All Seminars
| 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). |
| Title: Arithmetic Geometry and Stacky Curves |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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. |