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added to Jones polynomial pointers to Edward Witten’s slides for his recent talks at the Clay meeting in Oxford.
Thanks to Bruce Bartlett for providing them!!
He seems to have a strikingly different style to your own Hilbertian-Hegelian one
Dieses Ergebnis scheint uns fast auf den Hegelschen Standpunkt zu führen, wonach aus blossen Begriffen alle Beschaffenheit der Natur rein logisch deduziert werden kann.
I seem to recall a discussion we started once about when the concrete particular disguised within it the general abstract. Take a couple of themes that always seem to be at hand for Witten: analytic continuation and Morse theory. Is there something general abstract going on in his use of them?
There is group theory and then there are the exceptional groups and the most exceptional, the monster. Similarly, there is local prequantum field theory, and then there are the exceptional such culminating in the M-onster, the brane bouquet. This is the exceptional concrete particular that Witten is and has been exploring. It’s a huge M-onster, which sometimes makes it hard to see the wood for the trees. But it is one single exceptional thing: ( 19:40: “I promise you that by the end of the talk we have just one big theory.”)
Please don’t mistake as “my style” what may be my preoccupation at some point of time. To me the general abstract and the concrete particular are two sides of nature and I am equally interested in both, but there are only so many hours in a day and some priorities to be chosen.
By the way, also Witten develops general theory when he needs it and it is missing. Last year he had a foundational discussion of the theory of supergeometry here:
Ok, sorry names are a distraction. What I was wondering about is the difference between the abstract general being worked out in a particular case and what is just particular. So we may develop a theory in a certain logic and then interpret it in any category which supports that logic. But there may be special features of particular cases of those categories which are not generally provable.
But isn’t there a more subtle relation between the general and the particular, which has serious mathematicians reaching for Eastern mystic language? For instance, Michael Harris spoke at a conference I co-organised on Avatars: mathematical conjectures in the light of (re)incarnation. Maybe these are just places where the general hasn’t been made explicit yet, but the dream is there, as with motives.
Back on Morse theory, without really knowing why, I had a hunch that it should appear more generically. Coming to think of it, I think this was an impression gained way back when we discussed at the cafe, singularities and categories with duals. It felt like singularities were built very deep into the structure.
What is that statement by Harris exactly? I am not sure if I found the right link.
By the way that construction of categories with duals as fundamental categories of stratified manifolds seems (seems! I haven’t seen much detail yet) to be what Ayala and Rozenblyum use to defined $(\infty,n)$-categories with all adjoints here.
But I think Morse theory is just a tool here. On making Morse theory more of an abstract theory itself, I’d think that Witten’s very article Supersymmetry and Morse theory gives an answer: Morse theory is naturally interpreted as the study of the supersymmetric vacua in supersymmetric quantum mechanics (which in turn exists on more general abstract grounds).
It was just part of a talk whose video is linked to on Harris’s page. I’d need to see it again to know what he was saying precisely. But I think it’s not far removed from what Weil spoke about as the Rosetta stone. We don’t have a general theory for which we then turn the dial to different settings - Riemann surfaces, curves over finite field, arithmetic fields.
That Cafe discussion on fundamental categories with duals of statified spaces was the seed for a paper – Transversal homotopy theory – by Jon Woolf.
By the way, concerning this use of the word “theory”, I wanted to comment on something (which may be way off tangent to what you are after here, but maybe it does connect to what Harris may have said):
Some farily long time back we had a long discussion where I had claimed something outrageous sounding like that all mathematics is controled by category theory, and then people disagreed by pointing out that nothing general abstract helps to find a “theory of differential equations” or the like.
it just strikes me that in saying so, there is a different meaning given to “theory” than that given at theory.
With the term as defined in that entry (which is the standard formal definition), we do of course have a theory of differential equations, etc. (That’s precisely what Lawvere set up when introducing synthetic differential geometry, after all.) That notion of theory of course is a way to get the concrete general.
What people mean when saying “there is no general theory of partial differential equations” is that it is hard to classify the concrete particulars of this theory in a useful way. Every PDE seems to be its own topic.
But nevertheless, using category theory etc. we do have a “theory of PDEs”, in the sense of theory.
Not sure if it is useful that I say this here. This is effectively a hugely belated reply to something somebody said somewhere, none of which I remember precisely. ;-)
I’ve just heard Kobi Kremnitzer on geometry in closed symmetric monoidal categories (with limits/colimts of some kind) advocating the relative algebraic geometry approach.
I guess there’s a feeling of some kind of difference between the general theory taking you really quite a long way, and like with PDE where very rapidly you’re thrown into your own special world, requiring specially designed tools.
