The purpose of this letter is to warn the mathematical community of a new type of paper mill that could potentially appear as a consequence of the advances in artificial intelligence. I am a postdoc working on Math and AI, and I organize an international online seminar on math and machine learning.
Firstly, there exist groups that produce fake research papers and sell them to scientists; these groups are commonly called paper mills. According to this study, at least 2% of all research submissions in all sciences are submitted by paper mills.
Secondly, there also exist start-ups, companies and universities working on solving mathematical problems with machine learning. In particular, some use tools that take information from a mathematical problem and return, when possible, experiments to prove it. The level of human interaction varies: in some cases the user has to provide only a text description of the problem; while for other programs, the user must write code that evaluates the fitness of the solutions proposed by agents.1 AI agents are also capable of transforming the output of those experiments into a written research paper. While it is not clear how much interaction was needed to produce the preprint about the Navier-Stokes problem or to solve the Jacobian conjecture, in this post I want to give emphasis on the possibility of an artificial intelligence solving completely a problem only using a prompt as input.
With these two antecedents, I would like to ask the reader: what would happen if a group trained an algorithm to recognize important mathematical questions? The algorithm can work in parallel and mass produce meaningful questions. If we now give this list of questions as inputs to algorithms that try to solve them, and we furthermore assume that they manage to solve a small percentage of the input questions, then the final products are research papers that are publishable since they have solved meaningful questions for mathematicians. Finally, what if the company sells those finished results to mathematicians? This would create a paper mill that produces correct papers and sells the authorship. Moreover, if an AI that solves problems has a certain level of quality, an AI that formalizes the proof can further filter the papers.
One can wonder what the danger is if paper mills are producing knowledge, that is, publishable papers. Fraudulent credit is one of the problems; affecting hiring, promotion, grants, and reducing the credibility of the publication process.
In informal conversations, I asked some of the experts who attended the Mathematics × AI: Challenges and Opportunities event at the Korea Institute for Advanced Study and the International Conference on Machine Learning 2026 about the possibility of this new type of paper mill. Some of the people I talked to are collaborating with AI companies, while others were participants with different levels of experience in AI.
While I personally expected to be assured of the impossibility of this new type of paper mill, some of the experts considered this scenario plausible within a few years, and they could not think of a solution for journals to detect this problem. Some told me they have students working on an algorithm to predict important problems. I was encouraged by them to keep talking about this problem in order to raise awareness, which led me to write this letter.
Why would this be a very difficult problem for journals?
From the point of view of the journals, there are no steps in the review process to ensure that the person who submitted the paper did not buy the paper from a paper mill, or to certify that the person is aware of the content of the paper. Note also that using a detector to see if a paper is written by AI is useless: a person can buy finished papers and rewrite them.
We are now seeing talks in which people used AI in their work. Note that the quality of the talk may not be related to the use of AI. The speaker could be afraid of public speaking, perhaps the speaker does not have enough experience giving talks, faces a language barrier, or is new to the topic and knows just the minimum required for the result to hold.
I call on the community to discuss what changes journals would need to implement if we ever reach that level of technology.
Acknowledgement: I would like to thank Susana Lopez Moreno for related discussions.
- The following examples, listed alphabetically, are not exhaustive: Aletheia, AlphaEvolve, Astra, Iteris, OpenEvolve, Rethlas, and ShinkaEvolve.
︎
Received 11 September 2026.
, a torus 
, or a lattice 
, although many of the techniques can extend to more general domains (e.g., to Riemannian manifolds). While the Fourier transform often plays a prominent role (in particular, by setting the stage for time-frequency analysis and enabling various decompositions or other transforms that involve frequency space or phase space in addition to physical space), real-variable harmonic analysis is often focused on estimating other transforms or expressions that often interact well with the Fourier transform, but need not explicitly invoke it. Examples include 
) or the 
, where we use the normalization 
.
be an explicit transform of 
or maximal function 
). To what extent is the “size” of 
controlled by the “size” of 
being studied is linear or sublinear, in which case the natural type of bound to ask is a linear bound 
(e.g., 
norms), and 
is some bound. Depending on the application, we may be interested in various levels of precision regarding the bound 
constant for the Hilbert transform 
is exactly 
, and the optimal 
constant for 
in one dimension turns out to be 
(a 
or 
in auxiliary parameters, such as a scale parameter 
.
