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
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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: Mathematics for Remote Sensing and Earth Observation
Seminar: Numerical Analysis and Scientific Computing
Speaker: Cristina Sgattoni of CNR Florence
Contact: Matthias Chung, matthias.chung@emory.edu
Date: 2023-10-19 at 10:00AM
Venue: MSC N306
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Abstract:
FORUM (Far-infrared Outgoing Radiation Understanding and Monitoring) is a satellite mission selected in 2019 as the ninth ESA (European Space Agency) Earth Explorer mission. FORUM will provide interferometric measurements in the spectral interval encompassing the Far-InfraRed (FIR) part of the spectrum, responsible for about 50% of the outgoing longwave flux lost by our planet into space. While more accurate measurements of the Top Of the Atmosphere (TOA) resolved spectrum in the FIR are necessary for reducing uncertainty in climate models, existing instruments are insufficient, necessitating the use of innovative computational techniques. The new observations will also improve the knowledge of several atmospheric variables, such as tropospheric water vapor, ice cloud properties and, especially, surface emissivity in the FIR. In the early stages of the mission development, an End-to-End Simulator (E2ES) was devised to demonstrate proof-of-concept and to evaluate the impact of instrument characteristics and scene conditions on the accuracy of the reconstructed atmospheric properties. The atmospheric components retrieval is obtained through inversion of the radiative transfer equation, in which the atmospheric state that best reconstructs the simulated measured spectrum is determined at each step. This is a severely ill-conditioned problem and requires the application of the Optimal Estimation (OE) approach, a specialized Tikhonov regularization scheme based on a Bayesian formulation. Additional regularization, based on the Iterative Variable Strength (IVS), is often necessary to regularize unphysical oscillations that may arise during the retrieval process. In the first part of this seminar, I will focus on the retrieval of the surface emissivity, in particular on the choice of the retrieval grid step and the IVS parameters, using the FORUM simulated measurements in different latitude bands. In the second part, I will discuss the sensitivity of the FORUM simulated measurements to surface emissivity across all latitudes in clear sky conditions and in the presence of clouds in Antarctica. Moreover, I will present procedures for the assimilation of observed data and Bayesian techniques for deriving a database of surface emissivity estimates to adopt as apriori data in the OE procedure. Finally, I will conclude by introducing my future work at Emory, which consists of the use of a fast neural network approach combined with autoencoders to face both the radiative transfer problem and its inversion.
Title: Reduction of post-critically finite polynomials
Seminar: Algebra
Speaker: Bella Tobin of Agnes Scott College
Contact: Andrew Kobin, andrew.jon.kobin@emory.edu
Date: 2023-10-17 at 4:00PM
Venue: MSC W301
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Abstract:
Post-critically finite rational maps are often thought of as a dynamical analog of CM abelian varieties. We study reduction properties of post-critically finite polynomials in contrast to the reduction properties of CM abelian varieties.
Title: Reduced-Order Models for Parametrized PDE Models with Constraints
Seminar: Numerical Analysis and Scientific Computing
Speaker: Howard Elman of University of Maryland, College Park
Contact: Matthias Chung, matthias.chung@emory.edu
Date: 2023-10-13 at 10:00AM
Venue: MSC N304
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Abstract:
Model order reduction techniques are effective for solving parametrized models involving PDEs, including models of incompressible flow, where the constraint is the incompressibility constraint, and in optimal control, where the constraints themselves are PDEs. However, reduced models may fail to be inf-sup stable. We present a new approach for generating reduced bases in this scenario, using a so-called stacked reduced basis, which avoids some of the difficulties associated with inf-sup stability. We show that this approach is effective although in some circumstances it also requires stabilization, which can be done using either classic methods of penalization or through Petrov-Galerkin methods. Computational tests are presented for models based on PDE-constrained optimization and incompressible flow. This is joint work with Kayla Davie, Applied Mathematics Program, University of Maryland at College Park
Title: Regularity estimates for elliptic transmission problems
Seminar: Analysis and Differential Geometry
Speaker: Pablo Raúl Stinga of Iowa State University
Contact: Maja Taskovic, maja.taskovic@emory.edu
Date: 2023-10-13 at 2:00PM
Venue: MSC W301
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Abstract:
We present regularity estimates for solutions to transmission problems driven by second order elliptic equations with curved interfaces. First, we consider a transmission problem for harmonic functions and use the mean value theorem to prove sharp $C^{1,\alpha}$ estimates up to the transmission surface. Then, we show various up to the boundary Hölder regularity estimates for viscosity solutions to transmission problems for fully nonlinear uniformly elliptic equations depending on the regularity of the interface. Among the main tools, we introduce an ABP estimate for the problem and new constructive stability results. These are joint works with Luis A. Caffarelli (UT Austin) and María Soria-Carro (Rutgers)
Title: Global existence for an isotropic Boltzmann-type model
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: Everywhere unbalanced configurations
Seminar: Algebra
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: 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
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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
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
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.