Seminars and Colloquia Schedule

Probability and variational methods in PDEs — optimal transport, regularity, and universality

Series
Job Candidate Talk
Time
Tuesday, December 12, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/96443370732
Speaker
Tobias Ried – Max Planck Institute for Mathematics in the Sciences, Liepzig, Germany –
In this talk I will present an overview of my research, highlighting in more detail two topics: 
1. A purely variational approach to the regularity theory of optimal transportation, which is analogous to De Giorgi’s strategy for the regularity theory of minimal surfaces. I will show some interesting connections to Wasserstein barycenters, branched transport, and pattern formation in materials science, as well as applications in density functional theory. 
2. Variational methods for a singular stochastic PDE describing the magnetization ripple, a microstructure in thin-film ferromagnets triggered by the poly-crystallinity of the sample. I will describe how the universal character of the magnetization ripple can be addressed using variational methods based on Γ-convergence.

Sums of odd-ly many fractions and the distribution of primes

Series
Number Theory
Time
Wednesday, December 13, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Vivian Kuperberg – ETH –

Abstract: In this talk, I will discuss new bounds on constrained sets of fractions. Specifically, I will discuss the answer to the following question, which arises in several areas of number theory: For an integer $k \ge 2$, consider the set of $k$-tuples of reduced fractions $\frac{a_1}{q_1}, \dots, \frac{a_k}{q_k} \in I$, where $I$ is an interval around $0$.
How many $k$-tuples are there with $\sum_i \frac{a_i}{q_i} \in \mathbb Z$?

When $k$ is even, the answer is well-known: the main contribution to the number of solutions comes from ``diagonal'' terms, where the fractions $\frac{a_i}{q_i}$ cancel in pairs. When $k$ is odd, the answer is much more mysterious! In ongoing work with Bloom, we prove a near-optimal upper bound on this problem when $k$ is odd. I will also discuss applications of this problem to estimating moments of the distributions of primes and reduced residues.