### Ubiquitous Duality

#### 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
## 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).