All Seminars
| Title: Everywhere unbalanced configurations |
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| Seminar: Combinatorics |
| Speaker: Jeck Lim of Caltech |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2023-10-11 at 4:00PM |
| Venue: MSC E406 |
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| Abstract: An old problem in discrete geometry, originating with Kupitz, asks whether there is a fixed natural number $k$ such that every finite set of points in the plane has a line through at least two of its points where the number of points on either side of this line differ by at most $k$. We give a negative answer to the pseudoline variant of this problem. Joint work with David Conlon. |
| Title: Global existence for an isotropic Boltzmann-type model |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Stanley Snelson of Florida Institute of Technology |
| Contact: Maja Taskovic, maja.taskovic@emory.edu |
| Date: 2023-10-11 at 4:00PM |
| Venue: White Hall 111 |
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| Abstract: The Boltzmann equation is a kinetic differential equation that plays a central role in thermal and statistical physics. Global existence for this equation is a challenging open problem, and in this talk, we will discuss a new model equation—called the "isotropic Boltzmann equation"—that is more tractable than the Boltzmann equation while still encapsulating many of the key mathematical difficulties. After briefly deriving the model equation, we will discuss the proof of global existence, which features a surprising application of a fractional Hardy inequality. |
| Title: Have you Ever Meta-Conjectured? |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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). |