Mathematics Dept.
  Events
Stanford University
Wed 3 Jan 2018, 12:00pm SPECIAL
Topology and related seminars
ESB 4133 (PIMS Lounge)
Stability in the homology of Torelli groups
ESB 4133 (PIMS Lounge)
Wed 3 Jan 2018, 12:00pm-1:00pm

Abstract

The Torelli subgroups of mapping class groups are fundamental objects in low-dimensional topology, through some basic questions about their structure remain open. In this talk I will describe these groups, and how to use tools from representation theory to establish patterns their homology. This project is joint with Jeremy Miller and Peter Patzt. These "representation stability" results are an application of advances in a general algebraic framework for studying sequences of group representations.
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University of Adelaide
Wed 3 Jan 2018, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
TBA (provisional time slot)
ESB 4133 (PIMS Lounge)
Wed 3 Jan 2018, 3:15pm-4:15pm

Abstract


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Stanford University
Tue 9 Jan 2018, 3:00pm SPECIAL
Topology and related seminars
ESB 4133 (PIMS Lounge)
Totally geodesic submanifolds of Teichmuller space and the Kontsevich-Zorich cocycle
ESB 4133 (PIMS Lounge)
Tue 9 Jan 2018, 3:00pm-4:00pm

Abstract

One of the ways we understand Teichmuller space, endowed with the Teichmuller metric, is by studying Teichmuller discs. They exist in great abundance: There is a Teichmuller disc through any point and in any direction. Typically, their projection to moduli space is dense, and yet infinitely often their projection is a closed subvariety of moduli space called a Teichmuller curve. Recently, in joint work with Eskin, McMullen, and Mukamel, we discovered the first non-trivial examples of higher dimensional analogues of Teichmuller discs, namely totally geodesic submanifolds.

In this talk, we will explain that such higher dimensional totally geodesic submanifolds are much more rigid and rare than Teichmuller discs: Each must cover a closed subvariety of moduli space, and only finitely many such subvarieties exist in each moduli space. This result is an application of joint work with Eskin and Filip on the Kontsevich-Zorich cocycle. One of the goals of the talk will be to explain what this cocycle is and why it lies at the heart of Teichmuller dynamics.
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