All Seminars

Title: Nonlocal PDEs and Quantum Optics
Colloquium: Analysis and Differential Geometry
Speaker: John Schotland of Yale University
Contact: Yiran Wang, yiran.wang@emory.edu
Date: 2023-10-20 at 2:00PM
Venue: MSC W301
Download Flyer
Abstract:
Quantum optics is the quantum theory of the interaction of light and matter. In this talk, I will describe a real-space formulation of quantum electrodynamics with applications to many body problems. The goal is to understand the transport of nonclassical states of light in random media. In this setting, there is a close relation to kinetic equations for nonlocal PDEs with random coefficients.
Title: Have you Ever Meta-Conjectured?
Seminar: Combinatorics
Speaker: Ron Gould of Emory University
Contact: Liana Yepremyan, liana.yepremyan@emory.edu
Date: 2023-10-04 at 4:00PM
Venue: MSC E406
Download Flyer
Abstract:
The study of cycles in graphs has a long history. In 1971 A. Bondy noted a tie linking hamiltonian graphs and pancyclic graphs. He stated his famous meta-conjecture: Almost any nontrivial condition on a graph which implies the graph is hamiltonian also implies the graph is pancyclic. There may be some simple family of exceptional graphs. A cycle contains a chord if there exists an edge between two vertices of the cycle that is not an edge of the cycle. A cycle is said to be chorded if it has one or more chords. In this talk I will extend Bondy's meta-conjecture in several ways to a broader class of cycle problems in graphs, namely to finding conditions that imply the existence of chorded cycles in graphs. I will offer supporting evidence to these meta-conjectures.
Title: Bayesian Filtering Methods for Dynamic Parameter Estimation in Differential Equations
Seminar: Numerical Analysis and Scientific Computing
Speaker: Andrea Arnold of Worcester Polytechnic Institute
Contact: Matthias Chung, matthias.chung@emory.edu
Date: 2023-10-03 at 10:00AM
Venue: MSC N306
Download Flyer
Abstract:
Estimating and quantifying uncertainty in unknown system parameters from partial, noisy system measurements remains a challenging inverse problem. In addition to constant parameters, a variety of systems stemming from real-world applications include unobservable parameters that change with time but have unknown evolution models. In this talk, we present several approaches using Bayesian filtering techniques to estimate time-varying parameters in deterministic dynamical systems governed by differential equations.
Title: Local-global principles for reductive groups over finitely generated fields
Seminar: Algebra
Speaker: Igor Rapinchuk of Michigan State University
Contact: Andrew Kobin, andrew.jon.kobin@emory.edu
Date: 2023-10-03 at 4:00PM
Venue: MSC N302
Download Flyer
Abstract:
One of the major results in the arithmetic theory of algebraic groups is the validity of the cohomological local-global (or Hasse) principle for simply-connected and adjoint semisimple groups over number fields. Over the last several years, there has been growing interest in studying Hasse principles for reductive groups over arbitrary finitely generated fields with respect to suitable sets of discrete valuations. In particular, we have conjectured that for divisorial sets, the corresponding Tate-Shafarevich set, which measures the deviation from the local-global principle, should be finite for all reductive groups. I will report on recent progress on this conjecture, focusing in particular on the case of algebraic tori as well as on connections to groups with good reduction. This talk is based on joint work with V. Chernousov and A. Rapinchuk.
Title: Weighted X-ray mapping properties on the Euclidean and Hyperbolic Disks
Seminar: Analysis and Differential Geometry
Speaker: Joey Zou of Northwestern University
Contact: Yiran Wang, yiran.wang@emory.edu
Date: 2023-09-29 at 11:00AM
Venue: Atwood 240
Download Flyer
Abstract:
We discuss recent works studying the sharp mapping properties of weighted X-ray transforms and weighted normal operators. These include a C^\infty isomorphism result for certain weighted normal operators on the Euclidean disk, whose proof involves studying the spectrum of a distinguished Keldysh-type degenerate elliptic differential operator; we also describe mapping properties for the weighted normal operator in terms of Sobolev-type spaces adapted to this distinguished differential operator. In addition, we discuss ongoing work which applies these results to the X-ray transform on the hyperbolic disk by using a projective equivalence between the Euclidean and hyperbolic disks. Joint works with N. Eptaminitakis, R. K. Mishra, and F. Monard.
