Seminars and Colloquia by Series

Egyptian fractions: problems and progress

Series
School of Mathematics Colloquium
Time
Thursday, April 27, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Thomas Bloom – University of Oxford –

The study of Egyptian fractions, representing rational numbers as the sum of distinct unit fractions, is one of the oldest areas of number theory. In this talk we will discuss some fascinating problems in the area, including both open problems and some recent progress, such as the solution to the Erdos-Graham conjecture: 1 can be written as the sum of unit fractions with denominators drawn from an arbitrary set of integers of positive density.

Geometry and dynamics of compressible fluids

Series
School of Mathematics Colloquium
Time
Thursday, March 2, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Boris Khesin – University of Toronto –

Please Note: Live-stream link: https://gatech.zoom.us/j/93100501365?pwd=bWFEeURxek5pWG1BRjN4MHcvYllYQT09 Passcode provided in talk announcement

We describe a geometric framework to study Newton's
equations on infinite-dimensional configuration spaces of
diffeomorphisms and smooth probability densities. It turns out that
several important PDEs of hydrodynamical origin can be described in
this framework in a natural way. In particular, the so-called Madelung
transform between the Schrödinger-type equations on wave functions and
Newton's equations on densities turns out to be a Kähler map between
the corresponding phase spaces, equipped with the Fubini-Study and
Fisher-Rao information metrics. This is a joint work with G.Misiolek
and K.Modin.

From Artin monoids to Artin groups

Series
School of Mathematics Colloquium
Time
Friday, December 9, 2022 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ruth Charney – Brandeis University –

Braid groups belong to a broad class of groups known as Artin groups, which are defined by presentations of a particular form and have played a major role in geometric group theory and low-dimensional topology in recent years. These groups fall into two classes, finite-type and infinte-type Artin groups. The former come equipped with a powerful combinatorial structure, known as a Garside structure, while the latter are much less understood and present many challenges. However, if one restricts to the Artin monoid, then much of the combinatorial structure still applies in the infinite-type case. In a joint project with Rachael Boyd, Rose Morris-Wright, and Sarah Rees, we use geometric techniques to study the relation between the Artin monoid and the Artin group.

Learning to Solve Hard Minimal Problems

Series
School of Mathematics Colloquium
Time
Thursday, October 13, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anton Leykin – Georgia Tech –

The main result in this talk concerns a new fast algorithm to solve a minimal problem with many spurious solutions that arises as a relaxation of a geometric optimization problem. The algorithm recovers relative camera pose from points and lines in multiple views. Solvers like this are the backbone of structure-from-motion techniques that estimate 3D structures from 2D image sequences.  

Our methodology is general and applicable in areas other than computer vision. The ingredients come from algebra, geometry, numerical methods, and applied statistics. Our fast implementation relies on a homotopy continuation optimized for our setting and a machine-learned neural network.

(This covers joint works with Tim Duff, Ricardo Fabbri, Petr Hruby, Kathlen Kohn, Tomas Pajdla, and others.

The talk is suitable for both professors and students.)

A proof of the Erdős–Faber–Lovász conjecture

Series
School of Mathematics Colloquium
Time
Thursday, May 6, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/87011170680?pwd=ektPOWtkN1U0TW5ETFcrVDNTL1V1QT09
Speaker
Tom Kelly – University of Birmingham –

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.

Macdonald and Schubert polynomials from Markov chains

Series
School of Mathematics Colloquium
Time
Thursday, April 29, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/87011170680?pwd=ektPOWtkN1U0TW5ETFcrVDNTL1V1QT09
Speaker
Lauren K. Williams – Harvard University –

Two of the most famous families of polynomials in combinatorics are Macdonald polynomials and Schubert polynomials. Macdonald polynomials are a family of orthogonal symmetric polynomials which generalize Schur and Hall-Littlewood polynomials and are connected to the Hilbert scheme.  Schubert polynomials also generalize Schur polynomials, and represent cohomology classes of Schubert varieties in the flag variety. Meanwhile, the asymmetric exclusion process (ASEP) is a model of particles hopping on a one-dimensional lattice, which was initially introduced by Macdonald-Gibbs-Pipkin to provide a model for translation in protein synthesis.  In my talk I will explain how two different variants of the ASEP have stationary distributions which are closely connected to Macdonald polynomials and Schubert polynomials, respectively.  This leads to new formulas and new conjectures.

This talk is based on joint work with Corteel-Mandelshtam, and joint work with Donghyun Kim.

A modern take on Huber regression

Series
School of Mathematics Colloquium
Time
Thursday, April 22, 2021 - 12:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/87011170680?pwd=ektPOWtkN1U0TW5ETFcrVDNTL1V1QT09
Speaker
Po-Ling Loh – University of Cambridge –

Please Note: Note the unusual time: 12:00pm.

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.

Pages