Everything you wanted to know about machine learning but didn't know whom to ask

Sanjeev Arora
Princeton University; Visiting Professor, School of Mathematics
November 27, 2017

This talk is going to be an extended and more technical version of my brief public lecture https://www.ias.edu/ideas/2017/arora-zemel-machine-learning

I will present some of the basic ideas of machine learning, focusing on the mathematical formulations. Then I will take audience questions.

Open Gopakumar-Vafa conjecture for rational elliptic surfaces

Yu-Shen Lin
Harvard University
November 27, 2017
We will explain a definition of open Gromov-Witten invariants on the rational elliptic surfaces and explain the connection of the invariants with tropical geometry. For certain rational elliptic surfaces coming from meromorphic Hitchin system, we will show that the open Gromov-Witten invariants with boundary conditions near infinity (up to some transformation) coincide with the closed geodesic counting invariants defined by Gaiotto-Moore-Neitzke, which are integer-valued.

Automorphic forms and motivic cohomology II

Akshay Venkatesh
Stanford University; Distinguished Visiting Professor, School of Mathematics
November 21, 2017

In the lectures I will formulate a conjecture asserting that there is a hidden action of certain motivic cohomology groups on the cohomology of arithmetic groups. One can construct this action, tensored with $\mathbb C$, using differential forms. Also one can construct it, tensored with $\mathbb Q_p$, by using a derived version of the Hecke algebra (or a derived version of the Galois deformation rings).

Joint equidistribution of CM points

Ilya Khayutin
Princeton University; Veblen Research Instructor, School of Mathematics
November 21, 2017

A celebrated theorem of Duke states that Picard/Galois orbits of CM points on a complex modular curve equidistribute in the limit when the absolute value of the discriminant goes to infinity. The equidistribution of Picard and Galois orbits of special points in products of modular curves was conjectured by Michel and Venkatesh and as part of the equidistribution strengthening of the André-Oort conjecture. I will explain the proof of a recent theorem making progress towards this conjecture.

A practical guide to deep learning

Richard Zemel
University of Toronto; Visitor, School of Mathematics
November 21, 2017
Neural networks have been around for many decades. An important question is what has led to their recent surge in performance and popularity. I will start with an introduction to deep neural networks, covering the terminology and standard approaches to constructing networks. I will focus on the two primary, very successful forms of networks: deep convolutional nets, as originally developed for vision problems; and recurrent networks, for speech and language tasks.

Representations of Kauffman bracket skein algebras of a surface

Helen Wong
Carleton University; von Neumann Fellow, School of Mathematics
November 20, 2017
The definition of the Kauffman bracket skein algebra of an oriented surface was originally motivated by the Jones polynomial invariant of knots and links in space, and a representation of the skein algebra features in Witten's topological quantum field theory interpretation of the Jones invariant. Later, the skein algebra and its representations was discovered to bear deep relationships to hyperbolic geometry, via the $SL_2 \mathbb C$-character variety of the surface.

How to modify the Langlands' dual group

Joseph Bernstein
Tel Aviv University; Member, School of Mathematics
November 20, 2017

Let $\mathcal G$ be a split reductive group over a $p$-adic field $F$, and $G$ the group of its $F$-points.

The main insight of the local Langlands program is that to every irreducible smooth representation $(\rho, G, V )$ should correspond a morphism $\nu_\rho : W_F \to {}^\vee G$ of the Weil group $W_F$ of the field $F$ to the Langlands' dual group $^\vee G$.

Wild harmonic bundles and related topics II

Takuro Mochizuki
Kyoto University
November 17, 2017

Harmonic bundles are flat bundles equipped with a pluri-harmonic metric. They are very useful in the study of flat bundles on complex projective manifolds. Indeed, according to the fundamental theorem of Corlette, any semisimple flat bundle on a projective manifold has a pluri-harmonic metric. Moreover, Simpson generalized many important theorems for polarizable variation of Hodge structures, such as Hard Lefschetz Theorem, to the context of harmonic bundles.