NCG & Topological Quantum Field Theory

It’s humbling to realize how little you know.  As I mentioned in the last post I’ve become interested in Alain Connes both for his thoughts in the broad sense and also for noncommutative geometry.  While I’ve been enjoying “Triangle of Thoughts” I have come to realize that I am woefully underprepared for this subject matter.  I came across the writings of Masoud KhalKhali and in particular his “Beginner’s Guide to Noncommutative Geometry.”  He  also writes a “Very Basic Noncommutative Geometry” guide which is about a hundred pages  long and starts out with some heavy material.  It’s clear I need a much better understanding of category theory, a working understanding.  My friend Simon and I have decided to start chipping away at Saunder Maclane’s text “Categories for the Working Mathematician” at about a section or so a week.  Think of it as a New Year’s resolution: learn some category theory.

This was something we’d toyed with a few months ago when we read Barry Mazur’s piece “When is one thing equal to some other thing?” which has some really good writing about categories from both practical and philosophical viewpoints.  

The attractive aspect of Noncommutative Geometry is that it sheds light on quantum mechanics and physics as a whole.  Bummed that this would be delayed quite a while (as I chip away at prerequisites) I went on another path.  A topologist in my department studies Conley index theory and in particular generalizations of it.  I need to finish my master’s at some point and the idea of doing something with Morse Theory has been tempting, but I need to fit it into my grand scheme.  So I went looking for connections between Morse Theory and Quantum Mechanics and came across something I didn’t expect: Topological Quantum Field Theory.  Instead of repeating what little I’ve read so far, I’ll recommend John Baez’s Higher-Dimensional Algebra 
and Planck-Scale Physics “ and Quantum Quandaries:A Category-Theoretic Perspective” which are really good.  This stuff is really interesting partially because it’s topological and I’m really interested in metric independent physics but it’s also interesting because not only is category theory useful, it’s essential and seems to be the most natural language for this area.  I’ll write more after I digest a couple of articles/notes (link below).  Anyways, it looks like I might be able to hook this up with Morse theory and draw the interest of the abovementioned faculty member, who I’ll refer to has Dr. Bob.

In summary, NCG is going to be a long haul (check out this reading list) but category is 2014.


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