All Seminars
| Title: Hyper-Differential Sensitivity Analysis with Respect to Model Discrepancy |
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| Seminar: Computational and Data-Enabled Science |
| Speaker: Joseph Hart of Sandia National Laboratories |
| Contact: Elizabeth Newman, elizabeth.newman@emory.edu |
| Date: 2023-03-23 at 10:00AM |
| Venue: MSC W201 |
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| Abstract: Mathematical models are a building block for computational science and are a foundational tool to support decision-making. Outer loop analysis such as optimization is crucial to support decisions related to system design/control or estimation of unobservable processes. However, models are imperfect representations of complex physical processes and often require simplifications to achieve computational efficiency. The discrepancy between models and the physical system is frequently amplified by outer loop analysis such as optimization. As a result, the optimal solution determined from simplified or reduced models is insufficient to support critical decisions. We present a novel approach to compute the sensitivity of optimization problems with respect to model discrepancy and use this information to improve the optimal solution. By posing a Bayesian inverse problem to calibrate the discrepancy, we compute a posterior discrepancy distribution and then propagate it through post-optimality sensitivities to compute a posterior distribution on the optimal solution. In this presentation, we will introduce the mathematical foundations of hyper-differential sensitivity analysis with respect to model discrepancy, discuss its computational benefits, present results showing how limited high-fidelity data can significantly improve the optimal solution, and as time permits, discuss ongoing work exploring optimal data collection strategies to maximize improvements in the optimal solution. |
| Title: A local-global principle for adjoint groups over function fields of p-adic curves |
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| Defense: Dissertation |
| Speaker: Jack Barlow of Emory University |
| Contact: Jack Barlow, jack.barlow@emory.edu |
| Date: 2023-03-23 at 2:30PM |
| Venue: MSC E406 |
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| Abstract: Let $k$ be a number field and $G$ a semisimple simply connected linear algebraic group over $k$. The Kneser conjecture states that the Hasse principle holds for principal homogeneous spaces under $G$. Kneser's conjecture is a theorem due to Kneser for all classical groups, Harder for exceptional groups other than $E_8$, and Chernousov for $E_8$. It has also been proved by Sansuc that if $G$ is an adjoint linear algebraic group over $k$, then the Hasse principle holds for principal\\ homogeneous spaces under $G$.\\ \par Now let $p\in\mathbb{N}$ be a prime with $p\neq 2$, and let $K$ be a $p$-adic field. Let $F$ be the function field of a curve over $K$. Let $\Omega_F$ be the set of all divisorial discrete valuations of $F$. It is a conjecture of Colliot-Thélène, Parimala and Suresh that if $G$ is a semisimple simply connected linear algebraic group over $F$, then the Hasse principle holds for principal homogeneous spaces under $G$. This conjecture has been proved for all groups of classical type. In this talk, we ask whether the Hasse principle holds for adjoint groups over $F$, motivated by the number field case. We give a positive answer to this question for a class of adjoint classical groups. |
| Title: Low-Rank Exploiting Optimization Methods for Inverse Problems and Machine Learning |
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| Defense: Dissertation |
| Speaker: Kelvin Kan of Emory University |
| Contact: Kelvin Kan, kelvin.kan@emory.edu |
| Date: 2023-03-22 at 12:00PM |
| Venue: MSC E308A |
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| Abstract: Due to rapid technological development, datasets of enormous size have emerged in various domains, including inverse problems and machine learning. Many important applications in these domains, e.g. PDE parameter estimation, data classification and regression, are formulated as optimization problems. These problems are often of large-scale and can be computationally intractable to solve. Fortunately, it has been empirically observed that large datasets can be accurately estimated by low-rank approximation. Specifically, they can be approximately expressed using a relatively compact representation whose computation is less demanding. Therefore, an effective way to circumvent the computational obstacle is to exploit the low-rank approximation. In addition, low-rank approximation can serve as a regularization technique to filter out irrelevant features (e.g. noise) from the data since it can capture essential features while discarding less pertinent ones.\\ \\ This dissertation presents three applications of low-rank exploiting optimization methods for inverse problems and optimization. The first application is a projected Newton-Krylov method which efficiently exploits the low-rank approximation to the Hessian matrix to compute the projection for bound-constrained optimization problems. The second application is a modified Newton-Krylov method geared toward log-sum-exp minimization. It is scalable to large problem sizes thanks to its utilization of the low-rank approximation to the Hessian. In the third application, we apply hybrid regularization, which synergistically combines iterative low-rank approximation schemes and Tikhonov regularization, to effectively and automatically avoid an undesirable phenomenon in machine learning. |
