Description

The goal of this summer school is to introduce participants to tools and ideas from algebraic topology and homotopy theory that are used in the study of fixed point theory. This will be a problem set focused summer school surrounding four mini-courses: 

1) Fixed point theory and Nielsen theory

2) Categorical Approach to Duality

3) Spectra

4) Trace Methods

See more details on the content of each mini-course below.

Each day there will be:

• three sessions, each of which consists of a 15/20 minute talk introducing the basic ideas followed by a problem session. 

• an hour or so for a more conventional lecture, when members of the scientific committee or participants will present on current research

While the design of this summer school is mainly aimed at second and third year graduate students and/or those who are familiar with the material in Hatcher's Algebraic Topology

https://pi.math.cornell.edu/~hatcher/AT/AT.pdf

(except the appendices), you should not feel that these are firm barriers to entry. If you feel that you would benefit from this summer school, please apply.

Topics Covered

1) Fixed point theory / Nielsen theory - Malkiewich and Yeakel

This course covers the Lefschetz number and fixed point index, the Lefschetz Fixed Point Theorem, the Poincaré-Hopf and Lefschetz-Hopf Theorems, the Nielsen number and the Reidemeister trace.

2) Categorical Approach to Duality - Ponto

This course covers symmetric monoidal categories, duality and traces in a symmetric monoidal category, bicategories, duality and traces in a bicategory, and string diagrams. 

3) Spectra - Campbell and Lind

This course covers Spanier-Whitehead (n-)duality for closed smooth manifolds, finite complexes or compact ENRs, the Pontryagin-Thom construction (framed case), the Spanier-Whitehead category and its basic properties, methods for computing traces and transfers, diagram spectra and their homotopy category, the smash product, and fiberwise Pontryagin-Thom and Costenoble-Waner duality for closed smooth manifolds. 

4) Trace Methods - Klang and Zakharevich

This course covers additivity of traces, the Hattori-Stallings trace, HH∗ of a ring, Morita invariance (using traces), explicit K_0 of a Waldhausen category, trace map to HH_0, K of a Waldhausen category (S_• construction), and the trace map to THH.

Schedule

The schedule can be viewed here.

 Scientific Leaders and Speakers

Jonathan Campbell, Center for Communications Research, La Jolla

Inbar Klang, Columbia University

Kate Ponto, University of Kentucky

Cary Malkiewich, Binghamton University

John Lind, California State Chico

Sarah Yeakel, UC Riverside

Dylan Wilson, Harvard

Inna Zakharevich, Cornell University

Organizers

Agnès Beaudry (CU Boulder agnes.beaudry@colorado.edu), Kate Ponto (University of Kentucky) and  Dylan Wilson (Harvard)

Funding

The workshop is partially funded by  Foundation Nagoya Mathematical Journal  and by the National Science Foundation under the award DMS-2153772.