really used them for my own work.

So I started using Anthropic’s Claude (no particular reason for this choice, except that some colleagues had had positive coding experiences with it), figured out how to wire it up at the command line, and started talking to it. (After, yes, paying a subscription fee.)

My first project with one of these newly-capable AIs was a minor reanalysis of our data from the Planck satellite, seeing how the cosmological inferences respond to small changes in the data, part of the work for a recent paper. I knew exactly what I needed to do, but it was a lot of plumbing: getting disparate bits of software written by other people to work together in a way different from their authors had intended. I figured I could do it in a few days of solid work.

Instead, I pointed Claude to the draft of the paper, along with publicly available Planck data and code repositories, and asked it to implement the paper’s algorithms with Planck’s data. A few hours later, there was code, alongside tables, figures, and lots of tests to make sure I could trust — and understand — the results. It wasn’t (we weren’t) just able to write and debug the code quickly, it was able to run it, again and again, making small tweaks to the inputs and the code itself, and to the figures it generated, now part of our recent paper.

Next was something more involved: colleagues and I have created a program called Almanac to analyse specific kinds of cosmological data. We wanted to apply Almanac to some new results, in a regime in which it hadn’t really been tested (a very small patch, around 1% of the total sphere of the sky). Almanac helps us measure a curve called the power spectrum, which I’ve written about before.

I pointed Claude to our code, our papers, and to the new data, and explained the problem. Even ensuring that the (poorly documented) data was in a form that our code could understand would have taken me a few hours, but Claude suggested and implemented a series of tests to ensure that everything was self-consistent.

Almanac is a Monte Carlo sampler: because we are trying to understand the probability distribution of matter in the Universe, using noisy and incomplete data, the answer to our questions can only be given as probability distributions. Almanac is essentially a very complicated random number generator, and you can do self-consistency checks to see whether it is producing random numbers with the right properties.

Almanac’s results failed these tests. Could we understand why? Could we fix it? Now, rather than just plumbing, I needed Claude to help me diagnose the problem. It took a while.

Was it a simple bug? We did a series of tests showing that Almanac does work, essentially perfectly, on simpler datasets covering much more of the sky. In fact, this gave me the opportunity to ask Claude to write some new software, based on a paper and related code that I first wrote, with Dick Bond and Lloyd Knox, about 30 years ago. This older algorithm (“BJK”, from our initials) answered the same statistical question as Almanac, using a very different technique. On large areas of sky, Almanac and this older algorithm got the same answer — the code works.

We went on a long rabbit-hole modifying the details of Almanac’s setup, making it more similar to other state of the art samplers. This change also didn’t solve our problem, although it seems to help on the margins. I made one suggestion that I thought would help, based on our long-ago experience with the BJK algorithm, bundling up some of the numbers we were trying to determine into “bands”. I don’t know if Claude would have come up with this idea on its own, and it took a while to get the details right. In fact, Claude would sometimes declare premature victory, admitting its mistakes only when I pointed them out.

It worked, eventually. After a lot of iterations, we transformed a problem unsolvable with the previous version of the code to one that was, well, easy.

But it only worked because my knowledge and experience — literally decades working on problems of this sort — meshed with Claude’s own “talents” — quick turnaround, patience, and encyclopaedic, if not always discriminating, knowledge of computing and of at least some aspects of the underlying science, statistics, and mathematics.

And it was fun! I thought that I liked programming, but I am very happy to have Claude do most of the grunt-work for me. The quick turnaround, and not having to sweat the details of writing and running re-writing and re-running program after program, was a delight.

In many way, working with Claude was like working with a junior colleague. But Claude is not a colleague, but a machine. And, as David Hogg has advocated, the point of doing astrophysics, a beautiful but useless field of science, is exactly the training and fulfilment of the people doing it. That has at least two implications. First, given how much my own experience was necessary to getting good results, that means we had better make sure that we are training humans, not just better LLMs. Second, no matter how delightful the interactions, they mustn’t replace training our students and collaborating with our colleagues.

(This post was written by me, not by Claude, though I did ask it to suggest a title — and this sentence.)

Big screen, big crowd to watch USMNT against Belgium. Team USA, sadly, didn’t do much to make it festive. The three North American host countries knocked out in three straight days. Sure was a nice night by the lake, though.

For non-US folks, feel free to skip. 

For US folks:  The Office of Management and Budget, which for much of its history has been a comparatively uncontroversial element of the executive branch, has set rules and guidelines for how many executive-branch agencies conduct business and interact with, e.g., universities.  For the purposes of how the research ecosystem operates, the most relevant is OMB's "Uniform Guidance" about how grants and contracts work.  Periodically these rules are updated for various reasons, including the goals and policies of the presidential administration.  The standard way this works is that the proposed changes are published in the Federal Register; there is a public comment period; OMB makes revisions and then publishes the new rules.  In principle, Congress can act to override or prevent rule changes, but without the agreement of the President, this is an extremely challenging path.

OMB has proposed sweeping changes to the Uniform Guidance, summarized here.  These proposed rule changes are huge deviations from previous practice.  For example, they would have all final grant decisions made by political appointees or hires of the executive branch (rather than, e.g., agency subject matter experts); grants could be cancelled at any time for essentially any reason (completely undefined insufficient support of the president's priorities), with no appeal process; international collaborations would be severely curtailed. That's just three for starters.  Note that this would also go beyond just the public research enterprise - it would allow the executive branch to cancel funding for things like bridges, roads, schools, agriculture, etc. for undefined political reasons.  It would be a huge transfer of power from Congress to the presidency.  Here is another summary by the AAU.  Here is an editorial essay from ars technica.  

