## Combinatorics of the amplituhedron

Lauren Williams

University of California, Berkeley; von Neumann Fellow, School of Mathematics

January 23, 2017

Lauren Williams

University of California, Berkeley; von Neumann Fellow, School of Mathematics

January 23, 2017

Nathaniel Bottman

Member, School of Mathematics

December 12, 2016

The Fukaya category of a symplectic manifold is a robust intersection theory of its Lagrangian submanifolds. Over the past decade, ideas emerging from Wehrheim--Woodward's theory of quilts have suggested a method for producing maps between the Fukaya categories of different symplectic manifolds. I have proposed that one should consider maps controlled by compactified moduli spaces of marked parallel lines in the plane, called "2-associahedra".

Thomas Church

Stanford University; Member, School of Mathematics

November 28, 2016

Representation theory over $\mathbb Z$ is famously intractable, but "representation stability" provides a way to get around these difficulties, at least asymptotically, by enlarging our groups until they behave more like commutative rings. Moreover, it turns out that important questions in topology/number theory/representation theory/... correspond to asking whether familiar algebraic properties hold for these "rings".

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.

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.

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.

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.

Nicholas Sheridan

Princeton University; Member, School of Mathematics

October 17, 2016

I will give a brief overview of symplectic mapping class groups, then explain how one can use homological mirror symmetry to get information about them. This is joint work with Ivan Smith.

Alex Kontorovich

Rutgers University; Member, School of Mathematics

April 11, 2016

We will discuss some natural problems in arithmetic that can be reformulated in terms of orbits of certain "thin" (semi)groups of integer matrix groups.

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.