| Title: A volcanic approach to CM points on Shimura curves |
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| Seminar: Algebra |
| Speaker: Freddy Saia of University of Georgia |
| Contact: Andrew Kobin, ajkobin@emory.edu |
| Date: 2023-03-21 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: A CM component of the $\ell$-isogeny graph of elliptic curves has a particular structure, that of an $\ell$-volcano, at least away from certain CM orders. The structure of “isogeny volcanoes’’ has seen much use in the study of CM elliptic curves over finite fields, originating with 1996 PhD thesis work of Kohel. Recent work of Clark—Saia leverages infinite depth versions of these graphs to study moduli of isogenies of CM elliptic curves over $\overline{\mathbb{Q}}$. We will discuss an analogue of this work for abelian surfaces with quaternionic multiplication. A main result is an algorithm to compute the $o$-CM locus on the Shimura curve $X_0^D(N)$ over $\mathbb{Q}$, for $o$ any imaginary quadratic order and $\textrm{gcd}(D,N) = 1$. As an application, we give an explicit list of pairs $(D,N)$ for which the Shimura curves $X_0^D(N)$ and $X_1^D(N)$ may fail to have a sporadic CM point. |
| Title: Random graphs and Suprema of stochastic processes |
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| Colloquium: Combinatorics |
| Speaker: Huy Pham of Stanford University |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2023-03-20 at 4:00PM |
| Venue: Atwood 215 |
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| Abstract: The threshold phenomenon is a central direction of study in probabilistic combinatorics, particularly the study of random graphs, and in theoretical computer science. The threshold of an increasing graph property (or more generally an increasing boolean function) is the density at which a random graph (or a random set) transitions from unlikely satisfying to likely satisfying the property (or the function). Kahn and Kalai conjectured that this threshold is always within a logarithmic factor of the expectation threshold, a natural lower bound to the threshold which is often much easier to compute. In probabilistic combinatorics and random graph theory, the Kahn—Kalai conjecture directly implies a number of difficult results, such as Shamir’s problem on hypergraph matchings. I will discuss joint work with Jinyoung Park that resolves the Kahn—Kalai conjecture. I will also discuss recent joint work with Vishesh Jain that resolves a conjecture of Johansson, Keevash, and Luria and Simkin on the threshold for containment of Latin squares and Steiner triple systems, and joint work with Ashwin Sah, Mehtaab Sawhney, Michael Simkin on thresholds in robust settings. Zooming into finer details of random graphs beyond the threshold phenomenon, I will touch on nonlinear large deviation results for subgraph counts and connections to sparse regularity obtained in joint work with Nicholas Cook and Amir Dembo. Interestingly, the proof of the Kahn—Kalai conjecture is closely related to our resolution of a conjecture of Talagrand on extreme events of suprema of certain stochastic processes driven by sparse Bernoulli random variables (known as selector processes), and a question of Talagrand on suprema of general positive empirical processes. These conjectures play an important role in generalizing the study of suprema of stochastic processes beyond the Gaussian case, and given recent advances on chaining and the resolution of the (generalized) Bernoulli conjecture, our results give the first steps towards Talagrand’s last ``Unfulfilled dreams’’ in the study of suprema of general stochastic processes. |
| Title: Dr. Screencast or How I Learned to Stop Worrying and Love Making Videos |
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| Seminar: Teaching and Pedagogy |
| Speaker: Michael Santana of Grand Valley State University |
| Contact: Neha Gupta, neha.gupta@emory.edu |
| Date: 2023-03-17 at 10:00AM |
| Venue: MSC E406 |
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| Abstract: As the animated movie, An American Tail, once said "Never say never". In 2013, I swore I would never use videos in my classroom, and now 10 years later, I am one of the most prolific screencast creators at Grand Valley State University. In this talk, I'll briefly discuss my journey from hating the idea of videos to where I am now, and I hope to give a lot of different ideas on how to use videos in and out of the classroom. |
| Title: Robust sublinear expanders |
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| Colloquium: Combinatorics |
