Categories and TQFT beginnings

It was happenstance that I stumbled across these lectures on Youtube on Category Theory which were given by Steve Awodey, a professor of philosophy at Carnegie-Mellon.  There are only a couple (I’ve watched the first one) and they were given during a short summer school on computer science (I still haven’t figured out how or why computer scientists use category theory).  I watched the first one after reading the first couple sections of Maclane’s book and as it turns out Awodey was Maclane’s student!  The lecture was quite good, well paced with good examples.  I’m waiting to watch the next one as I try and pace myself on this topic.  Simon and I are working the exercises at the end of the section on functors at the moment.  No haste needed, we’re just taking our time.

Along these lines I’ve been starting my literature search on TQFTs and figuring out what order to read/work these.  Baez has a couple of articles that are largely expository.  I’ve read “Higher Dimensional Algebra and Plank scale physics” and it’s a fun read, not too taxing.  I also came across an article by Atiyah entitled “Introduction to Topological Quantum Field Theories” and it’s also quite readable though I’ll need to spend a bit more time on it to work through a couple parts. His original article “Topological Quantum Field Theory” I’ve yet to read.

I’m still mulling over what I’ll say in my Pizza Pi talk at the end of February.  The topic is certainly Penrose tiles but there are a number of ways of going after them.  It’ll be light on proofs (sketches at best) and more of a story with highlighted results and properties that are surprising and novel.  I would like to start with Wang and Logic and then work towards Penrose tiles, discuss some properties, and then finish off with a nod to noncommutative geometry.  As a space or mathematical object it has a lot of surprises and links together some interesting characters like Wang, Penrose and Connes.  I’ll post more as it becomes clearer to me.  Martin Gardner has an excellent book “Penrose Tiles to Trapdoor Ciphers” which features Penrose tiles for the first couple of chapters.

I’m still getting the hang of this writing & posting business.  Hopefully I’ll have something of substance to write soon.

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About because0fbeauty

Fascinated by the way mathematics and physics interact, captivated by visual and tactile mathematics and hoping to become a better expositor of these things is why I blog...occasionally...when I remember.
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6 Responses to Categories and TQFT beginnings

  1. S. D. Browne says:

    Thanks Kevin for the helpful post. The Baez article seems especially accessible to the non-mathematician.

  2. sheabrowne says:

    I just finished my first read-through of Baez’s “Higher Dimensional Algebra and Plank scale physics”….holy shit, it’s great stuff. I could follow the analog between (n-1)-dimensional manifolds and dimensional cobordisms and Hilbert spaces and operators (loved the clear example of non-commutative cobordisms!). I got lost a bit on the category theory and higher dimensional algebras (I need to have my pen in hand before I read), but my mind was blown by the idea of spin-foams…just the name is cool enough! This could get very distracting!

    • Glad you liked it. Baez is particularly clear. Do you have a DropBox account? I can email you some other articles if you don’t. Atiyah’s original article is also quite good though more technical and Wittens is much more technical but you know a lot more field theory than I so you might be able to use it sooner. I’ve also come across some undergraduate lecture notes which is a physics first approach. So it introduces the necessary qft before tackling the topology.

      The cobordisms are really interesting. You can decompose them over and over again which makes them neat examples of ‘arrows’ in categories that are not functions. They nicely capture creation and annhiliation as well. Consider something like a sphere the boundary of which is the empty set. I’m pretty sure you could think of this as the case where something is born (whatever a circle would stand for) and then is annihilated). I hope to put up more soon!

  3. Pingback: Quantum, Local, and Background Free | Fields, Particles & Space-time

  4. Pingback: Quantum, Local, and Background Free | because0fbeauty

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