![{[2^k, 2^{k+1}]}](https://s0.wp.com/latex.php?latex=%7B%5B2%5Ek%2C+2%5E%7Bk%2B1%7D%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
for 
all potentially contribute an equal amount that “interfere constructively” to cause a logarithmic divergence. This can be unnecessarily conservative when one is in regime (c) (which is for instance the situation in modern topics such as restriction theory or the Kakeya conjecture); nevertheless, the general skills gained by trying to not lose even a logarithmic factor in the bounds are often valuable in these other types of analysis.
by some decomposition 
, so that the transformed function 
in various ways (e.g., via the triangle inequality, by Bessel type inequalities, or by the more modern technique of decoupling inequalities). In short, the subject tends to proceed by a divide and conquer philosophy: it is generally preferable to replace a simple-looking but hard-to-estimate expression with a large, messy-looking combination of expressions that are easier to estimate. As such, the aesthetics of the subject are almost the reverse of those in the more algebraic portions of mathematics, in which progress is often made by making the expressions involved look as simple and unified as possible.
provides the singularity, but the sign change from 
to 
provides the oscillation. Classically, the interplay between these two phenomena can be tamed by analyzing the behavior of this operator both in the time (or “physical”) domain and in frequency (or “Fourier”) domain; in particular, the fact that singular integral operators such as the Hilbert transform are simultaneously a well-behaved Fourier multiplier and is “pseudo-local” in physical space lies at the heart of the standard Calderón–Zygmund theory for such operators. This dovetails nicely with more modern “time-frequency analysis” approaches to the subject, which can also handle other interesting operators, such as restriction or Bochner–Riesz operators via tools such as the wave packet decomposition, although these will be outside the scope of this course.
, which are function spaces that are useful for measuring how “singular” or “decaying” various functions are, but do not pay attention to how they oscillate or where their mass is distributed. As such, these spaces do not capture the underlying geometry of the domain, which also plays an essential role in the subject; but it is nevertheless essential to have a good base understanding of the rearrangement-invariant theory before moving on to the more delicate aspects of harmonic analysis that are sensitive to rearrangements.
, 
, or 
denotes the assertion that 
for some constant 
for 
. If we permit this constant 
or 
denotes a bound of the form 
for some constant 
that can depend on 
and 
. As indicated above, in this course we will generally not dwell much on exactly what these constants 
on some measure space 
. (Here we will follow the common practice if identifying functions that agree almost everywhere; in particular, we will be content to work with functions that are undefined on a set of measure zero. Also, while we work here with complex-valued functions throughout, most of the discussion here is also valid for real-valued or vector-valued functions.) Informally speaking, to measure how “big” such a function is, there are two (imprecisely defined) basic statistics to be aware of:
is for 
in the “dominant” component of the support of 
and 
, this function has magnitude 
on the ball 
, which has volume 
(if we allow constants in the informal 
notation to depend on the dimension 
on 
, which is implicitly involved in the definition of the Hilbert transform 
to denote the 
(equal to 
when 
otherwise), then each component 
and width 
. Thus, while this function can be viewed as a superposition of components of various heights and widths, rather than a single such component. 
norms of a function 
as 
as 
denotes the 
with respect to the measure 
. Often we abbreviate 
as 
, 
, 
, or just 
as 
) when the missing arguments are clear from context. (For instance, when working with Euclidean spaces 
the Lebesgue 
-algebra, unless otherwise specified.) In terms of the width 
and height 
is equal to 
(where now 
is the indicator function of a measurable set 
for which the 
is discrete and 
, or even just 
.
. On a Euclidean space 
lies in 
(with a norm of 
) if and only if 
, while the function 
lies in 
. The function 
does not lie in any 
. Thus we see that control in 
, a 
, and a 
. 
obeys the scaling condition 
(here we adopt the usual conventions 
). In particular, if 
and 
, then 
. 
is the conjugate exponent to 
.
when 
(with the additional hypothesis that 
), but in practice the relation 
when 
is absolutely convergent almost everywhere, and obeys the above inequality (so in particular is in 
for any complex numbers 
, which in turn ultimately stems from the complex triangle inequality 
and concave nature of 
for 
. In particular, for a finite sum 
, another application of Hölder’s inequality gives the quasi-triangle inequality 
is not too large. 
in 
, and that 
, where 
. For this reason, the measure of the support of 
norm of 
power of the 
and 
of the functions, but also under the homogeneity 
of the underlying measure. This latter symmetry demonstrates why the condition 
is necessary. (The first two symmetries demonstrate why 
.)
is equivalent to the scaling symmetry 
for 
, as the Jacobian of this map is 
. But the point is that by manipulating the measure directly, one still enjoys this symmetry even when no scaling operation is present. 
have heights 
, 
, 
and widths 
, 
, 
respectively. Then one expects the heights to be related by the formula 
concentrates in ought to be a subset of the region that 
by 
equal infinity as appropriate). Discuss how your conclusions align with the heuristic analysis presented above. 
