Members Seminar

Fun with finite covers of 3-manifolds: connections between topology, geometry, and arithmetic

Nathan Dunfield
University of Illinois, Urbana-Champaign
November 23, 2015
Following the revolutionary work of Thurston and Perelman, the topology of 3-manifolds is deeply intertwined with their geometry. In particular, hyperbolic geometry, the non-Euclidean geometry of constant negative curvature, plays a central role. In turn, hyperbolic geometry opens the door to applying tools from number theory, specifically automorphic forms, to what might seem like purely topological questions.

The $\mathrm{SL}(2,\mathbb R)$ action on moduli space

Alex Eskin
University of Chicago; Member, School of Mathematics
November 16, 2015
There is a natural action of the group $\mathrm{SL}(2,\mathbb R)$ of $2 \times 2$ matrices on the unit tangent bundle of the moduli space of compact Riemann surfaces. This action can be visualized using flat geometry models, which allows one to make an analogy with homogeneous spaces, such as the space of lattices in $\mathbb R^n$. I will make the basic definitions, and mention some recent developments. This talk will be even more introductory than usual.

Hard Lefschetz theorem and Hodge-Riemann relations for combinatorial geometries

June Huh
Princeton University; Veblen Fellow, School of Mathematics
November 9, 2015
A conjecture of Read predicts that the coefficients of the chromatic polynomial of a graph form a log-concave sequence for any graph. A related conjecture of Welsh predicts that the number of linearly independent subsets of varying sizes form a log-concave sequence for any configuration of vectors in a vector space. In this talk, I will argue that two main results of Hodge theory, the Hard Lefschetz theorem and the Hodge-Riemann relations, continue to hold in a realm that goes beyond that of Kahler geometry.

Quantum Ergodicity for the uninitiated

Zeev Rudnick
Tel Aviv University; Member, School of Mathematics
October 26, 2015
A key result in spectral theory linking classical and quantum mechanics is the Quantum Ergodicity theorem, which states that in a system in which the classical is ergodic, almost all of the Laplace eigenfunctions become uniformly distributed in phase space. There are similar statements which are valid for some integrable and pseudo-integrable systems, such as flat tori and rational polygons. I will give an introduction to these notions, including explanations of the undefined terms above, and describe some connections with number theory.

3-manifold groups

Ian Agol
University of California, Berkeley; Distinguished Visiting Professor, School of Mathematics
October 12, 2015
I'll review recent progress on properties of 3-manifold groups, especially following from geometric properties. Then I'll discuss some open questions regarding 3-manifold groups, including their profinite completions, torsion in covers, orderability, and $PD(3)$ groups.

Extending the Prym map

Samuel Grushevsky
Stony Brook University
February 10, 2015
The Torelli map associates to a genus g curve its Jacobian - a $g$-dimensional principally polarized abelian variety. It turns out, by the works of Mumford and Namikawa in the 1970s (resp. Alexeev and Brunyate in 2010s), that the Torelli map extends to a morphism from the Deligne-Mumford moduli of stable curves to the Voronoi (resp. perfect cone) toroidal compactification of the moduli of abelian varieties. The Prym map associates to an etale double cover of a genus g curve its Prym - a principally polarized $(g-1)$-dimensional abelian variety.

Twisted matrix factorizations and loop groups

Daniel Freed
University of Texas, Austin; Member, School of Mathematics and Natural Sciences
February 9, 2015
The data of a compact Lie group $G$ and a degree 4 cohomology class on its classifying space leads to invariants in low-dimensional topology as well as important representations of the infinite dimensional group of loops in $G$. Previous work with Mike Hopkins and Constantin Teleman brought the twisted equivariant topological K-theory of $G$ into the game, but only on the level of equivalence classes. After reviewing these ideas I will describe ongoing work with Teleman which gives a geometric construction of the representation categories.

Ball quotients

Bruno Klingler
Université Paris Diderot; Member, School of Mathematics
December 8, 2014
Ball quotients are complex manifolds appearing in many different settings: algebraic geometry, hyperbolic geometry, group theory and number theory. I will describe various results and conjectures on them.

Graphs, vectors and integers

Noga Alon
Tel Aviv University; Visiting Professor, School of Mathematics
December 1, 2014
The study of Cayley graphs of finite groups is related to the investigation of pseudo-random graphs and to problems in Combinatorial Number Theory, Geometry and Information Theory. I will discuss this topic, describing the motivation and focusing on several results that illustrate the interplay between Graph Theory, Geometry and Number Theory.