The public comment period on this runs until July 13.  Here is a link where you can make a comment.  Here is a guide for how to be effective at this from Stand Up for Science.  The APS has a tool for helping people to comment about specific aspects of the rule changes.  It is also a good idea to contact congressional delegations (representatives, senators).  

It's important to have a clear public record about the proposed changes.  They may try to implement these regardless, but if so, there will be a continued fight over this in Congress and through the courts.  

I love the 1812 Overture as much as the next man, indeed substantially more, but I think it’s weird that Peacock’s “Star Spangled Sunday” baseball broadcast is celebrating America’s 250th birthday with a song about the Russian army repelling a French advance just short of Moscow.

Oh, hmm, apparently this is a thing.

I’m at the New Jersey shore with family and friends, where we’ve spent this Fourth of July eating hot dogs, playing miniature golf, and wading into the ocean that my great-grandparents crossed to escape calamities they knew about and much greater calamities that they didn’t. Tonight we’ll see the fireworks, weather permitting.

And yes, on the crowded beach today you can find people sporting MAGA and “45-47” hats, and even a giant “Trump 2028” flag—a stark reminder of the millions who would redefine the meaning of our 250-year-old experiment to something dark and authoritarian, the opposite of what its founders intended. Of course, those forces find mirror images on the left end of the political spectrum, where one can find millions more who fully agree with MAGA about the failures of liberalism and the Enlightenment, differing only on the secondary question of which racist thugs should rule instead.

Despite everything, I don’t believe that both factions together constitute a majority. Even on the beach, the MAGA hats are vastly outnumbered by “250” banners and girls in stars-and-stripes bikinis, Americans who just want to celebrate.

Despite everything, I remain thoroughly American if I’ve ever been anything, and invested in the country’s future if I’ve ever been invested in anything.

JD Vance and his friends, who might rule the country after the predictable failure of “Trump 2028,” make a huge deal about “Heritage Americans.” Of course the point of such phrases is to exclude those like me, and recent immigrants, and even (incredibly) JD’s own wife. On reflection, though: could I, too, count as a Heritage American at this point? After all, my family has now been here for half the country’s history. My grandfather, who grew up in poverty, became a professional boxer in Philadelphia and Atlantic City during the Great Depression. He then joined the Army and ended up clearing German mines in North Africa and Italy in WWII. He was assigned to a company of Southerners, who had never met a Jew and were shocked that my grandfather didn’t have horns—but by the war’s end, my grandfather and the relatively few others in his company who remained alive had become best friends. My grandfather told me that he could understand the German POWs who they captured tolerably well, since German was similar enough to Yiddish, but who he could never understand was the British.

As for me, I grew up in the town of Washington Crossing, PA, maybe a mile’s walk from where this happened (and where it’s still reenacted every Christmas):

My earliest childhood hero (that I can remember) was Ben Franklin, whose institute in Philadelphia I visited often. I didn’t even recognize as unusual at the time how the founding of the country didn’t feel like a remote abstraction to me, but was all around me, as if it was yesterday.

That the founders of the United States created the model for all time of how to bend the arc of human history a little bit away from its usual horribleness, of how to overthrow a despotism without instituting an even worse despotism in its place, of how to found a new civilization on ideas and principles rather than raw power … is one of those things that seemed true to me as a child and that still seems true to me today.

May this greatest experiment continue for another 250 years. May it triumph against all those within and without who would see it destroyed.

This week was Amplitudes, my old subfield’s big yearly conference. This year, it’s at Queen Mary University of London.

I’m too busy to attend these days, now that nobody is paying me a salary to do that sort of thing. But it’s still a good chance to keep up with the field, which helps me find stories. And I know I have a few readers who are interested. So I read the slides when I can, and fill you all in.

As of writing this, I’ve read through slides from the first few days of talks. I’ll likely post on the rest next week. As usual when I’m conference-blogging, this post will be quite a bit more technical than my average post, so readers beware: I’ll be mentioning a lot of amplitude-ish ideas without much explanation. I’m happy to explain in the comments if you’re curious, though!

Before I launch into talking about the content, I should mention something I’ve heard about the venue. Apparently this year registration filled up surprisingly quickly. The rumor is that the folks at Queen Mary weren’t able to find a venue with enough space to host the full community, leading to a smaller conference than usual. As the Amplitudes subfield grows, I suspect this will be more and more of a challenge. I remember back when I was organizing in 2021, rooms that could hold enough people were shockingly expensive to book. I wouldn’t be surprised if this becomes more of an issue going forward.