, determine the cases for which 
? 
with respect to a measure 
for some suitable exponents 
.) 
twice with respect to 
and show that this is non-negative (take 
to equal one. Thus the task is now to show that if 
, then 
for all 
between 
, or more precisely the estimate 
. We split 
, then 
, and when 
, then 
. Thus we end up with 
by its 
power 
using the product measure construction, and similarly replace 
, defined by 
instead of 
and we obtain 
, which is of at most double-exponential growth, or more precisely 
for some 
. Suppose that we have the bounds 
when 
and 
when 
. Then we have 
for all 
in the strip. 
shows. Note in this hypothesis that we allow the implied constants in the asymptotic notation to depend on 
appropriately). So we may normalise 
. Similarly, the hypotheses and conclusion of the lemma are invariant under the operation of multiplying 
for some real 
. Using this, one can also normalise 
. So now 
the complex-analytic function 
.
by ![{f(z) \exp(\varepsilon e^{i [(\pi-\varepsilon)z + \varepsilon/2]})}](https://s0.wp.com/latex.php?latex=%7Bf%28z%29+%5Cexp%28%5Cvarepsilon+e%5E%7Bi+%5B%28%5Cpi-%5Cvarepsilon%29z+%2B+%5Cvarepsilon%2F2%5D%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002)
; a little complex arithmetic shows that this converts the almost double-exponentially growing function 
to conclude the claim. 
on the sides of the strip. Show that it obeys the polynomial bound 
and 
, and hence (by a trivially rescaled version of the three lines lemma) bounded by 
, which gives the claim for simple functions. The claim for more general functions (dropping the qualitative assumption of simpleness and finite measure support) then follows by a standard limiting argument (using for instance monotone convergence) which we leave as an exercise.
, one of these extremes is not necessary. If the measure space is finite in the sense that 
(thus prohibiting functions from being arbitrarily broad), then higher 
, and the general case then follows by convexity. The bound 
for 
, then we see that higher 
one can refine 
for all sets 
for 
. For 
norms are comparable to some extent, but the comparability gets worse as 
be the 
for 
, then we have granularity with constant 
, and now lower 
exponent), while if 
sums with 
we have 
and any 
. 
of a function 
, we obtain 
.
to be the space of all functions with finite 
. Indeed one can think of a weak 
. 
for 
by 
norm. We shall abbreviate 
by 
, 
, or even 
when there is no chance of confusion.
. The most important values of 
, 
, and 
; the other cases essentially never occur in applications. 
is inconsequential, but is traditional in order to maintain compatibility with the strong 
is of course multiplicative Haar measure on 
. One can interpret the equivalence of (i)-(iii) below by making the change of variables 
, so that the Haar measure just becomes Lebesgue measure in 
(modulo an inessential constant) and then passing from continuous 
is a monotone non-increasing function, show that 
norm of 
for any 
and 
almost everywhere and 
. (Thus 
.)
almost everywhere on 
. (Thus 
.)
within a definition; if multiple quasi-step functions appear in an argument, the question then arises as to whether the implied constants are uniform. In our applications, the implied constants here are true absolute constants (like 
) so this will not be an issue. 
we see that a non-negative sub-step function 
where 
is an actual step function of height 
, while a quasi-step function has 
– almost exactly like actual step functions. In the converse direction, it turns out that every 
sum of “very different” 
, and let 
. Then the following are equivalent up to changes in the implied constants:
. 
where each 
is a quasi-step function of height 
and some width 
, with the 
denotes the variable that the 
, with 
where each 
and some height 
, with the 
non-increasing in 
, the bounds 
on the support of 
, where 
and 
. But if 
of quasi-step functions of height 
, and either the heights or the widths are sufficiently variable in 
. 
