# Upcoming Seminars

Title: Extremal problems for uniformly dense hypergraphs
Seminar: Combinatorics
Speaker: Mathias Schacht of Hamburg University
Contact: Liana Yepremyan, liana.yepremyan@emory.edu
Date: 2023-03-27 at 4:30PM
Venue: MSC E408
Abstract:
Extremal combinatorics is a central research area in discrete mathematics. The field can be traced back to the work of Turán and it was established by Erd?s through his fundamental contributions and his uncounted guiding questions. Since then it has grown into an important discipline with strong ties to other mathematical areas such as theoretical computer science, number theory, and ergodic theory. We focus on extremal problems for hypergraphs, which were introduced by Turán. After solving the analogous question for graphs, Turán asked to determine the maximum cardinality of a set E of 3-element subsets of a given n-element set V such that for any 4 elements of V at least one triple is missing in E. This innocent looking problem is still open and despite a great deal of effort over the last 80 years and our knowledge is still somewhat limited. We consider a variant of the problem by imposing additional restrictions on the distribution of the 3-element subsets in E. These additional assumptions yield a finer control over the corresponding extremal problem. In fact, this leads to many interesting and more manageable subproblems, some of which were already considered by Erd?s and Sós in the 1980ies. The additional assumptions on the distribution of the 3-element subsets are closely related to the theory of quasirandom discrete structures, which was pioneered by Szemerédi and became a central theme in the field. In fact, the hypergraph extensions by Gowers and by Rödl et al. of the regularity lemma provide essential tools for this line of research.
Title: Extremal problems for uniformly dense hypergraphs
Seminar: Combinatorics
Speaker: Mathias Schacht of Hamburg University
Contact: Liana Yepremyan, liana.yepremyan@emory.edu
Date: 2023-03-27 at 4:30PM
Venue: MSC E408
Abstract:
Extremal combinatorics is a central research area in discrete mathematics. The field can be traced back to the work of Turán and it was established by Erd?s through his fundamental contributions and his uncounted guiding questions. Since then it has grown into an important discipline with strong ties to other mathematical areas such as theoretical computer science, number theory, and ergodic theory. We focus on extremal problems for hypergraphs, which were introduced by Turán. After solving the analogous question for graphs, Turán asked to determine the maximum cardinality of a set E of 3-element subsets of a given n-element set V such that for any 4 elements of V at least one triple is missing in E. This innocent looking problem is still open and despite a great deal of effort over the last 80 years and our knowledge is still somewhat limited. We consider a variant of the problem by imposing additional restrictions on the distribution of the 3-element subsets in E. These additional assumptions yield a finer control over the corresponding extremal problem. In fact, this leads to many interesting and more manageable subproblems, some of which were already considered by Erd?s and Sós in the 1980ies. The additional assumptions on the distribution of the 3-element subsets are closely related to the theory of quasirandom discrete structures, which was pioneered by Szemerédi and became a central theme in the field. In fact, the hypergraph extensions by Gowers and by Rödl et al. of the regularity lemma provide essential tools for this line of research.
Title: Counting low degree number fields with almost prescribed successive minima
Seminar: Algebra
Speaker: Sameera Vemulapalli of Princeton University
Contact: Andrew Kobin, ajkobin@emory.edu
Date: 2023-03-28 at 4:00PM
Venue: MSC W301
Abstract:
The successive minima of an order in a degree n number field are n real numbers encoding information about the Euclidean structure of the order. How many orders in degree n number fields are there with almost prescribed successive minima, fixed Galois group, and bounded discriminant? In this talk, I will address this question for $n = 3,4,5$. The answers, appropriately interpreted, turn out to be piecewise linear functions on certain convex bodies. If time permits, I will also discuss a geometric analogue of this problem: scrollar invariants of covers of $\mathbb{P}^1$.
Title: Balancing the stability-accuracy Trade-off in Neural Networks for Ill-conditioned Inverse Problems
Seminar: CODES@Emory
Speaker: Davide Evangelista of University of Bologna
Contact: Jim Nagy, jnagy@emory.edu
Date: 2023-03-30 at 10:00AM
Venue: MSC W201
Abstract:
Deep learning algorithms have recently become state-of-art in solving Inverse Problems, overcoming the classical variational methods in terms of both accuracy and efficiency. On the other hand, it is still unclear if neural networks can compete in terms of reliability and a rigorous complete analysis still lacks in the literature. Starting from the brilliant works of N.M.Gottschling, V.Antun (2020) and M.J.Colbrook, V.Antun (2021), we will try to understand the relationship between the accuracy and stability of neural networks for solving ill-conditioned inverse problems, deriving new theoretical results shedding light on the trade-off between accuracy and stability. Following the study of M.Genzel, J.Macdonald (2020), we will find that, under some conditions, neural networks can be more unstable the more they are accurate, and we will propose new regularization techniques with provable increase in stability and minumum accuracy loss.
Title: Motivic Euler characteristics and the transfer
Seminar: Algebra
Speaker: Roy Joshua of Ohio State University
Contact: Andrew Kobin, ajkobin@emory.edu
Date: 2023-04-04 at 4:00PM
Venue: MSC W301
Abstract:
In the first part of the talk, we will consider motivic Euler characteristics of certain homogeneous spaces as they relate to splittings in the motivic homotopy category. In the second part of the talk, we will discuss certain applications of these to computations in Algebraic K-Theory and Brauer groups.
Title: The Calderon Problem: 40 Years Later
Colloquium: Analysis and Differential Geometry
Speaker: Gunther Uhlmann of University of Washington
Contact: Yiran Wang, yiran.wang@emory.edu
Date: 2023-04-07 at 10:00AM
Venue: MSC W303
Abstract:
Calderon's inverse problem asks whether one can determine the conductivity of a medium by making voltage and current measurements at the boundary. This question arises in several areas of applications including medical imaging and geophysics. I will report on some of the progress that has been made on this problem since Calderon proposed it, including recent developments on similar problems for nonlinear equations and nonlocal operators.
Title: TBD
Seminar: Algebra
Speaker: Manami Roy of Fordham University
Contact: Andrew Kobin, ajkobin@emory.edu
Date: 2023-04-11 at 4:00PM
Venue: MSC W301
Abstract:
TBD
Title: TBD
Seminar: Algebra
Speaker: James Phillips of Seton Hall University
Contact: Andrew Kobin, ajkobin@emory.edu
Date: 2023-04-18 at 4:00PM
Venue: MSC W301
Abstract:
TBD