Mathoverflow

The site mathoverflow is a forum populated by professional mathematicians in which posts are up voted if they are considered good answers to posted questions. In this post I will collect postings that I hope to follow up on in future posts and my own private study.

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Quantum physics and representation theory

Unfortunately Grothendieck didn’t examine, that we know of, the foundations of physics although some people suggest that he avoided the subject because he found the foundations to be too non-mathematical, or to put it more directly nonexistent.

Peter Woit’s blog not even wrong is a very interesting site that not only talks about mathematical physics but also briefly makes forays into pure mathematics. Here are some interesting recent posts:

  • George Mackey was (he died in 2006) a mathematician was a functional analyst in the intersection of  representation theory and quantum physics. Within this article is a reference to  paper on the Stone-John von Neumann theorem that, according to Woit

    is a theorem which essentially says that once you choose Planck’s constant, up to unitary equivalence there is only one possible representation of the Heisenberg commutation relations. This uniqueness theorem is what allows one to just define quantum theory in terms of the operator commutation relations, and not worry about which explicit construction of the representation of these operators on a Hilbert space one uses. The theorem is only true for a finite number of degrees of freedom, and thus doesn’t apply to quantum field theory, one reason why quantum field theory is a much more subtle business than quantum mechanics. Stone and von Neumann put their work in the context of representation theory of the Heisenberg group (actually due to Weyl) and this was of great interest to mathematicians since it was one of the first results about the representation theory of non-compact Lie groups.

    Another article written in memorium of Mackey’s passing is one written by S. Varadarjan titled George Mackey and his work on representation theory and foundations of physics.

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Deformations

The word deformation is in reference to the notion introduced initially by Kodaira and Spencer in their important work in the late 1950s. My primary interest is in Grothendieck’s reformation of this theory in the context of scheme theory as described in his fondements de geometrie algebrique.

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Homotopy

The notion of homotopy, as originally introduced by Henri Poincare, has morphed into a vast framework that encompasses much of modern mathematics. It is my hope to explore this notion in the context of modern algebraic geometry as envisaged by Grothendieck in his so called derivators, a theory that is expounded in a 2000 page letter that he wrote in early 1990s and which is being transcribed into LaTeX by George Maltsiniotis and his collaborators.

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Arithmetic Deformation theory

The title of this post refers primarily to the work of Shinichi Mochizuki, in particular, his claim for a proof of the ABC conjecture.  He is no quack but appears to be very reputable mathematician who is, in fact,  quite brilliant. Unfortunately the papers he’s posted with respect to the ABC conjecture appears to be buried in vast jungle that he refers to as “Inter-universal Teichmuller theory” or as I prefer to refer to it,  arithmetic deformation theory. Some mathematicians have described reading these papers as reading papers from the future or from outer space. 

 

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Crystalline cohomology

As with all articles in this blog, this is a template where I will endeavour to motivate the development of the subject in the title of this note. In particular, an attempt to create a historically annotated bibliography will be one of the essential goals. Presumably one of the aims of p-adic methods for studying varieties defined in characteristic p is that \ell-adic methods escape characteristic p. Apparently this is what was so stunning about Dwork’s proof of the rationality of zeta functions of varieties defined over a finite field of characteristic p using actual p-adic methods. I guess the belief that not having a p-adic study means one may be losing information that is vital–trying to answer this will be one my starting points.

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Geometric representation theory

In this posting I will document my attempts to understand how representation theory in the “classical” sense can be reduced to a subtopic of algebraic geometry as is asserted by  a number of survey articles and books, particularly  Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups by Victor Ginzburg which is summary of a book he wrote along with N. Chriss. I found the article because it is cited in a note by David Ben-Zvi.

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