Members Seminar

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.

\(P = W\): a strange identity for \(\mathrm{GL}(2,\mathbb C)\)

Mark deCataldo
Stony Brook University; Member, School of Mathematics
November 24, 2014
Start with a compact Riemann surface \(X\) and a complex reductive group \(G\), like \(\mathrm{GL}(n,\mathbb C)\). According to Hitchin-Simpson's ``non abelian Hodge theory", the pair \((X,G)\) comes with two new complex manifolds: the character variety \(\mathcal M_B\) and the Higgs moduli space \(\mathcal{M}_\text{Dolbeault}\). When \(G= \mathbb C^*\), these manifolds are two instances of the usual first cohomology group of \(X\) with coefficients in the abelian \(\mathbb C^*\).

Hyperbolic groups, Cannon-Thurston maps, and hydra

Timothy Riley
Cornell University; Member, School of Mathematics
November 17, 2014
Groups are Gromov-hyperbolic when all geodesic triangles in their Cayley graphs are close to being tripods. Despite being tree-like in this manner, they can harbour extreme wildness in their subgroups. I will describe examples stemming from a re-imagining of Hercules' battle with the hydra, where wildness is found in properties of "Cannon-Thurston maps" between boundaries. Also, I will give examples where this map between boundaries fails to be defined. This is joint work with O. Baker.

Shot-noise random fields: some geometric properties and some applications for images

Agnès Desolneux
École normale supérieure de Cachan; Member, School of Mathematics
November 10, 2014
Shot-noise random fields can model a lot of different phenomena that can be described as the additive contributions of randomly distributed points. In the first part of the talk, I will give some properties of these random fields. And in a second part, I will discuss an application to images with the texture synthesis problem. The talk is intended for a wide mathematical audience: I will recall the needed notions of probability and show a lot of illustrations.

Apery, irrationality proofs and dinner parties

Francis Brown
Centre national de la recherche scientifique, Institut des Hautes Études Scientifiques
October 27, 2014
After introducing an elementary criterion for a real number to be irrational, I will discuss Apery's famous result proving the irrationality of \(\zeta(3)\). Then I will give an overview of subsequent results in this field, and finally propose a simple geometric interpretation based on a classical dinner party game. The talk is intended for a general mathematical audience.