All Seminars
| Title: Joints and Hypergraph Joints |
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| Seminar: Discrete Mathematics |
| Speaker: Ting-Wei Chao, PhD of Massachusetts Institute of Technology |
| Contact: Dr. Cosmin Pohoata, cosmin.pohoata@emory.edu |
| Date: 2024-11-08 at 10:00AM |
| Venue: MSC N306 |
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| Abstract: A joint in $\mathbb{R}^3$ is a common intersection of $3$ lines with linearly independent directions. Using the polynomial method, Guth and Katz showed that the number of joints of $n$ lines is $O(n^{3/2})$. We showed a tight bound on this problem, which translates to the Lov\'asz's version of the Kruskal—Katona theorem when restricting to certain configurations. The Kruskal—Katona theorem finds the maximum number of triangles in an $n$-edge graph. A generalization of this theorem by Friedgut and Kahn finds the order of the maximum number of any fixed hypergraph $H$ in an $n$-edge hypergraph. We combined the Friedgut—Kahn theorem and joints and gave a common generalization of them. This is a JOINT work with Hung-Hsun Hans Yu. |
| Title: The retrieval problem in remote sensing |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Luca Sgheri of CNR |
| Contact: Matthias Chung, matthias.chung@emory.edu |
| Date: 2024-11-07 at 10:00AM |
| Venue: MSC N306 |
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| Abstract: In this talk, we will present the retrieval problem in remote sensing, focusing specifically on data analysis from the FORUM spectrometer, which is the 9th ESA Earth-Explorer mission scheduled for launch in 2027. Data analysis from spectrometers is an interesting mathematical inverse problem. We will begin by introducing the direct problem, also known as radiative transfer, and demonstrate that the inverse problem, as is typical with most inverse problems, is ill-posed. We will then discuss the optimal estimation method commonly used in the remote sensing community to address the inverse problem. Finally, we will highlight some challenges related to inverse problems that could benefit from collaboration with mathematicians. |
| Title: Quantum Algorithms on Graphs |
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| Seminar: Discrete Mathematics |
| Speaker: Asaf Ferber of University of California, Irvine |
| Contact: Liana Yepremyan, lyeprem@emory.edu |
| Date: 2024-11-07 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: In this talk, I will provide a brief introduction to quantum computation and demonstrate its potential for accelerating classical graph algorithms. Specifically, I will present an asymptotically tight result for learning a Hamiltonian cycle using OR queries, as well as a significant polynomial improvement on the best-known quantum algorithm for ($\Delta$+1) -coloring a graph with maximum degree $\Delta$.\\ \\ This work is based on joint research with Liam Hardiman (UCI) and Xiaonan Chen (UCI). |
| Title: Zeros of Period polynomials for symmetric power L-functions |
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| Seminar: Algebra and Number Theory |
| Speaker: Robert Dicks of Clemson University |
| Contact: Santiago Arango, santiago.arango@emory.edu |
| Date: 2024-11-07 at 4:00PM |
| Venue: MSC E406 |
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| Abstract: Suppose that $k$ and $N$ are positive integers. Let $f$ be a normalized cuspidal Hecke eigenform on $\Gamma_0(N)$ of weight $k$ with $L$-function $L_f(s)$. Previous works have studied the zeros of the period polynomial $r_f(z)$, which is a generating function for the critical values of $L_f(s)$ and has a functional equation relating $z$ and $-1/Nz$. In particular, $r_f(z)$ satisfies a version of the Riemann hypothesis: all of its zeros are on the circle $\{z \in \mathbb{C} : |z|=1/\sqrt{N}\}$.\\ \\ In this paper, we define a natural analogue of period polynomials for the symmetric power $L$-functions of $f$ and prove the corresponding Riemann hypothesis when $k$ is large enough.\\ \\ This is joint work with Hui Xue. |
| Title: The dynamics of random soliton and soliton gasses |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Manuela Girotti, PhD of Emory University |
| Contact: Maja Taskovic, maja.taskovic@emory.edu |
| Date: 2024-11-07 at 4:00PM |
| Venue: MSC E408 |
