Members Seminar

Profinite rigidity and flexibility for compact 3-manifold groups

Alan Reid
University of Texas, Austin; Member, School of Mathematics
February 2, 2016
This talk will discuss the question: To what extent are the fundamental groups of compact 3-manifolds determined (amongst the fundamental groups of compact 3-manifolds) by their finite quotients. We will discuss work that provides a positive answer for fundamental groups of hyperbolic 1-punctured torus bundles.

The space of surface shapes, and some applications to biology

Joel Hass
University of California, Davis; Member, School of Mathematics
February 1, 2016
The problem of comparing the shapes of different surfaces turns up in different guises in numerous fields. I will discuss a way to put a metric on the space of smooth Riemannian 2-spheres (i.e. shapes) that allows for comparing their geometric similarity. The metric is based on a distortion energy defined on the space of conformal mappings between a pair of spheres. I'll also discuss a related idea based on hyperbolic orbifold metrics. I will present results of experiments on applying these techniques to biological data.

Locally symmetric spaces and torsion classes

Ana Cariani
Princeton University; Veblen Research Instructor, School of Mathematics
December 14, 2015
The Langlands program is an intricate network of conjectures, which are meant to connect different areas of mathematics, such as number theory, harmonic analysis and representation theory. One striking consequence of the Langlands program is the Ramanujan conjecture, which is a statement purely within harmonic analysis, about the growth rate of Fourier coefficients of modular forms. It turns out to be intimately connected to the Weil conjectures, a statement about the cohomology of projective, smooth varieties defined over finite fields.

Billiards in quadrilaterals, Hurwitz spaces, and real multiplication of Hecke type

Alexander Wright
Stanford University; Member, School of Mathematics
November 30, 2015

After a brief introduction to the dynamics of the $\mathrm{GL}(2,\mathbb R)$ action on the Hodge bundle (the space of translations surfaces), we will give a construction of six new orbit closures and explain why they are interesting. Joint work with Alex Eskin, Curtis McMullen, and Ronen Mukamel.

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.