All Seminars
| Title: Can computational math help settle down Morrey's and Iwaniec's conjectures? |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Wilfrid Gangbo, PhD of UCLA |
| Contact: Dr. Levon Nurbekyan, lnurbek@emory.edu |
| Date: 2025-02-14 at 11:00AM |
| Venue: MSC W303 |
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| Abstract: In 1987, D. L. Burkholder proposed a very simple looking and explicit energy functionals $I_p$ defined on $\mathbb{S}$, the set of smooth functions on the complex plane. A question of great interest is to know whether or not $\sup_{\mathbb{S}} I_p \geq 0$. Since the function $I_p$ is homogeneous of degree $p$, it is very surprising that it remains a challenge to prove or disprove that $\sup_{\mathcal{S}} I_p \geq 0$. Would $\sup_{\mathbb{S}} I_p \geq 0$, the so-called Iwaniec's conjecture on the Beurling--Ahlfors Transform in harmonic analysis would hold. Would $\sup_{\mathcal{S}} I_p > 0$, the so-called Morrey's conjecture in elasticity theory would hold. Therefore, proving or disproving that $\sup_{\mathbb{S}} I_p \geq 0$ is equally important. Since the computational capacity of computers has increased exponentially over the past decades, it is natural to hope that computational mathematics could help settle these two conjectures at once. |
| Title: TBA |
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| Seminar: Mathematics |
| Speaker: Andrew Lyons, PhD of University of North Carolina at Chapel Hill |
| Contact: Dr. Bree Ettinger, betting@emory.edu |
| Date: 2025-01-31 at 2:00PM |
| Venue: MSC W301 |
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| Abstract: TBA |
| Title: Sources, sinks, and sea lice: determining patch contribution and transient dynamics in marine metapopulations |
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| Seminar: Mathematics |
| Speaker: Peter Harrington, PhD of University of British Columbia |
| Contact: Dr. Bree Ettinger, betting@emory.edu |
| Date: 2025-01-27 at 9:00PM |
| Venue: MSC W303 |
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| Abstract: Sea lice are salmon parasites which threaten the health of both wild and farmed salmon. Open-net salmon farms act as reservoirs for sea lice in near coastal areas, which can lead to elevated sea louse levels on wild salmon. With a free-living larval stage, sea lice can disperse tens of kilometers in the ocean, both from salmon farms onto wild salmon and between salmon farms. This larval dispersal connects local sea louse populations on salmon farms and thus modelling the collection of salmon farms as a metapopulation can lead to a better understanding of which salmon farms are driving the overall growth of sea lice in a salmon farming region. In this talk I will discuss using metapopulation models to specifically study sea lice on salmon farms in the Broughton Archipelago, BC, and more broadly to better understand the transient and asymptotic dynamics of marine metapopulations. No ecological background will be assumed, and despite the biological motivation there will be plenty of mathematics in the talk. |
| Title: Merging Lanes and Mathematical Patterns |
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| Seminar: Mathematics |
| Speaker: Katie Johnson, PhD of Florida Gulf Coast University |
| Contact: Dr. Bree Ettinger, betting@emory.edu |
| Date: 2025-01-24 at 1:00PM |
| Venue: MSC W303 |
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| Abstract: What happens when vehicles approach a two-lane intersection with merging traffic and different driving behaviors? This simple scenario leads to fascinating connections in discrete mathematics, from lattice paths and coin flips to domino snakes and graph trails. Join us to explore these patterns and uncover surprising identities that emerge from a seemingly everyday situation. |
| Title: Vibrations, Structures, and Blood Flow: Unlocking the Power of Eigenvalues and Eigenvectors |
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| Seminar: Mathematics |
| Speaker: Jimena Martin-Tempestti, PhD of Emory University |
| Contact: Bree Ettinger, betting@emory.edu |
| Date: 2025-01-23 at 10:00AM |
| Venue: MSC W301 |
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| Abstract: Eigenvalues and eigenvectors are fundamental tools in understanding the behavior of complex systems, from structural vibrations to fluid flows. In this talk, we will examine their role in uncovering the intrinsic properties of matrices and how these insights extend to practical applications. Beginning with a mathematical foundation, we will explore how eigenvalues and eigenvectors emerge in modal analysis, a method used to analyze the vibrational behavior of physical structures, with examples ranging from the iconic Tacoma Narrows Bridge to finite element models of high-rise buildings. We will then pivot to their use in hemodynamics, the study of blood flow, and how mathematics helps us understand it better. A powerful tool called Proper Orthogonal Decomposition (POD) uses eigenvalues and eigenvectors to break down complex blood flow data into its most important patterns, which makes it possible to analyze real-world data from ultrasound imaging, even when the data is noisy, in a relatively short time. By applying POD, we can better understand how blood moves through vessels and compute quantities like Wall Shear Stress (WSS), which is crucial for studying cardiovascular health. |
