All Seminars

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
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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
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
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).
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
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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.
Title: How to accelerate learning tasks on big data with ANOVA-based NFFT matrix vector products
Seminar: Numerical Analysis and Scientific Computing
Speaker: Theresa Wagner of University of Technology Chemnitz
Contact: Matthias Chung, matthias.chung@emory.edu
Date: 2023-09-07 at 10:00AM
Venue: MSC N306
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Abstract:
Kernel matrices are crucial in many learning tasks and typically dense and large-scale. Depending on their dimension even computing all their entries is challenging and the cost of matrix vector products scales quadratically with the dimension, if no customized methods are applied. We present a matrix-free approach that exploits the computational power of the non-equispaced fast Fourier transform (NFFT) and is of linear complexity for fixed accuracy. The ANOVA kernel has proved to be a viable tool to group the features into smaller pieces that are then amenable to the NFFT-based summation technique. Multiple kernels based on lower-dimensional feature spaces are combined, such that kernel vector products can be realized by this fast approximation algorithm. Based on a feature grouping approach this can be embedded into a CG or GMRES solver within a learning method and we nearly reach a linear scaling. This approach enables to run learning tasks using kernel methods for large-scale data on a standard laptop computer in reasonable time without or very benign loss of accuracy. It can be embedded into methods that rely on kernel matrices or even graph Laplacians. In this talk, I will demonstrate how kernel ridge regression and support vector machine tasks can benefit from having the fast matrix vector products available and will give an outlook on further applications. This is joint work with Martin Stoll, Franziska Nestler, and John Pearson.
Title: Microlocal Methods in Hyperbolic Dynamics
Seminar: Analysis and Differential Geometry
Speaker: Guangqiu Liang of Emory University
Contact: Maja Taskovic, maja.taskovic@emory.edu
Date: 2023-09-01 at 11:00AM
Venue: Atwood 240
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Abstract:
Microlocal analysis, a toolbox in linear PDE theory, has brought some recent advancements in hyperbolic dynamics, namely the study of chaotic dynamical systems. Specifically, it provides an appropriate functional-analytic framework on which the dynamics exhibit nice spectral properties. In this talk, I will introduce the dynamical zeta function for smooth Anosov flows on compact manifolds, describe its meromorphic continuation using microlocal methods, and mention some work on extracting topological information of the underlying dynamical system from the zeta function's behavior at zero.
Title: Learning probabilistic graphical models with triangular transport and a Hessian score
Seminar: Numerical Analysis and Scientific Computing
Speaker: Rebecca Morrison of UC Boulder
Contact: Matthias Chung, matthias.chung@emory.edu
Date: 2023-08-31 at 10:00AM
Venue: MSC N306
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Abstract:
Probabilistic graphical models encode the conditional independence properties satisfied by a joint probability distribution. If the distribution is Gaussian, the edges of an undirected graphical model correspond to non-zero entries of the precision matrix. Generalizing this result to continuous non-Gaussian distributions, one can show that an edge exists if and only if an entry of the Hessian of the log density is non-zero (everywhere). But evaluation of the log density requires density estimation: for this, we propose the graph-learning algorithm SING (Sparsity Identification in Non-Gaussian distributions), which uses triangular transport for the density estimation step; this choice is advantageous as triangular maps inherit sparsity from conditional independence in the target distribution. Loosely speaking, the more non-Gaussian the distribution, the more difficult the transport problem. However, for a broad class of non-Gaussian distributions, estimating the Hessian of the log density is much easier than estimating the density itself. In this talk, I'll give examples of graphs that are relatively difficult and surprisingly easy to learn, and provide some theory that justifies the easy cases.
Title: Reconstructing Random Pictures
Seminar: Combinatorics
Speaker: Corrine Yap of Georgia Tech
Contact: Liana Yepremyan, liana.yepremyan@emory.edu
Date: 2023-08-30 at 4:00PM
Venue: MSC E406
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Abstract:
Reconstruction problems ask whether or not it is possible to uniquely build a discrete structure from the collection of its substructures of a fixed size. This question has been explored in a wide range of settings, most famously with graphs and the resulting Graph Reconstruction Conjecture due to Kelly and Ulam, but also including geometric sets, jigsaws, and abelian groups. In this talk, we'll consider the reconstruction of random pictures (n-by-n grids with binary entries) from the collection of its k-by-k subgrids and prove a nearly-sharp threshold for k = k(n). Our main proof technique is an adaptation of the Peierls contour method from statistical physics. Joint work with Bhargav Narayanan.
Title: Eigenvalues of the Laplace Operator on Quantum Graphs
Defense: Dissertation
Speaker: Haozhe Yu of Emory University
Contact: Haozhe Yu, haozhe.yu@emory.edu
Date: 2023-06-20 at 3:00PM
Venue: MSC W303
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Abstract:
This thesis focuses on estimates of eigenvalues on compact quantum graphs. On quantum graphs with all standard vertex condition, we prove an upper bound of eigenvalues based on the Davies inequality. We also prove some improvements of known upper bounds for eigenvalue gaps and ratios for metric trees. We finally establish a lower bound of eigenvalue gaps based on the idea of the weighted Cheeger constant on graphs with at least one Dirichlet vertex.
Title: Direct and inverse problems for elastic dislocations in geophysics
Seminar: Analysis and Differential Geometry
Speaker: Anna L. Mazzucato of Penn State University
Contact: Yiran Wang, yiran.wang@emory.edu
Date: 2023-04-21 at 11:00AM
Venue: MSC W303
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Abstract:
I will discuss a model for dislocations in an elastic medium, modeling faults in the Earth's crust. The direct problem consists in solving a non-standard boundary value/interface problem for in-homogeneous, possibly anisotropic linear elasticity with piecewise-Lipschitz coefficients. The non-linear inverse problem consists in determining the fault surface and slip vector from displacement measurements made at the surface. In applications, these come from GPS arrays and satellite interferometry. We establish uniqueness for the inverse problem under some geometric conditions, using unique continuation results for systems. We also derive a shape derivative formula for an iterative reconstruction algorithm. This is joint work with Andrea Aspri (Milan University, Italy), Elena Beretta (NYU-Abu Dhabi), and Maarten de Hoop (Rice).