Masterclass: High dimensional cohomology of moduli spaces

University Copenhagen, June 28 - July 2, 2021

In this Masterclass, we will learn about the high dimensional cohomology of moduli spaces such as the moduli space of curves, graphs, and lattices. These moduli spaces are classifying spaces of groups such as mapping class groups, automorphism groups of free groups, and arithmetic groups. We will learn about duality groups whose high dimensional cohomology of these moduli spaces is related to the low degree homology groups with twisted coefficients. We will also discuss graph homology and tropical curves.

The masterclass is aimed at advanced graduate students and postdocs with an interest in algebraic topology and geometric group theory. Connections with algebraic geometry and number theory will be mentioned but this is not the primary focus.

Graduate students and other early career researchers from US institutes can apply to be considered for financial support.

All graduate students and other early career researchers can apply for financial support to partly cover local expenses. The support should be roughly DKK 500 per day, at most DKK 2500 in total.

If you wish to apply for support, please indicate this on the registration form.

Please note: Due to a match of the UEFA Euro 2020 in Copenhagen on June 28, 2021, the Copenhagen Jazz Festival starting on July 2, 2021, and other events, there will limited accommodation available in Copenhagen. We advise to book accommodation early.

Lecturers/Invited speakers

Lecturers (three lectures each)

Tara Brendle (University of Glasgow)
Kai-Uwe Bux (University of Bielefeld)
Søren Galatius (University of Copenhagen)
Jennifer Wilson (University of Michigan)

Speakers (individual talks)

Benjamin Brück (ETH Zürich)
Mikala Ørsnes Jansen (University of Copenhagen)
Robin Sroka (University of Copenhagen)

Program

Abstracts for all lectures can be found here: [PDF]

Review package put together by Jennifer Wilson: [PDF]

Recordings of (most of) the lectures can be found on YouTube: [Playlist]

All notes for the lectures can be found by clicking on the title of the lecture in the schedule below.

	Monday	Tuesday	Wednesday	Thursday	Friday
Coffee
9:30-10:30	Patzt Introduction	Galatius High-dimensional cohomology of the moduli space of Riemann surfaces, Part I	Free	Galatius High-dimensional cohomology of the moduli space of Riemann surfaces, Part II	Galatius High-dimensional cohomology of the moduli space of Riemann surfaces, Part III
10:45-11:45	Jansen Expository: Borel-Serre Duality	Jansen The stratified homotopy type of the reductive Borel–Serre compactification	Free	Sroka On the high-dimensional rational cohomology of symplectic groups	Brück Free factor complexes as boundary structures
Lunch
13:30-14:30	Bux Outer space and its bordification	Brendle The Steinberg module of the mapping class group	Bux Dimension of the Torelli group of Out(F_n), Part I	Brendle Cohomology of congruence subgroups	Bux Dimension of the Torelli group of Out(F_n), Part II
14:45-15:45	Brendle The curve complex and duality for the mapping class group	Exercises	Exercises	Exercises	Exercises
Coffee
16:15-17:15	Wilson The top-degree rational cohomology of the special linear group of a number ring	Wilson A generating set for the Steinberg module of a Euclidean domain	Miller On not the dualizing module of Aut(F_n)	Problem Session
	Pizza and Drinks			Conference dinner

Venue and transportation

The conference/masterclass will take place at the Department of Mathematical Sciences, University of Copenhagen. See detailed instructions on how to reach Copenhagen and the conference venue.

The lectures will be held in Auditorium 4 and on Zoom. All registered participants will receive the information for the Zoom meeting. Breaks will be held in the foyer. And there will be pizza on monday in the lunch room on the fourth floor.

Tickets and passes for public transportation can be bought at the Copenhagen Airport and every train or metro station. You can find the DSB ticket office on your right-hand side as soon as you come out of the arrival area of the airport. DSB has an agreement with 7-Eleven, so many of their shops double as selling points for public transportation.

A journey planner in English is available.

More information on the "find us" webpage.

Accommodation

We kindly ask the participants to arrange their own accommodation.

We recommend Hotel 9 Små Hjem, which is pleasant and inexpensive and offers rooms with a kitchen. Other inexpensive alternatives are CabInn, which has several locations in Copenhagen: the Hotel City (close to Tivoli), Hotel Scandivania (Frederiksberg, close to the lakes), and Hotel Express (Frederiksberg) are the most convenient locations; the latter two are 2.5-3 km from the math department. Somewhat more expensive – and still recommended – options are Hotel Nora and Ibsen's Hotel.

An additional option is to combine a stay at the CabInn Metro Hotel with a pass for Copenhagen public transportation (efficient and reliable). See information about tickets & prices.

Supported by

PhD School of SCIENCE

Copenhagen Centre for Geometry and Topology

Organizers - Contact

Jesper Grodal (University of Copenhagen)
Jeremy Miller (Purdue University)
Peter Patzt (Unversity of Copenhagen/University of Oklahoma)
Nathalie Wahl (University of Copenhagen)