David Kosower opened the conference with a review of the state of the art in amplitudes for gravitational waves. I enjoyed an early slide showing how LIGO has increased in precision over the last ten years, which really helped illustrate the value that good theoretical predictions can bring as the experiment gets better. I also appreciated his attempt to get people to stop saying “post-Minkowskian” and start using a more normal term like Relativistic Perturbation Theory, though based on the other talks it doesn’t look like it’s catching on. After covering the overall state of the art (currently around four loops) he talked about his own work looking for ways to more directly get waveforms for orbiting black holes out of an amplitudes-style calculation, a theme his collaborator Donal O’Connell covered in more detail later that day. (The trick, apparently, is background fields!) Other gravitational wave-related talks came from Gustav Jakobsen, who covered four-loop results using the worldline method, Graham Brown, who explained how to calculate the Magnusian Jakobsen mentioned (it’s the log of the amplitude, basically), Canxin Shi, who talked about how a concept called Stratonovich-Weyl quantization helps explain structures that keep showing up in classical observables, Giulia Isabella, who covered a way to get six-loop contributions to gravitational waves via a wave equation, Lara Bohnenblust, who showed how to get strong-field information from a shockwave limit, and Gang Chen, whose slides were brief enough that I didn’t really get a clear idea of what he was up to. Rounding out the day were two talks that didn’t involve gravitational waves: Zihan Zhou, reporting on work with Nima on a water wave polytope called the hydrotope that they found a general formula for with help from Claude (the AI, not Duhr, to steal a recurring joke from Lancefest), and Michael Ruf, who probably snuck in due to the fact that he has done a lot of work with gravitational waves, but was reporting on a QCD calculation using a tool called Scorpio.

Tuesday had more of a QCD theme. Thomas Gehrmann gave the day’s opening review talk, where he pointed out that to get 1% precision for predictions for the LHC, we’ll likely need three-loop calculations. He pointed out the important role IR real emission calculations and improvements in parton distribution functions need to play, and pointed out that collider physics isn’t just the LHC: between reanalyses of older electron-positron collider data and the upcoming electron-ion collider and FCC, there are even more contexts where high-precision QCD will matter. Simone Zoia talked about one aspect of the current state of the art, five-particle processes involving massive particles, where the functions get strange and elliptical and one has to think hard about the methods one uses (for example, do you want canonical differential equations, or almost-canonical? Bulirsch-Stoer or AMFlow for numerics?) He also included an excellent Laplace quote, “Nature Laughs at the Difficulties of Integration”. Yang Zhang talked about a different calculational frontier, covering progress in the planar limit and with no masses, but for two-loop six-particle and three-loop five-particle scattering. The talk included some nice branding, like an “epsilon collaboration” proposing an algorithm to find epsilon forms for any Feynman integral and a program called Effortless to find symbol letters. Yang Zhang is also a skilled amateur photographer, and he mentions taking the conference photo for Amplitudes five times before. Personally, I’m surprised it’s only been five times! Dmitry Chicherin presented results bootstrapping QCD amplitudes. This was a dream of mine back in my hexagon function days, and while they aren’t quite at the point of being useful (my understanding is they’re still only getting the leading transcendentality, and none of the amplitudes they’re finding are new) it’s still pretty cool that this is even possible now. Bo Feng proposed what he claims is a general algorithm to find generating functions for IBP reduction. It’s not clear to me whether his setup bypasses the computational difficulties in existing methods (mostly involving solving large systems of equations), or whether it shifts the issue elsewhere, though. Pierre Vanhove’s talk also involved QCD, though at an effective field theory mediated step removed, with chiral perturbation theory, a theory of low-energy QCD that he used to shore up the lower energy ranges of lattice calculations for contributions to the muon anomalous magnetic moment (a natural calculation to involve him due to the presence of novel elliptic integrals). Overall, the QCD talks impressed me with the wide range of software packages mentioned, some of which already existed when I was in the field but many of which are new. I do wonder if this is just a result of many research paths maturing at the same time, or if people are finding it easier to write packages with tools like Claude Code.

The remaining three Tuesday talks were more mathematical or theoretical in focus. Henrik Johansson talked about black hole Compton scattering in N=8 supergravity, where it’s possible to use the wave equation to extract all-loop results. Cristian Vergu reported on his progress with Landau analysis. It’s been really fun watching this grow from a small reading group at NBI to what appears to be at this point quite a deep understanding, including a picture of how to understand amplitude singularities with quite a lot of breadth and detail, and the persistent hope that this could allow one to manipulate singular quantities without needing dimensional regularization. Michael Borinsky gave an update on tropicalization, where he showcased a theorem he proved demonstrating a way to compute certain classes of amplitudes in polynomial time in the loop order, even when the number of Feynman diagrams increases factorially. The examples he talked through were impressive, but I’m still a bit skeptical this could work for the Standard Model. I’d want to talk to him about it to figure it out, anyway!

Wednesday was a short day, as is tradition, to give people some time for tourism in the afternoon. Alexander Zhiboedov began the day with a talk on energy correlators in N=4 super Yang-Mills at finite coupling, where he is now able to do a bootstrap with two-sided bounds to hone in on the actual quantity even outside of the planar limit. Arthur Lipstein and Daniel Baumann both talked about cosmological correlators, the former with amplitudeologists’ favorite toy model of the conformally coupled scalar and the latter with Yang-Mills and gravity in de Sitter space. Dave Dunbar closed the day with a historical talk, walking through the milestone amplitudes papers before the Amplitudes conference existed. It’s the kind of talk he could have given at Lancefest the week before if others hadn’t already covered the material!

I’ll cover the rest of the conference (Thursday and Friday) in next week’s post, so see you then!

Scott’s foreword: Cynthia Dwork is Gordon McKay Professor of Computer Science at Harvard, and a pioneer in the fields of differential privacy and algorithmic fairness. On my recent travels to the SigmaWest science camp and then STOC, there was much talk about a recent Trump administration action that would ban not only differential privacy, but essentially all modern techniques for preserving privacy in large datasets, for example in the 2030 US Census. I realize that many of us have “outrage fatigue,” but this particular outrage hits extremely close to home for the CS theory community. So when Cynthia approached me at STOC to propose a guest post on the issue, of course I said yes. The post that she sent me, below, is cosigned by many other leaders in the field.