Title: Maximising copies of H in clique-free graphs
Seminar: Combinatorics
Speaker: Natasha Morrison of University of Victoria
Contact: Liana Yepremyan, liana.yepremyan@emory.edu
Date: 2023-09-27 at 4:00PM
Venue: MSC E406
Download Flyer
Abstract:
Let H be a graph. We show that if r is large enough as a function of H, then the r-partite Turán graph maximizes the number of copies of H among all Kr+1-free graphs on a given number of vertices. This confirms a conjecture of Gerbner and Palmer. This is joint work with JD Nir, Sergey Norin, Pawel Rzazewski, Alexandra Wesolek.
Title: Convex holes and almost uniform distribution in the unit cube
Seminar: Atlanta Discrete Analysis
Speaker: Boris Bukh of Carnegie Mellon University
Contact: Cosmin Pohoata, cosmin.pohoata@emory.edu
Date: 2023-09-25 at 4:00PM
Venue: MSC W301
Download Flyer
Abstract:
For $P \subset \mathbb{R}^{d}$, a hole is any set of convexly independent points whose convex hull contains no other points. We will discuss constructions of large finite sets that contain no large holes. The key role will be played by subsets of $[0,1]^d$ that contain about the same number of points in every dyadic box of a fixed volume. Based on joint works with Ting-Wei Chao and Ron Holzman.
Title: Recovery of time-dependent coefficients in hyperbolic equations on conformally transversally anisotropic manifolds from partial data
Seminar: Analysis and Differential Geometry
Speaker: Boya Liu of North Carolina State University
Contact: Yiran Wang, yiran.wang@emory.edu
Date: 2023-09-22 at 11:00AM
Venue: Atwood 240
Download Flyer
Abstract:
In this talk we discuss inverse problems of determining time-dependent coefficients appearing in the wave equation in a compact Riemannian manifold of dimension three or higher. More specifically, we are concerned with the case of conformally transversally anisotropic manifolds, or in other words, compact Riemannian manifolds with boundary conformally embedded in a product of the Euclidean line and a transversal manifold. With an additional assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove that the knowledge of a certain partial Cauchy data set determines time-dependent coefficients of the wave equation uniquely on a space-time cylinder. We shall discuss two problems: (1) Recovery of a potential appearing in the wave equation, with the Dirichlet value, in the Cauchy data, measured on only part of the lateral boundary of the space-time cylinder. (2) Recovery of both a damping coefficient and a potential appearing in the wave equation, with the Dirichlet value measured on the whole lateral boundary. This talk is based on joint works with Teemu Saksala (NC State University) and Lili Yan (University of Minnesota).
Title: Efficient solvers for Gaussian processes and Bayesian inverse problems
Seminar: Numerical Analysis and Scientific Computing
Speaker: Arvind Saibaba of North Carolina State University
Contact: Elizabeth Newman, elizabeth.newman@emory.edu
Date: 2023-09-21 at 10:00AM
Venue: MSC N306
Download Flyer
Abstract:
Gaussian processes (GPs) play an important role in many areas of scientific computing such as uncertainty quantification, reduced order modeling, and scientific machine learning. We consider the stochastic partial differential equation approach to GPs, where a major computational bottleneck is computing with fractional powers of elliptic differential operators that define the covariance operators of the GPs. We show how to address this computational challenge using an integral formulation for the fractional operator and efficient iterative methods for handling the resulting discretized system. The resulting approach makes it feasible to use GPs as priors in Bayesian inverse problems, which we demonstrate through synthetic and real-world inverse problems. We will also discuss a reduced basis approach for efficient sampling from GPs, where the covariance operator may be parameterized by multiple hyperparameters. This is joint work with Harbir Antil (George Mason).
Title: Arithmetic Geometry and Stacky Curves
Seminar: Algebra
Speaker: Andrew Kobin of Emory University
Contact: Andrew Kobin, ajkobin@emory.edu
Date: 2023-09-12 at 4:00PM
Venue: MSC N302
Download Flyer
Abstract:
Solutions to many problems in number theory can be described using the theory of algebraic stacks. In this talk, I will describe a few Diophantine equations, such as the ``generalized Fermat equation'' $Ax^{p} + Bx^{q} = Cz^{r}$, whose integer solutions can be found using an appropriate stacky curve: a curve with extra automorphisms at prescribed points. I will also describe how stacky curves can be used to study rings of modular forms both classically and in characteristic $p$. Parts of the talk are joint work in progress with Juanita Duque-Rosero, Chris Keyes, Manami Roy, Soumya Sankar and Yidi Wang, and separately with David Zureick-Brown.