### Deep Learning: Asymptotics and Financial Applications

## Abstract

Deep learning has revolutionized image, text, and speech recognition. Motivated by this success, there is growing interest in developing deep learning methods for financial applications. We will present some of our recent results in this area, including deep learning models of high-frequency data. In the second part of the talk, we prove a law of large numbers for single-layer neural networks trained with stochastic gradient descent. We show that, depending upon the normalization of the parameters, the law of large numbers either satisfies a deterministic partial differential equation or a random ordinary differential equation. Using similar analysis, a law of large numbers can also be established for reinforcement learning (e.g., Q-learning) with neural networks. The limit equations in each of these cases are discussed (e.g., whether a unique stationary point and global convergence can be proven).

16:00

### Least dilatation of pure surface braids

## Abstract

The $n$-stranded pure surface braid group of a genus g surface can be described as the subgroup of the pure mapping class group of a surface of genus $g$ with $n$-punctures which becomes trivial on the closed surface. I am interested in the least dilatation of pseudo-Anosov pure surface braids. For the $n=1$ case, upper and lower bounds on the least dilatation were proved by Dowdall and Aougab—Taylor, respectively. In this talk, I will describe the upper and lower bounds I have proved as a function of $g$ and $n$.

### Numerical Analysis meets Topology

## Abstract

One of the fundamental tools in numerical analysis and PDE

is the finite element method (FEM). A main ingredient in

FEM are splines: piecewise polynomial functions on a

mesh. Even for a fixed mesh in the plane, there are many open

questions about splines: for a triangular mesh T and

smoothness order one, the dimension of the vector space

C^1_3(T) of splines of polynomial degree at most three

is unknown. In 1973, Gil Strang conjectured a formula

for the dimension of the space C^1_2(T) in terms of the

combinatorics and geometry of the mesh T, and in 1987 Lou

Billera used algebraic topology to prove the conjecture

(and win the Fulkerson prize). I'll describe recent progress

on the study of spline spaces, including a quick and self

contained introduction to some basic but quite useful tools

from topology.