
Course Information

Instructor

Mona Merling
 Email: mmerling(at)math(dot)upenn(dot)edu
Meeting times
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 AtiyahSinger 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 QuillenLichtenbaum 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 preparationThe 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
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
Papers: Atiyah, $K$theory
Adams and Atiyah, $K$theory and the Hopf invariant

Jan 23, 28 
Jackson Goodman

AtiyahSinger index theorem
Papers: Atiyah and Singer, Index of Elliptic Operators

Jan 30 
Jacob Van Hook

$K_0$ and Wall finiteness obstruction
Paper: Wall, Finiteness conditions for CWcomplexes

Feb 4 
Yi Wang

$K_1$ and Whitehead torsion
Paper: Milnor, Whitehead torsion

Feb 6 
Bharath Palvannan

$J$homomorphism and the Adams conjecture
Paper: Quillen, The Adams conjecture

Feb 11, 13 
Elijah Gunther

Plus construction and the $K$theory of finite fields
Paper: 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
Paper: Quillen, Algebraic $K$theory I

Feb 20 
Marielle Ong

The $Q$ construction for exact categories and the fundamental theorems
Paper: Quillen, Algebraic $K$theory I

Feb 25 
Andrew Kwon

$K$theory of schemes
Paper: Quillen, Algebraic $K$theory I

Feb 27 
Tianyue Liu

The Plus=Q theorem
Paper: Grayson, Algebraic $K$theory II (after Quillen)

Mar 24, 26 
Man Cheung Tsui

Introduction to etale cohomology and the Quillen Lichtenbaum conjecture
Paper: Thomason, Algebraic Ktheory and etale cohomology

Mar 31 
Brett Frankel Thomas Brazelton

Milnor $K$theory and motivic cohomology
Papers: Milnor, Algebraic Ktheory and quadratic forms
Voevodsky, Motivic cohomology with $\mathbb{Z}/2$coefficients

Apr 2 
Zhaodong Cai

$K(\mathbb{Z})$ and the Vandiver conjecture
Paper: Kurihara, Some remarks on conjectures about cyclotomic fields and Kgroups of $\mathbb{Z}$

Apr 7 
Souparna Purohit

$K$theory and special values of zeta functions
Paper: Lichtenbaum Values of zeta functions, etale cohomology and algebraic Ktheory

Apr 9, 14 
Andres Mejia Jingye Yang

Waldhausen categories and the $S_\bullet$construction The additivity theorem and delooping Waldhausen $K$theory
Paper: Waldhausen, Algebraic $K$theory of spaces

Apr 16 
Elijah Gunther

Higher algebraic $K$theory of derived categories
Paper: Thomason and Trobaugh, Higher Algebraic Ktheory of schemes and derived categories

Apr 21, 23 
Jakob Hansen Hans Riess

Deloopings of categories
Papers: Segal, Categories and cohomology theories
May, Geometry of infinite loop spaces

Apr 28 
Darrick Lee

Multiplicative structure of $K$theory
Paper: Elmendorf and Mandell, Rings, modules and algebras in infinite loop space theory

Apr 30 
Thomas Brazelton

$K$theory of infinity categories and universal characterization
Papers: Blumberg, Gepner, Tabuada, Universal characterization of higher algebraic Ktheory
Barwick, On the algebraic $K$theory of higher categories