On June 4, 2026, the U.S. Secretary of Commerce issued a directive (DAO 216-26) relegating confidentiality protection in all Bureau of Economic Analysis (BEA) and U.S. Census Bureau publications to techniques dating back to the early 1970s, turning its back on over half a century of progress and protections for data subjects. Advances in confidentiality provision had enabled the Census Bureau to share increasing quantities of data at more granular detail. The order will result in less useful (or fewer available) statistics, weaker protection, or both. We write to illustrate the danger posed by the order and to mobilize the scientific community to speak out against it.

The acting force behind this order is political interest, not scientific merit. DAO 216-26 bypassed legally required administrative procedures. It fulfills a promise made by the architects of the Heritage Foundation’s Project 2025, and reflects both the rhetoric and misunderstandings of representatives of the Center for Renewing America (CRA), an organization founded by OMB Director Russell Vought. CRA’s explainer on the use of differential privacy in the 2020 Census is up-front about the stakes: “Even if the citizenship question is added to the Census, it will be impossible to ascertain the status of individuals so long as differential privacy is used.” But masking this sort of personal characteristics data is legally required by the Census Act (13 U.S. Code Section 9), which makes it a crime to “make any publication whereby the data furnished by any particular [individual] can be identified.” Confidentiality is also widely understood as critical to ensuring that people respond to the census. 

DAO-216-26 bans differential privacy and other modern (and not so modern) techniques. It restricts disclosure avoidance techniques to “coarsening,” which it describes as “reducing the level of detail or specificity of published statistics, such as through rounding, aggregating (grouping), and/or the use of ranges.” “Suppression” (“expressly redacting certain values”) may also be used, but only as a “last resort.” DAO-216-26 forbids “noise infusion”, described as “methods that involve modifying a dataset by adding random values, or noise.” 

Noise infusion was invented precisely to address the increasing demand for granular data in the face of confidentiality laws that forbid publishing reidentifiable data. Coarsening and suppression were satisfactory for most national, aggregate statistical series, like the Principal Federal Economic Indicators. However, these techniques failed when applied to business and demographic data at fine geographic or industrial detail. By forbidding noise infusion, the directive bans the disclosure avoidance techniques at the core of dozens of data releases over the last three decades. It bans input noise infusion, used in the Quarterly Workforce Indicators since 2002 and, until now, planned for the Bureau of Economic Analysis statistics [1].  It bans swapping, used for decennial census publications since 1990. It also bans differential privacy,  the best currently known approach for obtaining the most data utility for any given level of privacy. Differential privacy was used for sharing data on commuting patterns (OnTheMap) since 2008 and for publications based on the 2020 Census. Until the recent directive, differential privacy was planned for the 2030 Census too.  Many other products and procedures are implicated as well.

1.      Illustrations

DAO-216-26 is incompatible with the Census Bureau’s dual mandate to provide confidentiality and fitness for use. To illustrate this, we recall and expand on an example due to Nathan Goldschlag, inspired by the County Business Patterns (CBP) data, which provides statistics on business activity broken down by industry and geography. Goldschlag describes three scenarios, illustrating the tension between providing useful information and maintaining confidentiality of responses as required by the Census Act. 

·       “There is only one brewery in a small county. If the CBP published the exact count of brewery employees in that county, it would be disclosing the information of one business (how many workers it employs), a clear violation of the law.²

·        “There are two breweries in a small county, and the CBP again publishes the exact count of brewery employees. If I own one of those breweries, I could learn how many employees my competitor has, again violating the law.

·        “There are more than two breweries in a small county, but the CBP chooses not to publish the total number of brewery employees out of concern that it might compromise the privacy of the businesses. If I’m a prospective brewery owner, I may deem the project too risky to pursue without information about the market I’m entering.”

In Goldschlag’s example, coarsening makes the published statistics useless.  We now add a fourth scenario, showing that it also fails to maintain confidentiality.  To keep things simple, assume none of us owns any of the businesses in the new example. The County has two towns with one brewery each, North Bend and South Bend. Furthermore, North Bend has a mobile bottling company and South Bend has a stationary bottling company. That’s a total of four beer-related business entities in the County.  Two of these businesses, the North-Bend brewery and the South Bend bottling company, are publicly-owned.

• The CBP publishes five statistics:
  • (A) The total number of employees in beer-related businesses in North Bend: Because there is only one brewing company in North Bend and only one bottling company in North Bend, the category is coarsened to “beer-related”.
    
  • (B) The total number of employees in beer-related businesses in South Bend: Because there is only one brewing company in South Bend and only one bottling company in South Bend, the category is coarsened to “beer-related”.
    
  • (C) The total number of employees in brewing only:  Because there is only one brewing company in each of North Bend and South Bend, the statistic is coarsened to the total number of employees in brewing only in the County.
    
  • (D) The total number of employees in bottling only: Because there is only one bottling company in each of North Bend and South Bend, the statistic is coarsened to the total number of employees in bottling only in the County.
    
  • (E) The total number of employees at publicly owned companies: Because there is only one publicly owned company in each of North Bend and South Bend, the statistic is coarsened to the total number of employees in publicly owned companies in the County. 