I guess there’s a feeling of some kind of difference between the general theory taking you really quite a long way, and like with PDE where very rapidly you’re thrown into your own special world, requiring specially designed tools.
Yes, sure. What I was trying to say is: I find it unfortunate to use the word “theory” as in “there does not seem to be a general theory of PDEs”. What people should speak about is something like the “phenomenology” of PDEs etc. In the technical sense of “theory” of course there is a theory of PDEs and I think it is wrong to change that technical meaning of theory.
That PDEs do not have a good classification, hence that they have a very rich phenomenology is part of what makes nature interesting. It is like in biology, where there is no theory of birds, just a phenomenology of birds.
In fact I think of these two examples as being more closely related than it may seem: the world is rich and interesting because general abstract theory tends to have concrete general phenomenology which is rich, that’s how structure enters the world.
Instead of saying that “general abstract theory does not help with understanding PDEs” I think of this the other way around: “out of general abstract theory flows the richness of PDEs and thus of our physical world”.
No physicist would say “there is no theory of the universe” just because his fundamental theory only serves to directly explain fundamental particle interactions and does not help much with classifying bird species. The physicist nevertheless knows that his fundamental theory explains the existence of birds, while of course it is not a phenomenology of birds. And this is good. If all of nature followed on large scales the highly symmetric abstract laws on microscopic scales, life would be very dull.
Mathematicians should have the same attitude, and should not say that “there is no theory of PDEs”. They should say that there is a theory of PDEs which gives rise to a fascinating unchartably rich phenomenology of PDEs.
I think it is a success (scientifically) that immense unchartable richness flows out of general abstract laws. This is not something to lament, but to marvel at. This is how the richness of the world is created out of the general abstract.
And Anders Kock of course started developing the theory of at least first-order DEs in SDG e.g. the heat equation (which of course is a hugely important concrete particular, being used in formulations of important theorems in geometry).
Re Urs#6,
Morse theory is naturally interpreted as the study of the supersymmetric vacua in supersymmetric quantum mechanics (which in turn exists on more general abstract grounds).
So when people use Morse theory to do all sorts of things in standard algebraic topology, such as mentioned in this MO question – Poincare duality, Cup product, Kunneth isomorphism, Leray-Serre spectral sequence, Alexander duality, Steenrod operations, Massey products – that’s all interpretable as part of supersymmetry?
It’s all naturally interpretable in terms of supersymmetric quantum mechanics, yes. That was Witten’s insight, worth about 1/4th of a Fields medal. (I have just added pointers to some reviews here).
Witten showed that if the Morse function is interpreted as a “superpotential” for a bispinorial quantum particle propagating on the given manifold, then the Morse equalities etc. are part of the characterization of the supersymmetric vacuum structure of the theory.
(And this is only the tip of a vast iceberg which also includes analyis of BPS states in supersymmetric quantum field theory and all the modern geometry that goes with this.)
Now of course “just” because it is interpretable this way need not mean that in every single application of Morse theory this interpretation gains something.
Compare to other interpretation of pure maths in physics. For instance in the orbit method we learn that the representation theory of compact Lie groups is equivalently the quantum mechanics of certain topological particles (1d Wilson line TQFT). Clearly not every single time that a representation of a compact Lie group is mentioned, will it be desireable to highlight this interpretation. But sometimes, such as in the analysis of the Jones polynomial (to bring us back on topic, full circle! :-) it is paramout to do so.
Supersymmetry to the aid of topology again – arXiv:1310.5383.
I wonder which way the ‘interpretation’ goes. Perhaps it is both ways. The Gauss-Bonnet theorem is a lot easier to understand than supersymmetry (at least for me) but results like that in the Berwick-Evans paper help interpret aspects of the latter in terms that are nearer the way of thought of ordinary differential geometers. I understand Morse theory from the point of view of Milnor’s book and also for its applications in computer visualisation, neither of which benefits much from a super symmetric viewpoint.
On a related idea, does the analysis of Morse functions in the paper by Kapranov and Saito, in which they relate them to syzygies, fit into some of these insights from math. physics?
Two comments:
First, supersymmetry just means: invariance under an action of the super-Poincaré group.
Second, not all of super-geometry involves supersymmetry.
The article mentioned above uses supergeometry (or 0-dimensional supersymmetry, if you want). This is based on the classical observation that the internal hom from the odd line to a (super-) manifold in supergeometry produces the supermanifold whose function algebra is the $\mathbb{Z}_2$-graded de Rham complex. This is effectively the same algebraic phenomenon that also governs and motivates synthetic differential geoemtry, only that in supergeometry the infinitesimals are in addition odd graded.
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