and 
are trivial. To see that (i) implies (ii), set 
and 
. (This is the “vertically dyadic layer cake decomposition”.) The only thing that requires nontrivial verification is ![\displaystyle 2^m W_m^{1/p} \lesssim_{p,q} \| \lambda \mu( \{ |f| \geq \lambda \} )^{1/p} \|_{L^q([2^{m-2}, 2^{m-1}],\frac{d\lambda}{\lambda})}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2%5Em+W_m%5E%7B1%2Fp%7D+%5Clesssim_%7Bp%2Cq%7D+%5C%7C+%5Clambda+%5Cmu%28+%5C%7B+%7Cf%7C+%5Cgeq+%5Clambda+%5C%7D+%29%5E%7B1%2Fp%7D+%5C%7C_%7BL%5Eq%28%5B2%5E%7Bm-2%7D%2C+2%5E%7Bm-1%7D%5D%2C%5Cfrac%7Bd%5Clambda%7D%7B%5Clambda%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002)
(this comes from the hypothesis that 
![\displaystyle \begin{array}{rl} H_n 2^{n/p} &= (H_n^q 2^{nq/p})^{1/q} \\ &= (\sum_{k=0}^\infty (H_{n+k}^q - H_{n+k+1}^q) 2^{nq/p} )^{1/q} \\ &\lesssim_{p,q} (\sum_{k=0}^\infty 2^{-kq/p} \| \lambda 2^{(n+k)/p} \|_{L^q([H_{n+k+1}, H_{n+k}],\frac{d\lambda}{\lambda})}^q)^{1/q}\\ &\lesssim_{p,q} (\sum_{k=0}^\infty 2^{-kq/p} \| \lambda \mu( \{ |f| \geq \lambda \} )^{1/p} \|_{L^q([H_{n+k+1}, H_{n+k}],\frac{d\lambda}{\lambda})}^q)^{1/q} \end{array}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cbegin%7Barray%7D%7Brl%7D++H_n+2%5E%7Bn%2Fp%7D+%26%3D+%28H_n%5Eq+2%5E%7Bnq%2Fp%7D%29%5E%7B1%2Fq%7D+%5C%5C+%26%3D+%28%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%28H_%7Bn%2Bk%7D%5Eq+-+H_%7Bn%2Bk%2B1%7D%5Eq%29+2%5E%7Bnq%2Fp%7D+%29%5E%7B1%2Fq%7D+%5C%5C+%26%5Clesssim_%7Bp%2Cq%7D+%28%5Csum_%7Bk%3D0%7D%5E%5Cinfty+2%5E%7B-kq%2Fp%7D+%5C%7C+%5Clambda+2%5E%7B%28n%2Bk%29%2Fp%7D+%5C%7C_%7BL%5Eq%28%5BH_%7Bn%2Bk%2B1%7D%2C+H_%7Bn%2Bk%7D%5D%2C%5Cfrac%7Bd%5Clambda%7D%7B%5Clambda%7D%29%7D%5Eq%29%5E%7B1%2Fq%7D%5C%5C+%26%5Clesssim_%7Bp%2Cq%7D+%28%5Csum_%7Bk%3D0%7D%5E%5Cinfty+2%5E%7B-kq%2Fp%7D+%5C%7C+%5Clambda+%5Cmu%28+%5C%7B+%7Cf%7C+%5Cgeq+%5Clambda+%5C%7D+%29%5E%7B1%2Fp%7D+%5C%7C_%7BL%5Eq%28%5BH_%7Bn%2Bk%2B1%7D%2C+H_%7Bn%2Bk%7D%5D%2C%5Cfrac%7Bd%5Clambda%7D%7B%5Clambda%7D%29%7D%5Eq%29%5E%7B1%2Fq%7D+%5Cend%7Barray%7D+&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \| \lambda \mu( \{ |f| \geq \lambda \} )^{1/p} \|_{L^q((2^m,2^{m+1}],\frac{d\lambda}{\lambda})} \lesssim_{p,q} 2^m (\sum_{k=0}^\infty \mu(E_{m+k}))^{1/p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+%5Clambda+%5Cmu%28+%5C%7B+%7Cf%7C+%5Cgeq+%5Clambda+%5C%7D+%29%5E%7B1%2Fp%7D+%5C%7C_%7BL%5Eq%28%282%5Em%2C2%5E%7Bm%2B1%7D%5D%2C%5Cfrac%7Bd%5Clambda%7D%7B%5Clambda%7D%29%7D+%5Clesssim_%7Bp%2Cq%7D+2%5Em+%28%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5Cmu%28E_%7Bm%2Bk%7D%29%29%5E%7B1%2Fp%7D&bg=ffffff&fg=000000&s=0&c=20201002)
power we rewrite as 
we have 
are the modified heights 
for some 
and hence 
. The shifting trick and triangle inequality argument used previously shows that 
such that 
whenever 
is non-zero). Then the above theorem shows that two different Lorentz norms 
, 
with the same primary exponent 
. Similarly if the broadest width and narrowest width of a function differs by 
is equal to 
. Note also that for quasi-step functions, the 
; this can be seen for instance by using the equivalence of (i) and (iii). Another easy consequence is that the simple functions are dense in 
is a quasinorm on some function space 
, thus 
. Let 
. Show that 
, then 
. Then iterate this carefully to show that if 
and 
, then 
.) 
for 
obeys an exponential decay bound of the form 
and all 
. Show that 
, show that 
and 
obey 
and 
then 
, and drop the dependence of the implied constants on 
for brevity. By the equivalence of (i) and (v) in Theorem 
and 
where 
and 
component. Here we observe that 
has measure at most 
.
