Math 6190 Kan Seminar on Algebraic $K$-Theory

Course Information

Instructor

• Mona Merling
• Email: mmerling(at)math(dot)upenn(dot)edu

Meeting times

• TTh 5:15-6:45 pm in DRL 3C2

About the class

The class will run similarly to the MIT style Kan seminar, but it will be specifically focused on topics related to $K$-theory, and its connections to number theory and manifold theory. In particular, we will cover topological $K$-theory and the Hopf invariant 1 problem, the Atiyah-Singer index theorem, the definitions of the algebraic groups $K_0$ and $K_1$ and the geometric obstructions they encode, the plus and $Q$ constructions for higher algebraic $K$-theory, the proof that they agree, the fundamental theorems of $K$-theory (localization, devissage, etc.), the $K$-theory of schemes, the $K$-theory of finite fields, the Quillen-Lichtenbaum conjecture, the relationship between $K(\mathbb{Z})$ and the Vandiver conjecture, Thomason's work on $K$-theory and etale cohomology, the definition of Waldhausen $K$-theory via the $S_\bullet$-construction and the definition of $A$-theory, the universal characterization of algebraic $K$-theory, deloopings of $K$-theory, and the multiplicative structure of $K$-theory, ending with some newer frameworks and applications of $K$-theory to Lawvere theories and scissors congruence $K$-theory of graphs.

Talk preparation

The students will deliver the lectures, each talk covering one of the foundational papers (or part thereof). We will meet outisde of class to discuss and go over the plan for your lectures, and we will share resources on the Ed Discussion forum.

Material and notes

I will post expository material for each paper on our Ed Discussion page. If you find other good resources, you can also share them via Ed Discussion. We are lucky to have Mattie Ji livetexing the notes, which everyone will be invited to edited jointly on Overleaf.
UPDATE: The notes are now available on Github; if you have any suggested edits please make a pull request or contact Mattie Ji directly.


Reading responses

When you are not the speaker, your assignment is to post a brief reaction to the relevant paper being presented on our Ed Discussion message board. You should at least read the introduction and skim the paper, and your response should be at least a paragraph. It is fine to respond in a thread started by someone else's response if you want to continue a discussion. What connections do you see, how does the paper fit into the context of what you know, what do you find surprising, what do you wish to understand more? There is no fixed format for this, feel free to make your own rules for the response. To get an idea, you can look at some of the response correspondence from the MIT Kan seminar. Our format will be more interactive, since we will use the Ed Discussion message board, so everyone can answer to anyone, and the discussion is open to everyone in the class. Please feel encouraged to respond to other students. The more active a discussion, the better!



Kan Seminar learning objectives

(copied and edited slightly from the MIT Kan seminar page)

The main objective of this Kan Seminar is to acquaint students with the classic papers and the development of algebraic $K$-theory. More broadly, it provides experience and training in how to read a research paper. These papers are difficult, often long, and, as original sources, they sometimes embody what seems today like a peculiar perspective. The students learn to focus on critical arguments and ideas and distill them to an essence that can be presented in one lecture. The seminar also develops student's ability to scan an article quickly, to glean the essential points and relate them to the rest of their evolving intellectual infrastructures, and to express this understanding.

But there are hidden objectives as well. Usually, participants in the Kan Seminar spend a lot of time together, explaining things to each other and trying to puzzle things out. Trust and the habit of working together greatly enhance the graduate student experience, and this subject is designed to encourage the development of these characteristics.

Schedule of talks

The tentative talk schedule is as follows. This will possibly be pushed back as some lectures will take longer than planned.