We now have 5 equations in 4 unknowns. Using only 4 of these (A, B, C, and E), we can solve for the exact number of employees at each of the four companies with high school algebra.

In the above (fictional but realistic) scenario, the County Business Patterns were released with good-faith coarsenings for the geographical, business, and ownership categories.  Nonetheless, even without inside knowledge of one of the companies’ number of employees, we can completely reconstruct all four numbers.  What happened?  The coarsenings interacted poorly.  Noise infusion perturbs that set of equations, preventing exact reconstruction.

2.      Impediments to Implementation

The Commerce Department now claims the directive’s return to the outdated “tradstat” traditional statistical techniques of the 70s is good for data consumers: “This update to our disclosure limitation method protects respondents and provides the public with more essential economic information.”  (Emphasis added.)  As we saw from Goldschlag’s example, coarsening does just the opposite.

And it can’t be fixed. Coarsening by definition reduces access to fine-grained information.  Our example of three poorly interacting coarsenings shows that this sacrifice is for naught: without noise infusion, confidentiality is destroyed by elementary calculations.  For population surveys, this is precisely what formal noise infusion methods, like differential privacy, protect against; this is the “fancy math” that Goldschlag mentions in his post and that holds personal characteristics, like citizenship status, in confidence.  

3.      Confidentiality is Critical for Federal Statistics 

The scientific community continues to debate the best techniques for protecting the confidentiality of respondents’ data, but DAO-216-26 is not driven by science. It is driven by political interests. Those issuing this order are willing to risk the public’s trust in the process. We think that this is wrong-headed and dangerous. 

Civil servants will do their best to comply with this order while still following the laws that require them to protect the confidentiality of respondents’ data. To balance these competing mandates, they may seek to produce less data or coarsen data so much that it is unusable. Or they might be pushed by political actors to publish data that can be easily unmasked, like in the brewery examples above. Regardless of their choices, they will be hard-pressed to guarantee respondents’ confidentiality, which will prompt many businesses and individuals to simply not answer. This is devastating for an agency that delivers democracy’s data.

Conclusion

Rather than political actors overruling the government’s own statisticians, we need deep investment in our nation’s statistical agencies, ensuring that agencies have the staff and support to improve their methods using the best available tools. Regardless of how the scientific community feels about any specific privacy-enhancing technique, we must collectively reject this anti-scientific approach to governing federal statistics. Too much is at stake. 

How to Take Action

Share this post with others in your professional network and community.

Contact your Congressional representative and voice your concerns. Calling or writing to your representative is one of the most effective and easiest things a constituent can do that should only take a couple minutes of your time.

1. Find your representative contact information here.
2. State your concern. Here is a sample script: “My name is [Name], and I am a constituent from [City] in your district [ZIP CODE]. I am calling because I am concerned about the the U.S. Secretary of Commerce issued a directive (DAO 216-26) that wants to relegate confidentiality protection in all Bureau of Economic Analysis and U.S. Census Bureau data products and statistics to outdated and ineffective statistical techniques. If followed, this order will destroy the Commerce public data our nation relies on for important decisions, such as where to build necessary services for our community’s well-being. I want the DAO to be rescinded. I want proper administrative procedure to be followed. I want technical decisions such as the choice of method used to balance utility and confidentiality to be informed by professionals in the federal statistical agencies, not made unilaterally by political operatives.”
   1. Optional is stating what kind of constituent, such as a retired teacher or a working professional.

Volunteer to help preserve Census working papers and documentation. Pages explaining “noise infusion” and “differential privacy” are already going offline. Archive relevant methodology pages and technical documentation. You can also do this via the Internet Archive’s Wayback Machine (“Save Page Now”).

John Abowd 
Aloni Cohen
Cynthia Dwork
Jae June Lee
Jayshree Sarathy
Adam Smith
Salil Vadhan

[1] BEA Working Paper WP2026-9, now purged by the Department of Commerce.  As of 6/22 Google returns:

Bureau of Economic Analysis (BEA) (.gov)
https://bea.gov › files › papers › BEA-WP2026-9

An album of GoPro headcam footage from climbing the Aiguilles Crochues Traverse (PD, 2,840m) near Chamonix, France in 2022.

Full playlist (45 videos): https://youtube.com/playlist?list=PLM4i-DL0BZ0Q

Footage from our climbing trip to Frenchmans Cap, Tasmania (Australian grade 17, 380m) in 2022.

Full playlist (47 videos): https://youtube.com/playlist?list=PLT0z6qQjCS3c

GoPro headcam footage from climbing Triglav (2,864m), highest mountain in Slovenia, via ferrata. Climbed in 2022.

Full playlist (40 videos): https://youtube.com/playlist?list=PLZgovD57Nsr4

A scientist in Florence can’t avoid bumping into colleagues. 

When visiting the Renaissance’s birthplace last summer, I ran into a fellow physicist even on a Saturday morning. I was wandering around the Uffizi Gallery, a museum blessed with some of the greatest hits in western art. A familiar face arrested me on the first floor.

Another colleague cropped up outside the museum. (Some might classify him as an applied physicist or an engineer, but he exhibited a theoretical physicist’s overactive imagination.)