. Then for every 
, we have 
can be dropped if one instead assumes 
.
-normalised functions, and the worst inner product becomes the 
. This is fully consistent with the fact that the 
for some 
, then the following two statements are equivalent (up to changes in the implied constants):
. 
for all sets 
on 
). However, one does have a substitute:
. 
of 
such that 
on 
be functions with 
. Show that 
quasinorm only fails to be a norm “by a logarithm”. Show with an example that the 
, 
. In particular, weak 
, for instance simple functions with finite measure support.
part of this theorem, we simply estimate 
part, we normalise 
. It then suffices by homogeneity to find 
and 
.
. We will just give the proof in the case 
is trickier, and a partial argument is given in the exercises. By the equivalence of (i) and (ii) in Theorem 
where 
with disjoint supports such that the sequence 
has an 
. Now take 
when 
. Then (because of the disjoint supports) 
and so 
is the support of 
.
is necessarily a power of 
, and so the 
increase geometrically. It then suffices to show that 
as 
) and the 
we have 
are restricted to be non-negative can be deduced from the above result and a monotone convergence argument (representing 
is said to be 
with 
, there exists a measurable subset 
such that 
.
, then there exists a measurable subset 
of 
. (You may find it convenient to use Zorn’s lemma.) 
vanishes, or 
(not just bounded by 
and 
, which are also rearrangement-invariant and monotone, and which are occasionally useful. (There is a common generalisation of both, the Lorentz-Orlicz spaces, but these occur very rarely in applications.)
be a function (with some additional properties to be selected shortly) and ask if we can find a norm 
which obeys the property 
, then we need 
be increasing. Also to deal with the zero norm case one typically requires 
.
needs to be convex. Looking at 
.
. The space 
is not really an Orlicz space, but can be viewed the limiting case where 
is infinite for 
and zero for 
(or more informally, 
). Aside from the Lebesgue spaces, the most common Orlicz spaces which appear are
, defined as the Orlicz space with 
; 
, defined as the Orlicz space with 
; 
, defined as the Orlicz space with 
.
are comparable then their Orlicz norms are comparable also; a little more generally, if 
, then 
. It is the behaviour of 
. We can then define the Young dual 
of 
to be the smallest function for which one has the inequality 
is 
. Show also that the Young dual of 
takes the form 
for 
. What does the Young duals of 
and 
look like? 
is also increasing, convex, and super-linear, and so the Orlicz norm 
makes sense. From 
and 
.
. Show that the following are equivalent (up to changes in the implied constants):
. 
for all 
for all 
, together with the obvious bounds 
for integer 
. Now we shall consider operators 
; the study of such operators is in fact a major focus of harmonic analysis. Ultimately we want to extend 
and 
. But it is also worth considering the more general sublinear operators, in which 
are a collection (possibly countably or uncountably infinite, though in the latter case one has to take some care in ensuring measurability) of linear or sublinear operators. The third most important example is a square function such as 
of operators indexed by some parameter 
, and take 
to be the norm in the 
in some suitable norm. But the above three examples of linear operators, maximal operators, and square functions already cover the vast majority of applications.
be exponents, and let 
(or simply type 
are fixed, strong-type implies weak-type and restricted strong-type, either of which imply restricted weak-type. In most applications, it is the strong-type bounds which are desired; however, we shall see in this section that the real interpolation method allows us to deduce strong-type bounds from weak-type, or even restricted weak-type bounds, as long as the strong-type bounds are an interpolant between the restricted weak-type bounds. This can be a very useful strategy, because (as we shall see in next week’s notes) weak-type or restricted weak-type bounds are easier to prove than strong-type estimate.
are simple functions with finite measure support. This is for instance the case if 
; thus in practice this assumption is easily satisfied. We observe that this form is non-negative, homogeneous and sublinear in both 
turns out to be a convenient way to understand the various types of 
for 
and 
of finite measure. Now suppose that 
, we can relax things further to restricted weak-type:
, 
, and 
, 
and 
be such that 
with constant 
(in the sense of 
. Then 
with constant 
for 
, where 
. 
and 
, then 
for all 
is inconvenient for many applications, since weak 
with 
such that 
in Corollary 
. 
and 
and 
. Then for any 
and 
with 
we have 
were defined in 
, then 
. 
