Ubiquitous Duality

January 11, 2007

Posted by David Corfield

I’m in one of those phases where everywhere I look I see the same thing. It’s Fourier duality and its cousins, a family which crops up here with amazing regularity. Back in August, John wrote:

So, amazingly enough, Fourier duality and the duality between syntax and semantics for algebraic theories are part of the same family of ideas.

With more work one could find a common generalization and prove a theorem which had both of these results as a special case. I don’t know if anyone has done this yet. If not, they should!

and told us something about Tannaka-Krein duality. (A good opportunity there for a contribution to Wikipedia.)

We had Urs telling us about Geometric Langlands as involving a form of categorified Fourier transform, which suggests that perhaps the whole Langlands program may likewise.

Then I quoted Michael Atiyah:

This replaces a space by its dual space, and in linear theories that duality is just the Fourier transform. But in non-linear theories, how to replace a Fourier transform is one of the big challenges. Large parts of mathematics are concerned with how to generalise dualities in nonlinear situations. Physicists seem to be able to do so in a remarkable way in their string theories and in M-theory…understanding those non-linear dualities does seem to be one of the big challenges of the next century as well.

We also have spoken about the Laplace transform as a twin of the Fourier transform, and their idempotent cousin the Legendre transform.

Elsewhere, I heard it said that work on arithmetic progressions of primes, by Tim Gowers and others, had something to do with a ‘quadratic’ Fourier analysis, mentioned in the May 18 entry here.

There must be some common framework, as John remarked. How much power does Brian Day’s construction pack?

Posted at January 11, 2007 1:43 PM UTC

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12 Comments & 1 Trackback

Re: Ubiquitous Duality

I have been thinking about a classification theory for a while in my own layman ways;
http://a-theory-of-nothingness.blogspot.com/
Fundamental categories seem to persist all the way to natural numbers. Factorial-Exponential duality for example. With multi-factorials, as defined by E.Barnes and others, factorials have a similar dual format to exponentials.
a^b
a!b
Where a!b means factorial of “a” to depth “b”. I think exponentials and factorials represent two independent class of fractals.

The two dualities seem to correspond;
{signal,FFT} — {factorial,exponential}

Life obeys a similar duality;
{body,mind}
Mind is essentially a compacting transformer of the outer reality.

Mediated dual-duality seems to be more common than mediated duality. Complex numbers obey this as well, {-i,+i,0,-1,+1}. Life have 5 senses and the universe have 5 forces (with Higgs).

Posted by: Akira Bergman on January 12, 2007 9:17 PM | Permalink | Reply to this

Re: Ubiquitous Duality

This is the Law of Fives all over again, and perfectly timed. (R.I.P., Robert Anton Wilson)

A lot of these things, especially the fives you list at the end, are there because you’re looking for them. Don’t worry, it just means you’re human. Humans are first and foremost pattern-matching machines. I don’t think it would be too wrong to say that humans think in PERL. The consequence is that we see patterns everywhere, even if they aren’t there.

The Law of Fives states that everything, everywhere, has something to do with the number five. The Strong Law of Fives says that the Law of Fives is more and more apparent the harder you look. Five is nicely convenient because we’ve got five fingers, five toes. We’ve identified five senses and leave out things like a kinesthetic sense of spatial orientation and motion to keep from going past five and breaking the nice pattern, just like you’ve collected what I think is a very odd set of complex numbers to count as in some way fundamental.

You see this sort of thing in spades in classical philosophy and metaphysics. Gematric methods (like some used in the Kabbalah) are entirely based around some mystical sense of language and number and numeral, which weren’t that far from each other at the time.

I think my advisor came up with the right balance of seriousness. He noted that the most common Gematric values found in the Hebrew litany of names for G-d were 26 and 10, and followed up by mentioning that if his mother thought he was getting religious she’d kill him. Patterns are fun, but meaning is a little more difficult to pin down.

Posted by: John Armstrong on January 13, 2007 12:21 AM | Permalink | Reply to this

Re: Ubiquitous Duality

I am not religious in the mainstream sense but I can also see some merits in the ancient wisdom. The nature is self similar and full of patterns. This enables us to understand, utilise and predict it. The patterns in our minds are a reflection of it. They are not arbitrary.

All small natural numbers play fundamental roles in nature. 5 fingers make 6 when the wrist is included. When the lost tail is included the human body also has a 6ness. Similarly balance is coupled with touch to form the center. Note that balance also seems missing like the tail. The fact that these things also appear in the religious culture does not diminish their value.

To see the 5ness in complex numbers, just consider the cartesian representation.

