Noncommutative Geometry Seminar

This should be a very informal learning seminar with the goal of learning some of the basics of noncommutative geometry! This is a super cool subject (see Connes and Marcolli "A walk in the noncommutative garden" )
for an overview of some of different ideas this connects to.
The plan is to go through a bit of stuff pulling from several resources, and beginning with the basics of C*-algebras. We're making a groupchat so contact me to be added. I'm linking several references below, and also will mention which are relevant to specific talks so everyone can follow along. We'll probably do a mix of presenting, discussing readings, and trying our hand at a couple of exercises, so there should be a lot for you regardless of how involved you are outside of the meeting times (exercises/ reading discussion will be kept at the end after the talks).
When? Fridays at 6pm? Probably
Where? Maybe Fine 322? Slightly TBD since we're just starting and it's pretty informal

Rough plan (everything may be changed based on time/ interests) (click on the titles for descriptions)

1. Intro to C*-algebras and the noncommutative torus

I'll try to type up some notes for this. Mostly discussed the definition, basic examples of C*-algebras, commutative C*-algebras and topology, some functional calculus (poorly unfortunately) and the noncommutative torus. Good references are Chapter 1 of Rordam et al, Chapter 1 of Khalkhali for general C*-algebras building towards the goals of this seminar. Section 2.5 of Khalkhali for the noncommutative torus, and also Davidson Chapter VI has more on the noncommutative torus. Khalkhali is probably better for seeing how we build towards differentials and such though, and Connes book is the best (we'll go over this topic in more detail later though). However, the GOALS lecture notes are probably the best intro to C*-algebras in general, and I think they do the functional calculus and Gelfand duality well.

2. Examples of C*-algebras, and projections and unitaries (prep for K-theory)

I might recap the noncommutative torus a bit as well. But for the most part it'll be new examples/ more in depth ones, and discussing Chapter 2 of Rordam et al is by far the most relevant reference. Possibly Chapter VI of Davidson too, since I might go into more depth on the structure of the noncommutative torus as an example.

3. K-theory

4. The Atiyah-Singer Index theorem

5. Groupoids, foliations, group actions...

This will be slightly determined by interests. Probably connect briefly with stacks tho. And this is where crossed products get more time to shine... and Fourier related things/ duals of groups... And applying a bit of K theory stuff. Probably should include KMS states/ C*-dynamical systems stuff since doing all this.

6. Cyclic and Hochschild cohomology

7. Spectral triples

8. Depending on interests

Maybe more on Connes index. Maybe application to something else. Can be up to the audience! I'd be very interested in \Q-lattices or Bost-Connes systems more specifically since it ties this stuff together a bit with number theory and quantum stat mech, but we'll see. Also, I think to do any of this in depth it might be good to spill a bit over into the next semester if people are interested.

References

Rørdam, Larsen, Laustsen An Introduction to K-Theory for C*-Algebras. This is a good introduction to K-theory and introduces basic C*-algebras at the beginning link
The GOALS lecture notes on C*-algebras by KRISTIN COURTNEY, ELIZABETH GILLASPY, AND LARA ISMERT found here. I strongly recommend these as a beginning resource on operator algebras as they were designed with accessibility as a primary goal and have a lot of exercises to try out. link
Masoud Khalkhali, Lectures on Noncommutative Geometry is a shorter but fairly accessible set of notes on noncommutative geometry parts of which follow some of what we're doing. link
Kenneth R. Davidson, C*-algebras by example especially chapter VI so far, since it covers noncommutative tori , link
Connes Noncommutative Geometry. This is THE canonical reference. A bit overwhelming in terms of content tho, but we'll be using parts of this. link
José M. Gracia-Bondía , Joseph C. Várilly , Héctor Figueroa Elements of Noncommutative Geometry . I haven't read this, but I think it's the main alternative to Connes' book. Looking at the contents it covers a bit less, but does more introductory exposition so it might be good. link
Yuezhao Li notes on an Introduction to Index theory. Seems like a nice reference for seeing the connection of Atiyah Singer with C*-algebras a bit. link
Higson On the K-theory proof of the index theorem. Also some other cool articles in here, but most are slightly adjacent to what we're planning to talk about. link
Dhruv Goel's undergraduate thesis on The Joys of the Atiyah-Singer Index Theorem contains the content in his lecture plus more! link

If you want more, here are a couple other references that are potentially interesting/ cool, but not directly related to what we're talking about so far.

Dana P. Williams Crossed Products of C*-algebras . This is a cool topic, and we will touch on these briefly, but unless we get really interested in this and decide to change gears a bit we probably won't go into most of the content here. link
Michael Brannan Quantum Groups: What are they and what are they good for?. I haven't gone through these notes in depth, but quantum groups connect to this topic a bit (we can go in more depth on them if there's interest, but they'll at least be briefly mentioned) and I think Michael Brannan is generally good at explaining things. I might add a more formal reference on quantum groups at some point though. link
Nica, Speicher Lectures on the Combinatorics of Free Probability Theorey. We're not covering this topic in the current plan, but I think it's pretty cool! And also connects thematically a bit (although a bit of a different flavor). link
Maria Paula Gomez Aparicio, Pierre Julg, Alain Valette, The Baum-Connes conjecture: an extended survey. This is a topic we could discuss in our last talk. We'll probably discuss it a bit on the 5th talk too, and maybe mention on the 4th? This is still an open problem! link

I'll probably also add some more specific stuff that we might talk about later, and in general some more references.