. Observe that the statement and conclusion of the theorem have several homogeneity symmetries. The most obvious one is that we can multiply 
); similarly we may multiply 
by 
). Using these symmetries we may normalise 
(and hence 
for all 
). Our task is then to show that 
and 
a bit closer to 
of finite measure and 
and 
respectively. We can of course pick the better of the two estimates, leading to 
with 
sub-step functions of width 
respectively, with the height bounds 
are the sequences 
are non-zero by Remark 
, 
, and reduce to showing that 
, we can write the left-hand side as 
and some 
depending only on 
. We substitute 
(thus 
) and estimate this by 
uniformly in 
and recall that the 
norm will be dominated by the 
norm for 
, the last claim of the theorem follows. 
where the operator 
, let 
be measure spaces, and for 
let 
for some 
with the Loomis-Whitney inequality 
and 
is the canonical projection from 
. From this, deduce the weak isoperimetric inequality 
, where 
is the Lebesgue measure of 
is the 
-dimensional Hausdorff measure of the boundary of 
be such that 
. Show that 
be such that 
. Show that almost every 
is contained in only finitely many of the 
. 
means “greater than”; and 
, “greater than or equal to.” Don’t translate 
and 
.
. Evolve it under 
…”).
defines a regime, not a limit. The conditions 
and 
define limits and are inequivalent to 
Then Y.”
then Y.
. Substituting into Eq. (1) yields [equation].
, 
denote the measurement outcome.
denote the Pauli-
operator of qubit 
:
, encoding directionality.
-axis
, which means 
to physicists. Use 
instead, unless you’re writing for engineers (who denote 
(\ell) instead.
, –, 
, or 
. Find the symbol immediately rightward of the initial = sign. Begin the new line directly below that symbol.
in the example under (A). Put the corresponding 
means “approximately equals”; and ~, “scales as.” Approximations convey more information than scaling relations do.
For example, if 
is odd, then 
and 
fit the template.
isn’t a matrix; a matrix represents an operator in terms of a particular basis. Therefore, no equals sign should interrelate an 
of real numbers
and 
where 
:
has a 10-dimensional representation, the ‘vector’ representation 
, and also two 16-dimensional representations, the ‘left and right-handed spinor’ representations 
There’s an isomorphism of representations
but we might as well work with 
Here 
means the kth exterior power. For any vector space 
we have
for larger 
In fact this isomorphism is part of a pattern! The next one involves the vector and left-handed spinor representations of 
But it’s this:
while 14 is the dimension of 
is a subgroup of 
because 
were isomorphic to the 6th exterior power of the adjoint representation of 
with
to be a perfect square!
is an integer when 
is, so we need to find integer solutions of
are the Fibonacci numbers 0, 1, 1, 2, 3, … and 
are the 
with 
Apparently this happens iff 
I won’t think about this now… but this is the last hard step.
), electrons (
) or gluons and find the best info experiments have provided on properties like their mass and charge. It’s a great place to get an authoritative number for any particle physics property you’re interested in.
case had been proved by Euler over 200 years earlier, I decided that that would be a good place to start: once I had sorted that out, I was optimistic that I would be ready to tackle the general case.
had no solution. Rather, I just worked out some successive differences and stared at them, trying to get some idea of why none of them was a perfect cube. (I should be clear that this story is a reconstruction of what I think probably happened given the few memory traces that remain half a century later rather than a completely reliable account.) At some point, I had the idea of taking the difference sequence of the difference sequence, and discovered that it formed an arithmetic progression. That felt like progress, so I investigated difference sequences a bit more and discovered, purely empirically, the rule that if you start with 
.
_28_(cropped)).jpg)
Going up to 
gives the room to include one generation of fermions.
Bokor, Boyle and I tried using this to get three generations of fermions. There are things that make this tempting: not just the fact that 
is bigger, but the fact that the Jordan pair you get from it has a kind of three-fold symmetry. But we couldn’t get it to work.