One colleague, I’d been looking forward to meeting for over four years. Jae Dong Noh is a professor of physics at the University of Seoul in South Korea. He’d conducted the first numerical tests (classical-computer simulations) of an idea I’d helped midwife, the non-Abelian eigenstate thermalization hypothesis (NAETH). An earlier blog post described this mouthful, which predicts how certain quantum many-particle systems thermalize, or experience the flow of time. These systems’ dynamics conserve properties, analogous to energy, that are incompatible: one can’t measure the properties simultaneously, as one can’t measure a quantum particle’s position and momentum simultaneously. Because incompatibility helps distinguish quantum from classical physics, such systems’ thermodynamics qualifies as particularly quantum.

Jae Dong modeled such a system and others numerically in a paper. I admired his computational techniques and his grasp of symmetries (for experts: how non-Abelian symmetries affect chaotic quantum systems’ energy-level statistics). My postdoc Aleks Lasek was planning a more thorough numerical test of the NAETH, so I reached out to Jae Dong, and a collaboration crystallized. 

Seoul operates thirteen hours ahead of Maryland, but we managed to Zoom because Jae Dong is a night owl and I’m an early bird.¹ Zoom introduced me to a man perpetually dressed in a neat button-down shirt and sweater, silver overriding the black in his hair. The neatness extended to Jae Dong’s explanations: if Aleks and I didn’t understand one of his emails, he’d explain it quietly and calmly, untangling the confusion as though pulling a comb through wool.

The collaboration settled into a rhythm: I’d pose a question or propose a goal, Jae Dong would respond with an analytical calculation,² I’d find holes in the calculation, Jae Dong would plug the holes, I’d re-check the argument’s logic, and we’d repeat the cycle. Had I been in Jae Dong’s shoes, I’d have swallowed the constant objections as I’ve swallowed grape-flavored cough medicine,³ but he always responded with equanimity—sometimes even good cheer—and a possible solution. Meanwhile, Aleks and then-undergraduate Jade LeSchack checked our analytical arguments numerically.

So smoothly did the collaboration hum along that we coauthored two papers before ever meeting in person. One demonstrates numerically that two quantum many-body systems (for experts: nonintegrable Heisenberg models) obey the NAETH.⁴ In the other paper, we derive a symmetry relation from the NAETH. If the 17-syllable NAETH is a mouthful, the symmetry’s name is half a mouthful: a Kubo–Martin–Schwinger (KMS) relation. It’s important because (i) it enables us to calculate how rapidly a thermodynamic system responds to a stimulus, such as a weak magnetic field, and (ii) physicists go gaga over symmetries generally. 

The KMS relation constrains thermal states—essentially, systems that have temperatures. Your typical isolated many-particle quantum system looks thermal if you can observe just a small chunk of it at a time. Accordingly, Jae Dong and collaborators had proved that isolated many-particle quantum systems obey the KMS relation approximately. The larger the system, the more accurate the approximation. 

We extended his argument to systems whose dynamics conserve incompatible properties. Such an extension might sound simple, but its proof filled 24 pages of appendices. (For experts: Clebsch–Gordan coefficients are tricky blighters.) We discovered that, under certain conditions, incompatible conserved quantities can reduce the extent to which a quantum system obeys the KMS relation. Quantum incompatibility can augment deviations from conventional thermodynamics.

Jae Dong planned to present about our work at StatPhys, an international statistical-physics conference, which Florence was hosting in 2024. Throughout the two-and-a-half months before the conference, the KMS relation consumed our team. (For experts: Clebsch–Gordan coefficients are very tricky blighters.) I even hid in my hotel room, working and reworking our proofs, during another conference during that time. 

The toil paid off. We submitted our KMS manuscript for public scrutiny the day I flew to Florence—because not only Jae Dong would be representing our team at StatPhys. I was looking forward to meeting him there for the first time.

The StatPhys committee outdid itself. The opening ceremony unfolded in Florence’s Palazzo Vecchio, where members of the Medici dynasty once lived. Giorgio Parisi, who won a Nobel Prize for statistical physics in 2021, lectured at the ceremony.

The meat of the conference took place in two other palaces, the Palazzo dei Congressi and the Palazzo degli Affari. In one of them, I met Jae Dong. Although we’d shown that quantum incompatibility can defy thermodynamic predictions, he met my expectations.

We discovered another thermodynamic phenomenon challenged by incompatible conserved quantities, so stay tuned for another paper and blog post. Some colleagues, one can’t avoid; others are worth engaging with again and again.

¹ Aleks has confessed to night-owl habits, but physics motivates him to adapt. Some days, he’s emailed me results before even I’ve woken up. Who needs coffee when the thrill of discovery electrifies one minutes after one hops out of bed?

² An exact calculation written out on paper, as opposed to a numerical, or approximate, calculation performed by a silicon-based classical computer.

³ Does anyone like the grape flavor? Why do companies bother producing it?

⁴ Rohit Patil and Marcos Rigol, too, have checked numerically that a system obeys the NAETH.

 Some brief science and technology items heading into the weekend:

• IBM has reported making prototype chips for the "0.7 nm node".   As always, one should not interpret that size scale literally, since the effective diameter of a single silicon atom is around 0.2 nm.  The basic building block of their architecture here is the nanostack, which is a limiting case, somewhat 3D-integrated version of their nanosheet "gate all around" field effect transistors.  The fact that these structures can be made at this scale, reliably and en masse, is just phenomenal.  
   

