Georgia Institute of Technology College of Sciences

The Erdős–Faber–Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$.  In joint work with Dong Yeap Kang, Daniela Kühn, Abhishek Methuku, and Deryk Osthus, we proved this conjecture for every sufficiently large $n$.  In this talk, I will present the history of this conjecture and sketch our proof in a special case.

In the first part of the talk, we discuss the use of a penalized Huber M-estimator for high-dimensional linear regression. We explain how a fairly straightforward analysis yields high-probability error bounds that hold even when the additive errors are heavy-tailed. However, the parameter governing the shape of the Huber loss must be chosen in relation to the scale of the error distribution. We discuss how to use an adaptive technique, based on Lepski's method, to overcome the difficulties traditionally faced by applying Huber M-estimation in a context where both location and scale are unknown.

In the second part of the talk, we turn to a more complicated setting where both the covariates and responses may be heavy-tailed and/or adversarially contaminated. We show how to modify the Huber regression estimator by first applying an appropriate "filtering" procedure to the data based on the covariates. We prove that in low-dimensional settings, this filtered Huber regression estimator achieves near-optimal error rates. We further show that the commonly used least trimmed squares and least absolute deviation estimators may similarly be made robust to contaminated covariates via the same covariate filtering step. This is based on joint work with Ankit Pensia and Varun Jog.

A covering system of the integers is a finite collection of arithmetic progressions whose union is the integers. The study of these objects was initiated by Erdős in 1950, and over the following decades he asked a number of beautiful questions about them. Most famously, his so-called "minimum modulus problem" was resolved in 2015 by Hough, who proved that in every covering system with distinct moduli, the minimum modulus is at most $10^{16}$. 

In this talk I will present a variant of Hough's method, which turns out to be both simpler and more powerful. In particular, I will sketch a short proof of Hough's theorem, and discuss several further applications. I will also discuss a related result, proved using a different method, about the number of minimal covering systems.

Joint work with Paul Balister, Béla Bollobás, Julian Sahasrabudhe and Marius Tiba.

Piecewise polynomials with certain global smoothness can be given by traditional finite element methods and also by neural networks with some power of ReLU as activation function. In this talk, I will present some recent results on the connections between finite element and neural network functions and comparative studies of their approximation properties and applications to numerical solution of partial differential equations of high order and/or in high dimensions.