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.

Commitment to Diversity and Inclusivity

We are committed to fostering an inclusive environment during the summer school and to bring together a diverse group of participants. We especially encourage applications from those persons with marginalized aspects of their identity, including people of color; Black and Indigenous people; Asian people, Asian-Americans, and Pacific Islanders; women; nonbinary people; people with disabilities; members of the LGBTQIA+ community; people from low-income households; first-generation students; people who speak English as a second language; and any other identities which are underrepresented or disprivileged in mathematics. You do not have to be in a PhD program to apply to our summer school, or to be studying mathematics, as long as you are interested to learn algebraic topology beyond an introductory course.

In addition to establishing clear expectations for all participants with regards to being in community with one another, we hope to design a structure for this workshop that is anti-oppressive, and that removes, to whatever extent possible, the systemic aspects of a traditional conference which perpetuate oppression and exclusion. To help keep the organizers accountable in this endeavor, we will offer several avenues for feedback before, during, and after the event, including an anonymous feedback form which can be found here:

Anonymous Feedback Form

Please provide us with feedback about any aspect of the workshop, including our website and application process. Feel free to also contact us (agnes.beaudry@colorado.edu) if you have any questions.

Schedule

All talks will be in CASE. The schedule can be viewed here.

Getting to Boulder

There's an affordable bus from Denver International Airport to Boulder.  Click here for details. Tickets for the bus can be purchased at the airport. 

Reception

The reception is Wednesday at 6 pm at the Rayback Collective: https://www.therayback.com/ . Food will be provided and served between 6 and 6:30 pm. Additional food can also be purchased at food trucks at the venue. Beverages must be purchased at the bar.

 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)

Additional Information

• If you need help with finding childcare, or need assistance finding appropriate housing for your family, please do not hesitate to contact us.

• The summer school will adopt CU Boulders's Sexual Misconduct, Intimate Partner Violence and Stalking Policy and Discrimination and Harassment Policy . 

• Reporting to the Office of Institutional Equity and Compliance: Individuals who have been subjected to protected-class discrimination or harassment, sexual assault or other forms of sexual misconduct, intimate partner violence (including dating or domestic violence), or stalking have several options for reporting. Depending on the nature of the alleged conduct, individuals may pursue a university process, a criminal process, both processes, or neither. 

Funding

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

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.