Math 730 Kan Seminar on Algebraic $K$-Theory

Spring 2020



Course Information


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

Meeting times

  • TTh 3-4:30 pm in DRL 4C6

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.

    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 piazza.

    Material and notes

    I will post expository material for each paper on our piazza page. If you find other good resources, you can also share them via piazza. We will also add lecture notes on piazza. We will keep these private for now, but if anyone wants to type and polish their notes, we can later on post them on the webpage.

    Reading responses

    When you are not the speaker, your assignment is to post a brief reaction to the relevant paper being presented on our piazza 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 piazza 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.

Speaker Topic
Jan 16 Organizational meeting
Jan 21, 23 Tianyue Liu Topological $K$-theory and the Hopf invariant one problem

Atiyah, $K$-theory
Adams and Atiyah, $K$-theory and the Hopf invariant
Jan 23, 28 Jackson Goodman Atiyah-Singer index theorem

Atiyah and Singer, Index of Elliptic Operators
Jan 30 Jacob Van Hook $K_0$ and Wall finiteness obstruction

Wall, Finiteness conditions for CW-complexes
Feb 4 Yi Wang $K_1$ and Whitehead torsion

Milnor, Whitehead torsion
Feb 6 Bharath Palvannan $J$-homomorphism and the Adams conjecture

Quillen, The Adams conjecture
Feb 11, 13 Elijah Gunther Plus construction and the $K$-theory of finite fields

Quillen, On the cohomlogy and $K$-theory of the general linear groups over a finite field
Feb 18 Julian Gould Classifying spaces of categories and Quillen's theorems A and B

Quillen, Algebraic $K$-theory I
Feb 20 Marielle Ong The $Q$ construction for exact categories and the fundamental theorems

Quillen, Algebraic $K$-theory I
Feb 25 Andrew Kwon $K$-theory of schemes

Quillen, Algebraic $K$-theory I
Feb 27 Tianyue Liu The Plus=Q theorem

Grayson, Algebraic $K$-theory II (after Quillen)
Mar 24, 26 Man Cheung Tsui Introduction to etale cohomology and the Quillen Lichtenbaum conjecture

Thomason, Algebraic K-theory and etale cohomology
Mar 31 Brett Frankel
Thomas Brazelton
Milnor $K$-theory and motivic cohomology

Milnor, Algebraic K-theory and quadratic forms
Voevodsky, Motivic cohomology with $\mathbb{Z}/2$-coefficients
Apr 2 Zhaodong Cai $K(\mathbb{Z})$ and the Vandiver conjecture

Kurihara, Some remarks on conjectures about cyclotomic fields and K-groups of $\mathbb{Z}$
Apr 7 Souparna Purohit $K$-theory and special values of zeta functions

Lichtenbaum Values of zeta functions, etale cohomology and algebraic K-theory
Apr 9, 14 Andres Mejia
Jingye Yang
Waldhausen categories and the $S_\bullet$-construction
The additivity theorem and delooping Waldhausen $K$-theory

Waldhausen, Algebraic $K$-theory of spaces
Apr 16 Elijah Gunther Higher algebraic $K$-theory of derived categories

Thomason and Trobaugh, Higher Algebraic K-theory of schemes and derived categories
Apr 21, 23 Jakob Hansen
Hans Riess
Deloopings of categories

Segal, Categories and cohomology theories
May, Geometry of infinite loop spaces
Apr 28 Darrick Lee Multiplicative structure of $K$-theory

Elmendorf and Mandell, Rings, modules and algebras in infinite loop space theory
Apr 30 Thomas Brazelton $K$-theory of infinity categories and universal characterization

Blumberg, Gepner, Tabuada, Universal characterization of higher algebraic K-theory
Barwick, On the algebraic $K$-theory of higher categories