  

  
  
  
  
• I'd written previously about the Vesuvius Challenge, the attempt to use a combination of x-ray tomographic imaging and machine learning to read the carbonized ancient Roman scrolls found in a villa in Herculaneum, where they had been buried by the pyroclastic flow from the eruption in 79CE.  Well, they've managed to read a complete scroll - here's the preprint.  Very cool, and the hope is that among those scrolls might be books believed lost to history.
• At the beginning of the month, Microsoft unveiled the next iteration of their approach to implementing topological qubits based on superconductor/semiconductor hybrid devices, as described here.  The relevant preprint is this one.  Some reporting on this is here.  This week, Nature published a comment on the prior work as well as the reply.  
• There has been an explosion of research in recent years about trying to use electromagnetic cavities to tune the physical properties of condensed matter systems.  I'd discussed this here.  In the last couple of weeks, this preprint appeared, reporting that placing few-layer NbSe2 in an appropriate (THz) cavity can increase the superconducting transition temperature from 3.02 K to 3.41 K.  A 13% increase in \(T_{\mathrm{c}}\) is certainly interesting.
• The incoming president of the National Academy of Sciences has a nice statement in Science.  The key passage for me:  "By its charter, the Academy is nonpartisan and neither a progressive organization nor a conservative one. It is a scientific body that follows the evidence wherever it leads, even when the destination might be unwelcome. In heated and polarized discourse, it is the Academy’s obligation to be the most careful and trustworthy voice. But rigorous science that arrives too late, or speaks too quietly, serves no one."  

It’s been a while since I’ve said this: I’m at a conference this week!

Specifically, I’m at Lancefest, which is not just any old conference, but a birthday conference for Lance Dixon. When a renowned academic turns 60 or so, their students and collaborators hold a birthday party-flavored conference for them. The conferences are usually a mix of academic talks and reminiscences, with the occasional roast thrown in.

I went to my advisor’s birthday conference four years ago. Lance wasn’t my advisor, but in many ways he might as well have been. When my advisor took a sabbatical in the middle of my PhD, he sent me to work with Lance. It was my first real experience doing research in a team, not just puzzling away by myself with occasional feedback. And I was hooked: I spent the rest of my academic career in Lance’s field. We collaborated time after time, and even when I started to branch out he remained a frequent presence.

In part, that’s because Lance’s field was really Lance’s field. Amplitudeology has grown a lot since I started out, with several hundred people going to the field’s big yearly conference and subfields like Elliptics having their own yearly conferences. In such a world, it’s tough for anyone to feel like a truly central figure. But Lance tends to. He’s been able to keep up with that growing world, to keep finding important problems and keep understanding others’ ideas. While some have specialized, or stepped back, Lance seems to somehow manage to be a father figure for the whole field all at once. It’s a capability his advisor Jeff Harvey might have predicted, he mentioned in his talk that Lance’s interests were always broad. Over the years the young students I met who joined when the amplitudes field was already large saw Lance as a kind of mysterious titan, and were occasionally awed that I had worked with him. “What was that like?”

Well, it was like working with Lance. Lance isn’t a manager at heart, like some senior academics end up. He wants to understand everything he works on, and will happily dive in, Maple subscription in hand, and try to figure things out for himself. He wants you to keep up with him, understanding on your own terms, to keep him honest, to provide an independent check. But that often wasn’t possible, because the man is just so damn fast. He’d be miles ahead of me, with his Maple and his laptop, while I was churning overnight Mathematica runs on a twenty-machine cluster.

(Was the difference due to our taste in software? Partly. I did learn to use Maple later, it genuinely is faster at some things. But Lance is faster at almost everything.)

And he cared so damn much sometimes. About getting things right, like a scientist should. About making things nice, too: finding a pretty basis of functions, a better notation for the paper, something that might jostle out the next big insight. Working with him, you could feel like that one paper was the most important thing in the universe.

Others at the event have had similar stories. Fernando Febres Cordero remembers noticing a potential issue, emailing Lance about it, and in a few minutes hearing back with a potential explanation.

Lance is someone who became a leader without really being a politician. He doesn’t have the legions of students in tenured positions that some do. I trimmed that count by one, and it wasn’t huge to begin with. But for someone who isn’t “everywhere” in that sense, he manages to be “everywhere” all the same.

So Lance, happy birthday! You’re really the only person who could have had a birthday conference quite like this, a cross-section of the field, all with something kind to say. Thanks for putting up with any embarrassment associated with having this much attention for three days, and I wish you many Maple-fueled mysteries to come.

An album of one minute GoPro headcam clips from the Monte Rosa Tour (aka Spaghetti Tour) along the Swiss/Italian border in the Alps. This footage is from our expedition in 2022.

Full video playlist (101 clips): https://youtube.com/playlist?list=PLeRYUHyTyGxYa4zZnUIfiAScN1Hcl_HWe

I recently finished reading The Orphan Master’s Son, a (Pulitzer-winning, apparently) novel set in 2000’s-era North Korea. In one plot point, Kim Jong Il has agents steal a Japanese telescope designed to measure the cosmic microwave background radiation, under the mistaken impression that it will help him find uranium.

The novel plays it for (horrified) laughs, but I’ve seen this kind of misunderstanding crop up in the real world too. Sure, most people would realize that a telescope probably won’t help you find something buried under a mountain of rock. But there’s a deeper misunderstanding here. Ask yourself: what does “radiation” mean?

