Members Seminar

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.

Reciprocity laws for torsion classes

Ana Caraiani
October 31, 2016
The law of quadratic reciprocity and the celebrated connection between modular forms and elliptic curves over $\mathbb Q$ are both examples of reciprocity laws. Constructing new reciprocity laws is one of the goals of the Langlands program, which is meant to connect number theory with harmonic analysis and representation theory. In this talk, I will survey some exciting recent progress in establishing new reciprocity laws, namely how to construct Galois representations attached to torsion classes which occur in the cohomology of arithmetic hyperbolic $3$-manifolds.

Knot surgery and Heegaard Floer homology

Jennifer Hom
Member, School of Mathematics
April 4, 2016
One way to construct new 3-manifolds is by surgery on a knot in the 3-sphere; that is, we remove a neighborhood of a knot, and reglue it in a different way. What 3-manifolds can be obtained in this manner? We provide obstructions using the Heegaard Floer homology package of Ozsvath and Szabo. This is joint work with Cagri Karakurt and Tye Lidman.

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.