Nature imitates natural numbers in diminishing weight with size. But it likes some numbers more than others. And then it unfolds those symmetries like origami to imitate other numbers. A bit like black holes being at the boundary of reality and projecting outwards.

Posted by: Akira Bergman on January 13, 2007 2:30 AM | Permalink | Reply to this

Re: Ubiquitous Duality

Before we broaden this discussion out, let’s earn the right by making some progress on the problem set by John of finding a common framework for the dualities mentioned in the post. Otherwise, I’ll be for it when the other Café owners return from their travels.

Posted by: David Corfield on January 13, 2007 2:58 PM | Permalink | Reply to this

Re: Ubiquitous Duality

Well, as I understand it there are essentially two distinct senses of Fourier theory, the geometric and the representation-theoretical. Which flavor will go through I don’t know, but they are distinct away from the basic cases we all know.

I don’t know as much about the geometry, but in that case the dual space is the spectrum of the Laplacian on the space.

For representation theory, the dual space is the (unitary?) dual of the group: the space of all representations modulo equivalence. I’d lean towards saying that this is the right view for the problem, but I may well be biased.

Unfortunately, it’s been a while since I’ve looked at Fourier, and I’ve just woken up so the details are a bit hazy, and I’m blanking on the syntax/semantics duality Baez means. Hopefully someone can run with one flavor or the other and get a little farther.

Posted by: John Armstrong on January 13, 2007 4:34 PM | Permalink | Reply to this

Re: Ubiquitous Duality

Dear David,

I will be very very happy to see more details why the duality between syntax and semantics is related to Fourier duality.

Another mystery for me is why dualities are so important but we do not see, say, “trialities” which bring together three concepts rather than two. (Or even more than three…)

Posted by: gina on January 13, 2007 4:08 PM | Permalink | Reply to this

Re: Ubiquitous Duality

$MathML-enabled post (click for more details).$

What do we have to go on?:

Fourier duality takes locally compact Abelian groups such as $G = \mathbb{R}$ or $G = S^1$ and forms their duals $\hat{G} = Hom(G, U(1))$ . So $\hat{\mathbb{R}} = \mathbb{R}$ and $\hat{S^1} = \mathbb{Z}$ .

Then there is a duality between $L^2(G)$ and $L^2(\hat{G})$ .

In the syntax-sematics case, as TWF136 explains, take an algebraic theory such as that for groups, Th(Grp), so that Mod(Th(Grp), Set) = Grp. If $G$ is the object in Th(Grp) of which all other objects are products of copies of it, then there’s a functor from a model $F$ to the set $F(G)$ . The image of the left adjoint of this functor consists of free groups, and the opposite of that category is equivalent to Th(Grp).

In the latter case should we be thinking of the category of free groups as the $\widehat{Th(Grp)}$ , and then the isomorphism between functors from Th(Grp) to Set and contravariant functors from FreeGrp to Set, as the ‘Fourier’ transform?

Or is it that any contravariant functor from Grp to Set is already determined by one from FreeGrp, so we can have $\widehat{Th(Grp)}$ as Grp and still have a Fourier duality?

Then we’re looking at situations where there’s an equivalence between $A \to B$ and $(A \to B) \to B$ .

Posted by: David Corfield on January 13, 2007 4:50 PM | Permalink | Reply to this

Re: Ubiquitous Duality

“I’m in one of those phases where everywhere I look I see the same thing ”

probably a useful phase, David, if it is only a phase… and the other phase where everything look so much colorfully different is probably as useful if not more useful.

I got a very vague feeling of the analogy between Fourier transform and syntax/ semantic relation. Is this analogy really really useful?

And can you tell why just duality again and again and again, and not “triality” (or trinity)?

Posted by: gina on January 14, 2007 9:59 PM | Permalink | Reply to this

Re: Ubiquitous Duality

I got a very vague feeling of the analogy between Fourier transform and syntax/ semantic relation. Is this analogy really really useful?

Who knows? We’d only be able to tell if it led to a framework sufficiently general to allow a construction devised for one topic to be translated usefully to another.

In mathematics it seems better not to accept Wittgenstein’s point that when you get a network of linked entities, such as activitites designated by the term ‘a game’, that there’s unlikely to be a kind of generic concept of which they are all examples. While there is no quality shared by all games, when mathematical entities are analogous there usually is some generic construction operating behind the scenes.