We talk about radioactive elements like uranium releasing radiation. We talk about electromagnetic radiation, including everything from gamma rays to visible light to the 5G of your cell phone. We talk about cosmic radiation coming in from space, and about the cosmic background radiation that originated in the early universe. For someone who doesn’t know much about physics, it probably sounds like all of these are the same kind of thing.

But they’re not!

It’s helpful to break things down in terms of particles. Radioactive elements release three main types of radiation: alpha, beta, and gamma. Alpha radiation consists of helium nuclei: two protons stuck together with two neutrons. Beta radiation consists of electrons. Gamma radiation is a type of electromagnetic radiation, and consists of photons: particles of light.

Anything we call electromagnetic radiation is a wave in the electromagnetic field, a ripple that moves through space. That’s different from other shapes of electromagnetic fields, like a magnetic field that stays in place. From a particle perspective, an electromagnetic wave is made up of photons, and physicists will often describe all such waves as light. Some of that light is the familiar rainbow of visible light, while some has lower-energy photons, like microwaves and radio waves, or higher-energy photons, like gamma rays or X-rays.

Cosmic radiation (more often called cosmic rays), like radiation from radioactive elements, can be many types of particles again. Most of it consists of protons, while some consist of various nuclei, or electrons. A smaller fraction are antimatter, like antiprotons or positrons. Sometimes, physicists include neutrinos when they talk about cosmic rays, while sometimes they include gamma rays.

The cosmic background radiation is once again different. This is an overall hum of microwaves, electromagnetic radiation from the early universe that has gotten fainter and more diffuse over time. Cosmologists will sometimes talk about when the universe was “radiation-dominated” versus “matter-dominated”. They’re referring to times when most of the energy of the universe was in electromagnetic radiation, versus when it was mostly in other particles.

The only thing that ties all of these meanings together is the word’s literal meaning: radiation radiates. It starts in one place and travels outwards, having an effect at a distance. For the first scientists to observe phenomena like X-rays, this was almost all they knew about them, so they tossed them together in one category. Now, we know much more, but the names stuck.

So if you hear a physicist use the word “radiation”, try to avoid making any assumptions. You can’t know, just from that word, what they mean.

And please, don’t steal any Japanese space telescopes.

Paul Schwahn and I have come out with a new paper about octonions and the Standard Model:

• The Standard Model gauge group from the exceptional Jordan algebra

It builds on things I’ve discussed here, but it goes further. Let me explain a bit.

A bit is just a binary alternative: 1 or 0, true or false. That’s how it works in classical logic. We could also have a ‘trit’, meaning 3 alternatives.

In quantum physics we instead have qubits and qutrits.

Qubits and qutrits are usually described using complex numbers. The algebra of observables of a qubit is the Jordan algebra $\mathfrak{h}_2(\mathbb{C})$, consisting of $2 \times 2$ self-adjoint complex matrices. Similarly, the algebra of observables of an qutrit is the Jordan algebra $\mathfrak{h}_3(\mathbb{C})$, consisting of $3 \times 3$ self-adjoint complex matrices.

We can also study systems with more than 3 alternative ways to be. They work the same way, using the Jordan algebras $\mathfrak{h}_n(\mathbb{C})$ with $n \gt 3.$

But we can also do quantum mechanics using other number systems! The options have been mapped out, and the largest allowed number system for this purpose is the algebra of octonions.

A weird thing is that Jordan algebras built using octonions can describe qutrits, but not quantum systems with more than 3 alternative ways to be. The algebra of observables of an octonionic qutrit is the so-called ‘exceptional’ Jordan algebra $\mathfrak{h}_3(\mathbb{O})$, consisting of $3 \times 3$ self-adjoint octonion matrices. What makes it exceptional is that $\mathfrak{h}_n(\mathbb{O})$ is not a Jordan algebra when $n$ is bigger than 3.

So, there’s something special about octonionic qutrits — and it turns out that every symmetry in the gauge group of the Standard Model is a symmetry of an octonionic qutrit!

Not every symmetry of an octonionic qutrit is a symmetry of the Standard Model. But those that do have a simple description. They are those that restrict to give symmetries of an ordinary qutrit sitting inside the octonionic qutrit… and an ordinary qubit sitting inside that!

That sounds exciting, but also vague, so let me make it precise.

While lots of people say the gauge group of the Standard Model of particle physics is $\text{U}(1) \times \text{SU}(2) \times \text{SU}(3)$, in fact a certain subgroup of this acts trivially on all known particles. If we mod out by that, we’re left with

$$\begin{array}{ccl} \text{S}(\text{U}(2) \times \text{U}(3)) &= & \Big\{ x \in \text{SU}(5) : x = \left( \begin{array}{c c c c c} \ast & \ast & 0 & 0 & 0 \\ \ast & \ast & 0 & 0 & 0 \\ 0 & 0 & \ast & \ast & \ast \\ 0 & 0 & \ast & \ast & \ast \\ 0 & 0 & \ast & \ast & \ast \end{array} \right) \; \Big\}. \end{array}$$

and this is the group I’m talking about.

We proved two theorems describing this group in terms of the symmetries of an octonionic qutrit. The group of automorphisms of the exceptional Jordan algebra $\mathfrak{h}_3(\mathbb{O})$ is a 52-dimensional Lie group known affectionately as $\text{F}_4$ — so that’s what I mean by the symmetries of an octonionic qutrit.

Here’s our main result:

Theorem 1. Suppose $X,B$ are Jordan subalgebras of $\mathfrak{h}_3(\mathbb{O})$ such that