All Seminars
| Title: Bounds for subsets of F_p^n x F_p^n without L-shaped configurations |
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| Seminar: Discrete Analysis |
| Speaker: Sarah Peluse of University of Michigan |
| Contact: Cosmin Pohoata, cosmin.pohoata@emory.edu |
| Date: 2023-10-30 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: I will discuss the difficult problem of proving reasonable bounds in the multidimensional generalization of Szemeredi’s theorem and describe a proof of such bounds for sets lacking nontrivial configurations of the form (x,y), (x,y+z), (x,y+2z), (x+z,y) in the finite field model setting. |
| Title: Local and global boundary rigidity |
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| Colloquium: Analysis and Differential Geometry |
| Speaker: Plamen Stefanov of Purdue University |
| Contact: Yiran Wang, yiran.wang@emory.edu |
| Date: 2023-10-27 at 2:00PM |
| Venue: MSC W301 |
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| Abstract: The boundary rigidity problem consists of recovering a Riemannian metric in a domain, up to an isometry, from the distance between boundary points. We show that in dimensions three and higher, knowing the distance near a fixed strictly convex boundary point allows us to reconstruct the metric inside the domain near that point, and that this reconstruction is stable. We also prove semi-global and global results under certain an assumption of the existence of a strictly convex foliation. The problem can be reformulated as a recovery of the metric from the arrival times of waves between boundary points; which is known as travel-time tomography. The interest in this problem is motivated by imaging problems in seismology: to recover the sub-surface structure of the Earth given travel-times from the propagation of seismic waves. In oil exploration, the seismic signals are man-made and the problem is local in nature. In particular, we can recover locally the compressional and the shear wave speeds for the elastic Earth model, given local information. The talk is based on joint work with G.Uhlmann (UW-Seattle) and A.Vasy (Stanford). We will also present results for a recovery of a Lorentzian metric from red shifts motivated by the problem of observing cosmic strings. This work was featured in the news section of Nature and got recently a Frontiers of Science Award. |
| Title: A mean-field games laboratory for generative modeling |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Benjamin Zhang of University of Massachusetts Amherst |
| Contact: Lars Ruthotto, lruthotto@emory.edu |
| Date: 2023-10-26 at 10:00AM |
| Venue: MSC N306 |
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| Abstract: We demonstrate the versatility of mean-field games (MFGs) as a mathematical framework for explaining, enhancing, and designing generative models. We establish connections between MFGs and major classes of flow and diffusion-based generative models by deriving continuous-time normalizing flows, score-based models, and Wasserstein gradient flows through different choices of particle dynamics and cost functions. Furthermore, we study the mathematical structure and properties of each generative model by examining their associated MFG's optimality condition, which consist of a set of coupled forward-backward nonlinear partial differential equations. The optimality conditions of MFGs also allow us to introduce HJB regularizers for enhanced training of a broad class of generative models. We present this framework as an MFG laboratory which serves as a platform for revealing new avenues of experimentation and invention of generative models. |
| Title: The asymptotics of $r(4,t)$ |
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| Seminar: Combinatorics |
| Speaker: Sam Mattheus of UVB |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2023-10-25 at 4:00PM |
| Venue: MSC E406 |
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| Abstract: I will give an overview of recent work, joint with Jacques Verstraete, where we gave an improved lower bound for the off-diagonal Ramsey number $r(4,t)$, solving a long-standing conjecture of Erd\H{o}s. Our proof has a strong non-probabilistic component, in contrast to previous work. This approach was generalized in further work with David Conlon, Dhruv Mubayi and Jacques Verstraete to off-diagonal Ramsey numbers $r(H,t)$ for any fixed graph $H$. We will go over of the main ideas of these proofs and indicate some open problems. |
| Title: Additive smoothing in sets of small doubling |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Giorgis Petridis of University of Georgia |
| Contact: Cosmin Pohoata, cosmin.pohoata@emory.edu |
| Date: 2023-10-23 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: A useful principle is that taking convolutions tends to `smoothen’ functions. We will explore this principle in the context of characteristic functions of finite sets and get a glimpse of its applications to additive number theory. |
| Title: Nonlocal PDEs and Quantum Optics |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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) |