The question is whether there’s enough power in a category theoretic formulation, or ‘generalized logic’ in Lawvere’s words, to locate sufficiently deep structural properties in two situations. As Lawvere himself said:

Of course the really deep results in a subject depend very much on the particularity of that subject and the results we offer here in the field of metric spaces, taken individually, will justly appear shallow to those with any experience. Indeed for me the surprising aspect was that methods originally devised to deal with quite different fields of algebra and geometry could yield any significant known theorems at all… But there are many particularities, for example the special role of quadratic metrics, which I do not see how could be a result of ‘generalized logic’. p. 143

But in the Author Commentary 29 years later he adds:

Thus, contrary to the apology in the introduction of the 1973 paper, it appears that the unique role of the Pythagorean tensor does indeed have expression strictly in terms of the enriched category structure.

I can’t help feeling that mathematical development sometimes resembles a greedy search. John Baez once wrote:

As we’ve discussed before, a lot of exquisite math can be found in the brownfields left behind by overly hasty research programs.

I wonder if this isn’t a more pervasive problem, and whether much more recycling of already made constructions is not possible in much of mathematics.

As for “trialities”, I’ll have to think about it.

Posted by: David Corfield on January 15, 2007 11:34 AM | Permalink | Reply to this

Re: Ubiquitous Duality

And can you tell why just duality again and again and again, and not “triality” (or trinity)?

It seems obvious to me that dualities should be more common than trialities, because 2 is smaller than 3. (Indeed, there is only one famous triality that I know of in mathematics.)

Since 1 is smaller than 2, singularities are even more common. (Imagine if David had written “Everywhere I look I see individual things.” Well, duh!)

Outside of pure mathematics, where we have more associations open to us, you can easily go higher than 2. Thus Akira’s example of a purely mathematical instance of the Law of Fives was rather strained, but examples from ordinary life (starting with Aristotle’s count of human senses) are easy to find.

On the other hand, to find a larger number like 23, even outside pure mathematics, you usually have to cheat (by performing ad-hoc arithmetic operations, thereby exponentially increasing the range of possible results, instead of simply counting).

Posted by: Toby Bartels on January 15, 2007 9:01 PM | Permalink | Reply to this

Re: Ubiquitous Duality

Well, I guess one could go further with this and define partialities… That is, translate partially mathematical structures into other mathematical structures…

This is easily seen with binary relations:
http://en.wikipedia.org/wiki/Partial_function
http://www.cs.bham.ac.uk/~mmk/cade98-partiality/cfp.html

and I don’t see why this could not be extended to n-ary relations…

I don’t know whether something like this makes sense in the context here. But if it does, I guess partialities would be much more common than dualities…

Christine

Posted by: Christine Dantas on January 15, 2007 10:00 PM | Permalink | Reply to this

Re: Ubiquitous Duality

$MathML-enabled post (click for more details).$

A thread such as this always risks lapsing into mysticism. I have removed one comment, and shall continue doing so, if the mathematical content is too small.

So, to return to the question of why we might expect some forms to appear more than others, the edition of This Week’s Finds which made the greatest impact on me when I was getting interested in $n$ -categories was Week 84. It explained how adjunctions and monoidal products could be seen as arising from the way lines could be arranged in the plane. Adjunctions:

This concept of “doubling back” or “backtracking” is a very simple and powerful one, so it’s not surprising that it is prevalent throughout mathematics and physics. It is an essentially 2-dimensional phenomenon (though it occurs in higher dimensions as well), so it can be understood most generally in the language of 2-categories.

Monoidal product:

Next I will talk about another 2-dimensional concept, the concept of “joining” or “merging”…This is probably even more powerful than the concept of “folding”: it shows up whenever we add numbers, multiply numbers, or in many other ways combine things.

As to whether anything is gained by allowing more than three lines to meet at a point, perhaps it’s worth taking a look at Arnold’s Toronto lectures, especially Lecture 1: From Hilbert’s Superposition Problem to Dynamical Systems:

If you have a function in two variables $z(a, b)$ and you put inside, say instead of $a$ , a function in two variables $a(u, v)$ and continue in this way - you can get a function in any number of variables…Hilbert asked…whether you can represent any function in three variables as a superposition of functions in two variables - i.e. whether functions in three variables really do exist.

You’ll see there that for continuous functions the answer hung on the existence of a certain universal tree.

After reading TWF 84, I remember wondering whether a new construction, as rich as adjunction, might be found by thinking of ways surfaces could behave in 3-space.

Posted by: David Corfield on January 16, 2007 9:24 AM | Permalink | Reply to this

Read the post More on Duality
Weblog: The n-Category Café
Excerpt: Continuing our earlier discussion about duality, it's worth noting a distinction that Lawvere and Rosebrugh introduce in chapter 7 of their Sets for Mathematics between 'formal' and 'concrete' duality. Formal duality concerns mere arrow reversal in th...
Tracked: January 19, 2007 10:49 AM

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January 11, 2007