| Speaker: Matija Bucic of Institute of Advanced Study |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2023-03-17 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: Expander graphs are perhaps one of the most widely useful classes of graphs ever considered. In this talk, we will focus on a fairly weak notion of expanders called sublinear expanders, first introduced by Komlós and Szemerédi around 25 years ago. They have found many remarkable applications ever since. In particular, we will focus on certain robustness conditions one may impose on sublinear expanders and some applications of this very recent idea, which include: - recent progress on one of the most classical decomposition conjectures in combinatorics, the Erd?s-Gallai Conjecture, - essentially tight answer to the classical Erd?s unit distance problem in "almost all" real normed spaces of any fixed dimension and - Rainbow Turan problem for cycles, raised by Keevash, Mubayi, Sudakov and Verstraete, including an application of this result to additive number theory. |
| Title: New applications of inexact Krylov methods |
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| Seminar: CODES@Emory |
| Speaker: Malena Sabate Landman of Emory University |
| Contact: Matthias Chung, matthias.chung@emory.edu |
| Date: 2023-03-16 at 10:00AM |
| Venue: MSC W201 |
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| Abstract: In this talk I will present a new class of algorithms for separable nonlinear inverse problems based on inexact Krylov methods. In particular, I will focus on semi-blind deblurring applications, where we are interested in recovering an approximation of the original image and of a small number of parameters defining the blur. Classical methods in this setting involve solving a sequence of ill-posed and computationally expensive linear problems, and we propose using a new interpretation of inexact Krylov methods to solve this more efficiently. After giving a brief overview of the theoretical properties of these methods, as well as strategies to monitor the amount of inexactness that can be tolerated, the performance of the algorithms will be shown through numerical examples. Finally, I will also give an overview on current ongoing work on using inexact Krylov methods theory in a more general setting involving slowly varying linear systems. |
| Title: Stability and Statistical Inversion for Travel Time Tomography |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Hanming Zhou of University of California, Santa Barbara |
| Contact: Yiran Wang, yiran.wang@emory.edu |
| Date: 2023-03-16 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: In this talk, we consider the travel time tomography for conformal metrics on a bounded domain which consists of determining the conformal factor of the metric from the length of geodesics joining boundary points. We establish forward and inverse stability estimates for simple conformal metrics under some a priori conditions. We then apply the stability estimates to show the consistency of the statistical inversion of the travel time tomography with discrete, noisy measurements. This is based on joint work with Ashwin Tarikere. |
| Title: Topics in arithmetic statistics |
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| Defense: Dissertation |
| Speaker: Christopher Keyes of Emory University |
| Contact: Chris Keyes, christopher.keyes@emory.edu |
| Date: 2023-02-28 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Arithmetic statistics may be interpreted broadly to include questions in number theory and arithmetic geometry with a distinct quantitative flavor. To answer even simply stated such questions, we often employ diverse algebraic, analytic, or geometric techniques. This dissertation addresses several arithmetic statistical questions, and for its defense we focus on those related to superelliptic curves.\\ \\ A superelliptic curve is given by an affine algebraic equation of the form $C \colon y^m = f(x)$. For a fixed such curve $C$ and degree $n$, we ask how many number fields $K/\mathbb{Q}$ of degree $n$ arise as the minimal field of definition of an algebraic point on $C$, as counted by discriminant? For $n$ sufficiently large and subject to certain conditions, we find infinitely many of these fields, producing an asymptotic lower bound of the form $X^{\delta}$ for an explicit constant $\delta > 0$. In special cases, we are additionally able to count those extensions with prescribed Galois group.\\ \\ For certain degrees $n$, it is possible for a curve to have only finitely many points of degree $n$, or even none at all. Instead of fixing a curve $C$, one might ask how often a curve has (or lacks) points of certain degree, as it varies in some family. In the case of superelliptic curves, we make these questions precise by counting the defining polynomials $f$ by their coefficients. We then find that a positive proportion of superelliptic curves are everywhere locally soluble, a necessary condition for having a rational point, and pin down this proportion exactly in the trigonal genus 4 case. After placing conditions on the family, we also find that for certain degrees $n$, a positive proportion of curves have only finitely many points of degree $n$. |