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| Abstract: N. Zabusky coined the word "soliton" in 1965 to describe a curious feature he and M. Kruskal observed in their numerical simulations of the initial-value problem for a simple nonlinear PDE. The first part of the talk will be a broad introduction to the theory of solitons/solitary waves and integrable PDEs (the KdV and modified KdV equation in particular), describing classical results in the field. The second (and main) part of the talk will focus on some new developments and growing interest into a special case of solutions defined as "soliton gas". I will describe a collection of works done in collaborations with K. McLaughlin (Tulane U.), T. Grava (SISSA/Bristol), R. Jenkins (UCF) and A. Minakov (U. Karlova). We analyze the case of a regular, dense (modified) KdV soliton gas and its large time behaviour with the presence of a single trial soliton travelling through it. We show that the solution can be decomposed as the sum of the background gas solution -a modulated elliptic wave-, plus a soliton solution: the individual expressions are however quite convoluted due to the nonlinear interaction. We are able to derive a series of physical quantities that precisely describe the dynamics: the local phase shift of the gas after the passage of the soliton, the location of the soliton peak as the dynamics evolves, and the velocity of the soliton peak. In particular, we show that the peak velocity, while interacting with the gas, is highly oscillatory, while its leading-order, average value satisfies the kinetic velocity equation analogous to the one posited by V. Zakharov and G. El. I will finally present some ongoing work where we establish that the soliton gas is the universal limit for a large class of N-solutions with random initial data. |
| Title: Reflections and Perspectives: a look back at my teaching journey |
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| Seminar: Teaching |
| Speaker: Maryam Khaqan (she/her/hers) of Department of Mathematics, University of Toronto |
| Contact: Bree Ettinger, bree.d.ettinger@emory.edu |
| Date: 2024-11-01 at 10:00AM |
| Venue: MSC W307C |
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| Abstract: As a graduate student at Emory, I taught 6 semesters of Calculus 1 and 2 and one semester of Linear Algebra. I taught independently, as a part of a small loosely coordinated team, and as part of larger, more tightly coordinated teams of instructors with shared assessments and shared syllabi.\\ My teaching journey, just like many of yours, began with MATH590, and since then, I've taught both graduate and undergraduate courses on a small and large scale. Currently, I am a teaching postdoc at the University of Toronto, where I am co-coordinating a team of 3 instructors and 8 TAs as well as teaching my own course with ~140 students.\\ \\ Each new role has brought with itself its own unique challenges, opportunities for growth, and shifts in perspective. In this talk, I will share various stories from my teaching journey with an aim to demonstrate what is "out there" i.e., what kinds of teaching roles a MATH590 student might pursue after Emory and what skills they might learn along the way. |
| Title: Tensor Decomposition meets Reproducing Kernel Hilbert Spaces (RKHS) |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Tammy (Tamara) Kolda, PhD of MathSci.ai |
| Contact: Elizabeth Newman, elizabeth.newman@emory.edu |
| Date: 2024-10-31 at 10:00AM |
| Venue: MSC N306 |
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| Abstract: Tensor decompositions require that data live on a regular $d$-way grid, but many real-world datasets do not have this property. For example, time-evolving data may be measured at different intervals for different subjects and adaptive meshes in simulations are irregular by design. We can handle irregular grids by treating some modes as infinite-dimension rather than finite-dimensional; we refer to such tensors as quasi-tensors. For their decompositions, this means that we want the factors in the tensor decomposition to be smooth functions rather than vectors. This basic idea has appeared in myriad forms over the years, often using different terminology and with different applications. I will recall and build on these efforts. The result is a generic framework for incorporating continuous modes into the CP tensor decomposition. We focus on learning the infinite-dimensional modes from a reproducing kernel Hilbert space (RKHS) and present an alternating least squares algorithm that is computationally efficient. Including infinite-dimensional modes (1) enables practitioners to enforce common structural assumptions about data such as smoothness, (2) extends to situations in where the measurement times do not align by utilizing the framework of missing data, and (3) provides a more principled way to interpolate between observed points. |
