All Seminars
| Title: Motivic Euler characteristics and the transfer |
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| Seminar: Algebra |
| Speaker: Roy Joshua of Ohio State University |
| Contact: Andrew Kobin, ajkobin@emory.edu |
| Date: 2023-04-04 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: In the first part of the talk, we will consider motivic Euler characteristics of certain homogeneous spaces as they relate to splittings in the motivic homotopy category. In the second part of the talk, we will discuss certain applications of these to computations in Algebraic K-Theory and Brauer groups. |
| Title: Balancing the stability-accuracy Trade-off in Neural Networks for Ill-conditioned Inverse Problems |
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| Seminar: CODES@Emory |
| Speaker: Davide Evangelista of University of Bologna |
| Contact: Jim Nagy, jnagy@emory.edu |
| Date: 2023-03-30 at 10:00AM |
| Venue: MSC W201 |
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| Abstract: Deep learning algorithms have recently become state-of-art in solving Inverse Problems, overcoming the classical variational methods in terms of both accuracy and efficiency. On the other hand, it is still unclear if neural networks can compete in terms of reliability and a rigorous complete analysis still lacks in the literature. Starting from the brilliant works of N.M.Gottschling, V.Antun (2020) and M.J.Colbrook, V.Antun (2021), we will try to understand the relationship between the accuracy and stability of neural networks for solving ill-conditioned inverse problems, deriving new theoretical results shedding light on the trade-off between accuracy and stability. Following the study of M.Genzel, J.Macdonald (2020), we will find that, under some conditions, neural networks can be more unstable the more they are accurate, and we will propose new regularization techniques with provable increase in stability and minumum accuracy loss. |
| Title: Counting low degree number fields with almost prescribed successive minima |
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| Seminar: Algebra |
| Speaker: Sameera Vemulapalli of Princeton University |
| Contact: Andrew Kobin, ajkobin@emory.edu |
| Date: 2023-03-28 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: The successive minima of an order in a degree n number field are n real numbers encoding information about the Euclidean structure of the order. How many orders in degree n number fields are there with almost prescribed successive minima, fixed Galois group, and bounded discriminant? In this talk, I will address this question for $n = 3,4,5$. The answers, appropriately interpreted, turn out to be piecewise linear functions on certain convex bodies. If time permits, I will also discuss a geometric analogue of this problem: scrollar invariants of covers of $\mathbb{P}^1$. |
| Title: Extremal problems for uniformly dense hypergraphs |
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| Seminar: Combinatorics |
| Speaker: Mathias Schacht of Hamburg University |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2023-03-27 at 4:30PM |
| Venue: MSC E408 |
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| Abstract: Extremal combinatorics is a central research area in discrete mathematics. The field can be traced back to the work of Turán and it was established by Erd?s through his fundamental contributions and his uncounted guiding questions. Since then it has grown into an important discipline with strong ties to other mathematical areas such as theoretical computer science, number theory, and ergodic theory. We focus on extremal problems for hypergraphs, which were introduced by Turán. After solving the analogous question for graphs, Turán asked to determine the maximum cardinality of a set E of 3-element subsets of a given n-element set V such that for any 4 elements of V at least one triple is missing in E. This innocent looking problem is still open and despite a great deal of effort over the last 80 years and our knowledge is still somewhat limited. We consider a variant of the problem by imposing additional restrictions on the distribution of the 3-element subsets in E. These additional assumptions yield a finer control over the corresponding extremal problem. In fact, this leads to many interesting and more manageable subproblems, some of which were already considered by Erd?s and Sós in the 1980ies. The additional assumptions on the distribution of the 3-element subsets are closely related to the theory of quasirandom discrete structures, which was pioneered by Szemerédi and became a central theme in the field. In fact, the hypergraph extensions by Gowers and by Rödl et al. of the regularity lemma provide essential tools for this line of research. |
| Title: Extremal problems for uniformly dense hypergraphs |
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| Seminar: Combinatorics |
| Speaker: Mathias Schacht of Hamburg University |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2023-03-27 at 4:30PM |
| Venue: MSC E408 |
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| Abstract: Extremal combinatorics is a central research area in discrete mathematics. The field can be traced back to the work of Turán and it was established by Erd?s through his fundamental contributions and his uncounted guiding questions. Since then it has grown into an important discipline with strong ties to other mathematical areas such as theoretical computer science, number theory, and ergodic theory. We focus on extremal problems for hypergraphs, which were introduced by Turán. After solving the analogous question for graphs, Turán asked to determine the maximum cardinality of a set E of 3-element subsets of a given n-element set V such that for any 4 elements of V at least one triple is missing in E. This innocent looking problem is still open and despite a great deal of effort over the last 80 years and our knowledge is still somewhat limited. We consider a variant of the problem by imposing additional restrictions on the distribution of the 3-element subsets in E. These additional assumptions yield a finer control over the corresponding extremal problem. In fact, this leads to many interesting and more manageable subproblems, some of which were already considered by Erd?s and Sós in the 1980ies. The additional assumptions on the distribution of the 3-element subsets are closely related to the theory of quasirandom discrete structures, which was pioneered by Szemerédi and became a central theme in the field. In fact, the hypergraph extensions by Gowers and by Rödl et al. of the regularity lemma provide essential tools for this line of research. |
| 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. |