| Title: Multiscale Neural Networks for Approximating Green's Functions |
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| Seminar: CODES@emory |
| Speaker: Tianshi Xu, PhD of Emory University |
| Contact: Dr. Levon Nurbekyan, lnurbek@emory.edu |
| Date: 2024-12-05 at 10:00AM |
| Venue: MSC N306 |
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| Abstract: Neural networks have been widely used to solve partial differential equations (PDEs) in the applications of physics, biology, and engineering. One effective approach for solving PDEs with a fixed differential operator is learning Green's functions. However, Green's functions are notoriously difficult to learn due to their poor regularity, which typically requires larger NNs and longer training times. In this talk, we address these challenges by leveraging multiscale NNs to learn Green's functions. Through theoretical analysis using multiscale Barron space methods and experimental validation, we show that the multiscale approach significantly reduces the necessary NN size and accelerates training. We also discuss the potential application of this framework for solving Helmholtz equations. |
| Title: Higher rank antipodality |
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| Seminar: Combinatorics |
| Speaker: Márton Naszódi of Alfréd Rényi Institute of Mathematics, and Loránd Eötvös University, Budapest |
| Contact: Liana Yepremyan, |
| Date: 2024-12-05 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Motivated by general probability theory, we say that the set $X$ in $\mathbb{R}^d$ is \emph{antipodal of rank $k$}, if for any $k+1$ elements $q_1,\ldots q_{k+1}\in X$, there is an affine map from $\mathrm{conv} X$ to the $k$-dimensional simplex $\Delta_k$ that maps $q_1,\ldots q_{k+1}$ onto the $k+1$ vertices of $\Delta_k$. For $k=1$, it coincides with the well-studied notion of (pairwise) antipodality introduced by Klee.\\ \\ We consider the following natural generalization of Klee's problem on antipodal sets: What is the maximum size of an antipodal set of rank $k$ in $\mathbb{R}^d$? We present a geometric characterization of antipodal sets of rank $k$ and adapting the argument of Danzer and Gr{\"u}nbaum originally developed for the $k=1$ case, we prove an upper bound which is exponential in the dimension. We point out that this problem can be connected to a classical question in computer science on finding \emph{perfect hashes}, and it provides a lower bound on the maximum size, which is also exponential in the dimension. Joint work with Zsombor Szil{\'a}gyi and Mih{\'a}ly Weiner. |
| Title: Canonical Ramsey numbers for partite hypergraphs |
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| Seminar: Combinatorics |
| Speaker: Mathias Schacht of |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2024-12-03 at 4:00PM |
| Venue: MSC E408 |
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| Abstract: We consider quantitative aspects of the canonical Ramsey theorem of Rado for k-partite k-uniform hypergraphs. For the complete bipartite graph $K_{t,t}$ it was recently shown by Dobak and Mulrenin that these numbers grow exponential in $t-log(t)$ and considering random edge colourings shows that this bound is asymptotically optimal. We extend this result to k-uniform hypergraphs and obtain a bound exponential in $poly(t)$. This is joint work with Giovanne Santos and Matias Azocar |
| Title: New building blocks for $\mathbb{F}_1$-geometry: band and band schemes |
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| Seminar: Algebra |
| Speaker: Tong Jin of Georgia Institute of Technology |
| Contact: Santiago Arango-Piñeros, sarang2@emory.edu |
| Date: 2024-12-03 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Algebraic geometry over the ``field with one element'' was an idea first introduced by Jacquet Tits in the 1950s. After motivations from algebraic group theory and matroid theory, I will sketch the idea for band and band schemes along with their basic properties. There are various different topological spaces which one can associate to a band scheme $X$, and they correspond to various interesting combinatorial structures. This is based on a joint work with Matt Baker and Oliver Lorscheid. |
| Title: Averaging one-point hyperbolic-type metrics |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Professor Zair Ibragimov of California State University, Fullerton |
| Contact: Shanshuang Yang, syang05@emory.edu |
| Date: 2024-11-26 at 1:00PM |
| Venue: MSC E408 |
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| Abstract: It is known that the $\tilde j$-metric, the half-Apollonian metric, and the scale-invariant Cassinian metric are not Gromov hyperbolic. These metrics are defined as a supremum of one-point metrics (i.e., metrics constructed using one boundary point), and the supremum is taken over all boundary points. The aim of this talk is to show that taking the average instead of the supremum yields a metric that is Gromov hyperbolic. Moreover, we show that the Gromov hyperbolicity constant of the resulting metric does not depend on the number of boundary points used in taking the average. We also provide an example to show that the average of Gromov hyperbolic metrics is not, in general, Gromov hyperbolic. |