| Title: Off-diagonal Ramsey numbers for slowly growing hypergraphs |
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| Seminar: Combinatorics |
| Speaker: Jiaxi Nie, Postdoctoral Fellow of Georgia Tech |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2024-10-31 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: For a $k$-uniform hypergraph $F$ and a positive integer $n$, the Ramsey number $r(F,n)$ denotes the minimum $N$ such that every $N$-vertex $F$-free $k$-uniform hypergraph contains an independent set of $n$ vertices. A hypergraph is {\em slowly growing} if there is an ordering $e_1,e_2,\dots,e_t$ of its edges such that $|e_i \setminus \bigcup_{j = 1}^{i - 1}e_j| \leq 1$ for each $i \in \{2, \ldots, t\}$. We prove that if $k \geq 3$ is fixed and $F$ is any non $k$-partite slowly growing $k$-uniform hypergraph, then for $n\ge2$, \[ r(F,n) = \Omega\Bigl(\frac{n^k}{(\log n)^{2k - 2}}\Bigr).\]\\ \\ In particular, we deduce that the off-diagonal Ramsey number $r(F_5,n)$ is of order $n^{3}/\mbox{polylog}(n)$, where $F_5$ is the triple system $\{123, 124, 345\}$. This is the only 3-uniform Berge triangle for which the polynomial power of its off-diagonal Ramsey number was not previously known. Our constructions use pseudorandom graphs, martingales, and hypergraph containers. |
| Title: Off-diagonal Ramsey numbers for slowly growing hypergraphs |
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| Seminar: Discrete Mathematics |
| Speaker: Jiaxi Nie, PhD of Georgia Tech |
| Contact: Liana Yepremyan, lyeprem@emory.edu |
| Date: 2024-10-31 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: For a $k$-uniform hypergraph $F$ and a positive integer $n$, the Ramsey number $r(F,n)$ denotes the minimum $N$ such that every $N$-vertex $F$-free $k$-uniform hypergraph contains an independent set of $n$ vertices. A hypergraph is {\em slowly growing} if there is an ordering $e_1,e_2,\dots,e_t$ of its edges such that $|e_i \setminus \bigcup_{j = 1}^{i - 1}e_j| \leq 1$ for each $i \in \{2, \ldots, t\}$. We prove that if $k \geq 3$ is fixed and $F$ is any non $k$-partite slowly growing $k$-uniform hypergraph, then for $n\ge2$, \[ r(F,n) = \Omega\Bigl(\frac{n^k}{(\log n)^{2k - 2}}\Bigr).\] In particular, we deduce that the off-diagonal Ramsey number $r(F_5,n)$ is of order $n^{3}/\mbox{polylog}(n)$, where $F_5$ is the triple system $\{123, 124, 345\}$. This is the only 3-uniform Berge triangle for which the polynomial power of its off-diagonal Ramsey number was not previously known. Our constructions use pseudorandom graphs, martingales, and hypergraph containers. |
| Title: Sharp Detection of Low-Dimensional Structure in Bayesian Inverse Problems via (Dimensional) Logarithmic Sobolev Inequalities |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Matthew Li. PhD of Massachusetts Institute of Technology |
| Contact: Matthias Chung, matthias.chung@emory.edu |
| Date: 2024-10-28 at 1:00PM |
| Venue: E300 MSC |
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| Abstract: Statistical inference in high-dimensions - whether through measuretransport, sampling, or other algorithms—often hinges on identifying and leveraging low-dimensional structures. We present an approach that represents high-dimensional target measures as low-dimensional updates of a dominating reference measure. While determining the optimal "dimension reduction" in this sense is computationally intractable, we explore how logarithmic Sobolev and Poincaré inequalities, along with their generalizations, enable us to derive approximations with certifiable error guarantees that are typically satisfactory in practice. In the latter part of the talk, we address why such tools from Markov semigroup theory are relevant to dimension reduction. We demonstrate that the optimal functional inequality for the Kullback-Leibler divergence is the dimensional logarithmic Sobolev inequality, which intrinsically captures the low-dimensional update structure we aim to exploit. Finally, we illustrate applications of these concepts to modern Bayesian inverse problems, particularly those involving GAN-based generative priors. Time-permitting, we may also discuss the connection of these ideas to emergent feature learning properties empirically observed in deep neural networks. Based on joint work with Olivier Zahm, Youssef Marzouk, Tiangang Cui, and Fengyi Li. |