into the Lie algebra 
that fits between them:
of dimension 
acts on 
which is revealed when we put the Standard Model Lie algebra 
for the space of column vectors with two bioctonion entries.![[x,y,z]=\frac{1}{2}(x(y^{\dagger}z)+z(y^{\dagger}x))](https://s0.wp.com/latex.php?latex=%5Bx%2Cy%2Cz%5D%3D%5Cfrac%7B1%7D%7B2%7D%28x%28y%5E%7B%5Cdagger%7Dz%29%2Bz%28y%5E%7B%5Cdagger%7Dx%29%29+&bg=ffffff&fg=333333&s=0&c=20201002)
-graded real Lie algebra
The Lie algebra 
consists of all linear maps from 
to itself that are of this form:![x \mapsto [a,b,x] - [b,a,x]](https://s0.wp.com/latex.php?latex=x+%5Cmapsto+%5Ba%2Cb%2Cx%5D+-+%5Bb%2Ca%2Cx%5D+&bg=ffffff&fg=333333&s=0&c=20201002)
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 
and a Lie subalgebra 
whose Lie algebra is 
whose Lie algebra is 
is a nice kind of manifold called a 
Our original Jordan triple, 
is then the tangent space of that point! So, ![\mathfrak{e}_6 = \big[\mathfrak{so}(10) \textstyle{\oplus} \mathfrak{u}(1)\big] \textstyle{\oplus} \mathbb{O}_\mathbb{C}^2](https://s0.wp.com/latex.php?latex=%5Cmathfrak%7Be%7D_6+%3D+%5Cbig%5B%5Cmathfrak%7Bso%7D%2810%29+%5Ctextstyle%7B%5Coplus%7D+%5Cmathfrak%7Bu%7D%281%29%5Cbig%5D+%5Ctextstyle%7B%5Coplus%7D+%5Cmathbb%7BO%7D_%5Cmathbb%7BC%7D%5E2+&bg=ffffff&fg=333333&s=0&c=20201002)
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.
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.
In a Jordan triple 
their role is played by tripotents: elements 
with![[e,e,e] = e](https://s0.wp.com/latex.php?latex=%5Be%2Ce%2Ce%5D+%3D+e+&bg=ffffff&fg=333333&s=0&c=20201002)
![w \mapsto [e,e,w]](https://s0.wp.com/latex.php?latex=w+%5Cmapsto+%5Be%2Ce%2Cw%5D&bg=ffffff&fg=333333&s=0&c=20201002)
has eigenvalues 0, 1/2, and 1, so 
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.![[-,-,-]](https://s0.wp.com/latex.php?latex=%5B-%2C-%2C-%5D&bg=ffffff&fg=333333&s=0&c=20201002)
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:![[\alpha e, \alpha e, \alpha e] = \alpha \overline{\alpha} \alpha e = \alpha e](https://s0.wp.com/latex.php?latex=%5B%5Calpha+e%2C+%5Calpha+e%2C+%5Calpha+e%5D+%3D+%5Calpha+%5Coverline%7B%5Calpha%7D+%5Calpha+e+%3D+%5Calpha+e&bg=ffffff&fg=333333&s=0&c=20201002)
(even part in brackets) 
![\mathfrak{e}_6 = [\mathfrak{so}(10) \oplus \mathfrak{u}(1)] \oplus \mathbb{O}_\mathbb{C}^2](https://s0.wp.com/latex.php?latex=%5Cmathfrak%7Be%7D_6+%3D+%5B%5Cmathfrak%7Bso%7D%2810%29+%5Coplus+%5Cmathfrak%7Bu%7D%281%29%5D+%5Coplus+%5Cmathbb%7BO%7D_%5Cmathbb%7BC%7D%5E2&bg=ffffff&fg=333333&s=0&c=20201002)
![\mathfrak{so}(10) = [\mathfrak{su}(5) \oplus \mathfrak{u}(1)] \oplus \mathfrak{a}_5(\mathbb{C})](https://s0.wp.com/latex.php?latex=%5Cmathfrak%7Bso%7D%2810%29+%3D+%5B%5Cmathfrak%7Bsu%7D%285%29+%5Coplus+%5Cmathfrak%7Bu%7D%281%29%5D+%5Coplus+%5Cmathfrak%7Ba%7D_5%28%5Cmathbb%7BC%7D%29&bg=ffffff&fg=333333&s=0&c=20201002)
![\mathfrak{su}(5) = [\mathfrak{g}_{\mathrm{SM}}] \oplus \mathrm{M}_{3,2}(\mathbb{C})](https://s0.wp.com/latex.php?latex=%5Cmathfrak%7Bsu%7D%285%29+%3D+%5B%5Cmathfrak%7Bg%7D_%7B%5Cmathrm%7BSM%7D%7D%5D+%5Coplus+%5Cmathrm%7BM%7D_%7B3%2C2%7D%28%5Cmathbb%7BC%7D%29&bg=ffffff&fg=333333&s=0&c=20201002)
is the Jordan triple of antisymmetric 
complex matrices, 
is the Jordan triple of 
complex matrices, 
and
Descend the table twice:
which has real inner automorphism group 
Its Peirce 
-space is 
with real inner automorphism group 
(colinear with 
so living in 
). Its Peirce 
with real inner automorphism group exactly 
and 
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
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.
we have projections 
and 
onto its three eigenspaces: its so-called Peirce projectors. Since we get the Standard Model structure using two minimal tripotents 
, 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!
Six irreducible pieces, six particle types.
