All Seminars
| Title: Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Le Chen of Auburn University |
| Contact: Yiran Wang, yiran.wang@emory.edu |
| Date: 2023-11-03 at 11:00AM |
| Venue: Atwood 240 |
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| Abstract: This talk consists of two parts of about the same length. In the first part, we make a general introduction of stochastic partial differential equations (SPDEs) from our own perspective. We will particularly emphasize their deep relationship with statistical physics. This part is intended to be accessible to the general audience. In the second part, we will focus on the nonlinear parabolic stochastic PDEs on a bounded Lipschitz domain driven by a Gaussian noise that is white in time and colored in space, with Dirichlet or Neumann boundary condition. We establish existence, uniqueness and moment bounds of the random field solution under measure-valued initial data $\nu$. We also study the two-point correlation function of the solution and obtain explicit upper and lower bounds. For $C^{1, \alpha}$-domains with Dirichlet condition, the initial data $\nu$ is not required to be a finite measure and the moment bounds can be improved under the weaker condition that the leading eigenfunction of the differential operator is integrable with respect to $|\nu|$. As an application, we show that the solution is fully intermittent for sufficiently high level $\lambda$ of noise under the Dirichlet condition, and for all $\lambda > 0$ under the Neumann condition. The second part of the talk is based on a recent joint-work with David Candil and Cheuk-Yin Lee to appear at Stochastic PDE: Analysis and Computations, 2023 (preprint available at arXiv:2301:06435). |
| Title: Estimating Kernel Matrix Eigenvalues |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Mikhail Lepilov of Emory University |
| Contact: Matthias Chung, matthias.chung@emory.edu |
| Date: 2023-11-02 at 10:00AM |
| Venue: MSC N306 |
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| Abstract: Kernel matrices have appeared over the past few decades as intermediate structures when computing with "big data," such as during support vector machine classification or kernel ridge regression. Naive matrix algorithms quickly become too computationally intensive once such matrices reach moderate size; in fact, even explicitly forming such matrices is undesirable when the number of points is large. Hence, various low-rank approximations to such matrices become indispensable. If the underlying points come from the real world, however, it is a priori not often clear what the numerical rank of the resulting kernel matrix is for a given tolerance: existing methods like rank-revealing QR factorization or its randomized variants only apply in the case when the full matrix to be approximated has already been formed. In this work, we attempt to approximate the spectral decay of a kernel matrix that comes from a known distribution of points by that of a smaller matrix formed by sampling a few points from a related distribution. To do so, we use only information about the distribution and the analytical properties of the kernel. We explore how and when this may yield a useful approximation of the full spectrum using various sampling schemes. |
| Title: A Chebotarev Density Theorem over Local Fields |
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| Seminar: Algebra |
| Speaker: John Yin of University of Wisconsin |
| Contact: Andrew Kobin, ajkobin@emory.edu |
| Date: 2023-10-31 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: I will present an analog of the Chebotarev Density Theorem which works over local fields. As an application, I will use it to prove a conjecture of Bhargava, Cremona, Fisher, and Gajovi?. This is joint work with Asvin G and Yifan Wei. |
| 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. |