All Seminars

Title: Recovery of the nonlinearity from the modified scattering map
Seminar: Analysis and Differential Geometry
Speaker: Gong Chen of Georgia Institute of Technology
Contact: Maja Taskovic, maja.taskovic@emory.edu
Date: 2023-04-13 at 4:00PM
Venue: MSC W301
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Abstract:
We consider the problem of recovering the nonlinearity in a nonlinear Schrödinger equation from scattering data, a problem for which there is a relatively large literature. We consider a new situation in which the equation does not admit standard scattering, but instead features the modified scattering behavior with logarithmic phase correction. We prove that even in this case, the modified scattering data suffices to determine the unknown nonlinearity.
Title: High-Definition Hedgehogs
Seminar: Combinatorics
Speaker: Eoin Peter Hurley of
Contact: Liana Yepremyan, liana.yepremyan@emory.edu
Date: 2023-04-11 at 4:00PM
Venue: MSC E408
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Abstract:
Suppose I want to edge colour a complete hypergraph so that I avoid a fixed hypergraph H as a monochromatic subgraph. How large can my complete graph be? And how does the answer depend on the number of colours I use? Certainly I need to use 2 colours for the question to be non-trivial, but what if I use 3 or 5 or 173 colours, does this help? In the case of graphs, the answer is “not much”, but, in a breakthrough paper "Hedgehogs Are Not ColorBlind", Conlon, Fox and Rödl show that for a certain family of hypergraphs, called hedgehogs, it makes an exponential difference! This observation has ramifications for some of the most important questions in quantitative Ramsey theory. Further, Conlon, Fox and Rödl asked, can we replace the exponential difference by a difference of a tower of arbitrary height? We answer this in the affirmative, showing that Hedgehogs see in high definition. In this talk, I will discuss the main ideas and consequences of the result, including some surprising conjectures. Joint work with Quentin Dubroff, António Girão and Corrine Yap.
Title: Local data of elliptic curves under quadratic twists and isogeny
Seminar: Algebra
Speaker: Manami Roy of Fordham University
Contact: Andrew Kobin, ajkobin@emory.edu
Date: 2023-04-11 at 4:00PM
Venue: MSC W301
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Abstract:
For a minimal proper regular model of a rational elliptic curve E at a prime p, we can compute the local data at p, which includes the special fiber of the minimal model (i.e., Néron type), the exponent appearing at the prime p in the of the conductor of E, and the local Tamagawa number at p. We will discuss how the Kodaira-Néron types and the local Tamagawa numbers of rational elliptic curves change over isogeny graphs. To answer this question, we examine how local data of rational elliptic curves change under quadratic twists. Our aim is to answer an open problem on how the Kodaira-Néron types and the local Tamagawa numbers of isogenous rational elliptic curves with wild ramification change under 2- or 3-isogeny. This is an ongoing project with Alex Barrios, Nandita Sahajpal, Darwin Tallana, Bella Tobin, and Hanneke Wiersema.
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
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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: Range Restricted GMRES methods and some experimental considerations
Seminar: CODES@emory
Speaker: Lucas Onisk of Emory University
Contact: Matthias Chung, matthias.chung@emory.edu
Date: 2023-04-06 at 10:00AM
Venue: MSC W201
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Abstract:
The right-hand side vector of linear discrete ill-posed problems that can occur in science and engineering applications often represents an experimental measurement that is contaminated by error. The solution to these problems is typically very sensitive to this error. Previous works have shown that error propagation into the computed solution may be reduced by using specially designed iterative methods that allow the user to select the subspace in which the approximate solution is computed. Since the dimension of this subspace is often quite small, its choice is important for the quality of the computed solution. Iterative methods that modify the Generalized Minimal RESidual (GMRES) and block GMRES methods for the solution of appropriate linear systems of equations will be discussed, as well as experimental considerations.
Title: On Pisier type problems
Defense: Dissertation
Speaker: Marcelo Sales of Emory University
Contact: Marcelo Sales, marcelo.tadeu.sales@emory.edu
Date: 2023-04-05 at 4:00PM
Venue: MSC N301
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Abstract:
A subset $A$ of integers is \textit{free} if for every two distinct subsets $B, B'\subset A$ we have \[\sum_{b\in B}b\neq \sum_{b'\in B'} b'.\] Pisier asked if, for every subset $A$ of integers, the following two statements are equivalent:\\ \\ (1) $A$ is a union of finitely many free sets.\\ (2) There exists $\epsilon>0$ such that every finite subset $B\subset A$ contains a free subset $C\subset B$ with $|C|\geq \epsilon |B|$.\\ \\ In a more general framework, the Pisier question can be seen as the problem of determining if statements (1) and (2) are equivalent for subsets of a given structure with the prescribed property. We study the problem for several structures including $B_h$-sets, arithmetic progressions, independent sets in hypergraphs, and configurations in the euclidean space.
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
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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: 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
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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: 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
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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: 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
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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.