Those six match up one-to-one with the particle types:
1, +1/3)
and 
—all vanish, which is why we land on six pieces and not nine.
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 
you get the same result by choosing three mutually colinear tripotents instead of two—but for that, read our paper!
(where there are 
there exists an integer 
such that 
be two 
be a linear map from 
–isomorphism if there exists 
such that 
for every 
. By rescaling we can always take 
to be 1, in which case we have that 
for every 
, then this tells us that 
between 
from 
. (The idea of the proof is as follows: pick inside the unit ball of 
for any two 
and 
. The identity map between these two spaces is an 
Hadamard matrix, then the corresponding linear map is a ![p,q\in[1,\infty]](https://s0.wp.com/latex.php?latex=p%2Cq%5Cin%5B1%2C%5Cinfty%5D&bg=ffffff&fg=333333&s=0&c=20201002)
the Banach-Mazur distance between 
and 
is 
. It is also easy to show that 
, so 
-spaces hardly improve on the easy lower bound, and do not improve on it at all in dimensions 
,
for the set of all 
.) In words, there is a positive constant 
such that for every positive integer 
.
for the set of primes. And yet, Vinogradov’s result is unquestionably a theorem, while Gluskin’s result is unquestionably a counterexample, or at least an example.
, whereas for Gluskin’s theorem the statement to be proved is
and 
,
.
can be converted into an existentially quantifed statement 
. (For this to be an equivalence one needs the axiom of choice, but it is certainly a sufficient condition.) This is not just a piece of logical trickery, but it often reflects quite accurately how we think about some problems. For instance, it is more natural to think of Gluskin’s example as a recipe for constructing (or at least proving the existence of) a pair of suitable normed spaces for any given dimension 
to pairs of normed spaces by giving its value at each 
does not belong to the convex hull of a certain compact set 
and a 
such that 
and 
for every 
. In both these cases it feels natural to regard the result as a theorem that is proved via an existential statement, perhaps because it is the theorem that is ultimately what interests us. But using “what interests us” as a criterion to determine what counts as a counterexample seems a little vague, and is a difficult criterion to use if we want to explain convincingly why AI should be good at finding counterexamples.
, then I may well look for a property 
that follows from 
, it is often the case that the main question in our minds is less, “Why is 
is a counterexample to the universally quantified statement 
. However, we do not describe all existential statements as counterexamples. For example, if I were to say, “The 
are all separable, as is 
, but 
is not separable,” I would not describe the second part of that assertion as a counterexample to the claim that all Banach spaces are separable. Rather, I would present it as probably the most basic example of a non-separable space. The important point seems to be that there was no particular reason to think that all Banach spaces would be separable, and finding an example of a non-separable space is not very difficult.
of them, but that bound can be improved if instead you observe that a complete 5-partite graph can be written as a union of two triangle-free subgraphs, and therefore it is possible to write the complete graph as a union of 
triangle-free graphs. It is then tempting to try to do better, with triangle-free graphs that are less dense but that make up for it with unbounded chromatic number — a necessary condition if one wishes to use a sublogarithmic number of graphs, which is equivalent to showing a superexponential lower bound for 
. All this is to say that when I worked on the problem, my efforts were concentrated on what turned out to be the right direction, so for me OpenAI found an example of what I (weakly) expected, rather than a counterexample.
that is small enough for later arguments to work. But when we search for an example of an object 
), it is often not a good strategy to specify 
. If we are lucky we can describe in a nice way a very general class of objects 
and show that 
satisfies 
of a certain type. The problem is then reduced to finding 
, it can be surprisingly helpful to start by attempting to prove the statement 
. The reason this can be helpful is that using our standard methods of attempting to prove something, we may end up identifying a key lemma that would suffice: that is, we may find an intermediate property 
in a non-trivial way and thus reduce the problem 
. Turning things round again, it may well then be that finding a counterexample to 
not because we think it has a chance of working (if we did, then we would be using the first strategy), but because we hope that if 
that does not have that defect. Again, this strategy can either be iterated or combined with one or more of the other strategies.
, each of which is, individually, quite easy to satisfy. In such situations, we often “build” 
.
is easily checked to be 
. However, the map 
, 
and 
are homogeneous with respect to the grading where 
, 
and 
; their degrees are 
, 
and 
. I’m not sure what to make of this, but it surely matters.
, you get a cubic relation in the remaining variable. Here they are
cubic. Then the discriminants of the three cubics are 
, 
, 
where 
have no common zeroes. Roughly speaking, our map should have special behavior over the loci 
, 
, 
and 
. The fact that $p$, $q$ and $r$ each appear cubed means that the variables 
and 
should have three fold branching over the loci