
Course Information

Instructor

Mona Merling
 Email: mmerling(at)math(dot)jhu(dot)edu
Meeting times
 TTh 121:15 pm in Maryland 114
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 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 finite fields, the QuillenLichtenbaum conjecture, the definition of Waldhausen $K$theory via the $S_\bullet$construction and the definition of $A$theory, the stable parametrized $h$cobordism theorem, and the multiplicative structure of $K$theory.
Talk preparationI will send you resources for each talk, and we will meet outisde of class to discuss and prepare your lectures. The lectures will mostly be delivered by you.
Material and Notes
Our dropbox folder contains a bunch of material. In addition to this, I will individually send you some more material regarding your particular talk. I will also add pictures of talk notes and relevant email threads of our discussions to our dropbox. We will keep these private for now, but if anyone wants to type and polish the notes, we can later on post them.

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 
Sep 5 

Organizational meeting

Sep 7, 12, 14 
Apurv Nakade

Topological $K$theory and the Hopf invariant one problem
Papers: Atiyah, $K$theory
Adams and Atiyah, $K$theory and the Hopf invariant

Sep 19, 21 
Mona Merling

Overview of $K$theory of number rings and their relation to the Vandiver conjecture and the class number formula
Overview of questions related to the classification of manifolds and their relationship to $A$theory

Oct 3 
Keaton Stubis

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

Oct 5 
David Myers

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

Oct 10 
Martina Rovelli

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

Oct 12, 17 
Daniel FuentesKeuthan

Classifying spaces of categories and Quillen's theorems A and B
Paper: Quillen, Algebraic $K$theory I

Oct 17, 19 
Xiyuan Wang

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

Oct 19, 24 
Martina Rovelli

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

Oct 26, 31, Nov 2 
Xiyuan Wang

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

Nov 7, 9, 14 
Daniel FuentesKeuthan

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

Nov 17 
Tslil Clingman

Diagram spectra and the smash product
Paper: Mandell, May, Schwede, and Shipley, Model categories of diagram spectra

Nov 28, 30 
Tslil Clingman

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

Dec 5, 7 
Apurv Nakade

Computation of the $K$theory of finite fields
Paper: Quillen, On the cohomlogy and $K$theory of the general linear groups over a finite field


Announcements
Week of Sept 26, 28: There will be no lectures and we will work on preparing the upcoming talks.


