Members Seminar

Modular forms with small Fourier coefficients

Florian Sprung
Princeton University; Visitor, School of Mathematics
November 21, 2016
Computing the class number is a hard question. In 1956, Iwasawa announced a surprising formula for an infinite family of class numbers, starting an entire theory that lies behind this phenomenon. We will not focus too much on this theory (Iwasawa theory), but rather describe some analogous formulas for modular forms. Their origins have not been explained yet, especially when the $p$-th Fourier coefficient is small.

Eigenvalue bounds on sums of random matrices

Adam Marcus
Princeton University; Member, School of Mathematics
November 14, 2016
For certain applications of linear algebra, it is useful to understand the distribution of the largest eigenvalue of a finite sum of discrete random matrices. One of the useful tools in this area is the "Matrix Chernoff" bound which gives tight concentration around the largest eigenvalue of the expectation. In some situations, one can get better bounds by showing that the sum behaves (in some rough way) like one would expect from Gaussian random matrices.

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.