Members Seminar

Mirror symmetry via Berkovich geometry I: overview

Tony Yue Yu
Visitor, School of Mathematics
February 13, 2017
Berkovich geometry is an enhancement of classical rigid analytic geometry. Mirror symmetry is a conjectural duality between Calabi-Yau manifolds. I will explain (1) what is mirror symmetry, (2) what are Berkovich spaces, (3) how Berkovich spaces appear naturally in the study of mirror symmetry, and (4) how we obtain a better understanding of several aspects of mirror symmetry via this viewpoint. This member seminar serves also as an overview of my minicourses in the same week.

Local systems and the Hofer-Zehnder capacity

Alexandru Oancea
Université Pierre et Marie Curie; Member, School of Mathematics
February 6, 2017
The Hofer-Zehnder capacity of a symplectic manifold is one of the early symplectic invariants: it is a non-negative real number, possibly infinite. Finiteness of this capacity has strong consequences for Hamiltonian dynamics, and it is an old question to decide whether it holds for small compact neighborhoods of closed Lagrangians. I will explain a positive answer to this question for a class of manifolds whose free loop spaces admit nontrivial local systems.

Homological versus Hodge-theoretic mirror symmetry

Timothy Perutz
University of Texas, Austin; von Neumann Fellow, School of Mathematics
January 30, 2017
I'll describe joint work with Sheel Ganatra and Nick Sheridan which rigorously establishes the relationship between different aspects of the mirror symmetry phenomenon for Calabi-Yau manifolds. Homological mirror symmetry---an abstract, categorical statement---implies Hodge theoretic mirror symmetry, a concrete relation between counts of rational curves and variations of Hodge structure.

Combinatorics of the amplituhedron

Lauren Williams
University of California, Berkeley; von Neumann Fellow, School of Mathematics
January 23, 2017

Points and lines

Nathaniel Bottman
Member, School of Mathematics
December 12, 2016
The Fukaya category of a symplectic manifold is a robust intersection theory of its Lagrangian submanifolds. Over the past decade, ideas emerging from Wehrheim--Woodward's theory of quilts have suggested a method for producing maps between the Fukaya categories of different symplectic manifolds. I have proposed that one should consider maps controlled by compactified moduli spaces of marked parallel lines in the plane, called "2-associahedra".

Counting Galois representations

Frank Calegari
University of Chicago
November 4, 2016
One of the main ideas that comes up in the proof of Fermat's Last Theorem is a way of "counting" 2-dimensional Galois representations over $\mathbb Q$ with certain prescribed properties. We discuss the problem of counting other types of Galois representations, and show how this leads naturally to questions related to derived algebraic geometry and the cohomology of arithmetic groups. A key example will be the case of 1-dimensional representations of a general number field.