All Seminars
| 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: 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: 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: 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. |
| Title: Whitham modulation theory for a class of nonlinear dispersive wave equations |
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| Seminar: CODES@emory |
| Speaker: Patrick Sprenger of UC Merced |
| Contact: Levon Nurbekyan, levon.nurbekyan@emory.edu |
| Date: 2024-11-21 at 10:00AM |
| Venue: MSC N306 |
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| Abstract: Nonlinear periodic waves are central to a wide array of physical systems, from fluid dynamics to optics, where their stability and behavior under slow modulations reveal complex multiscale dynamics. This talk will explore the application of Whitham modulation theory, an asymptotic procedure that results in a system of conservation laws governing the slow evolution of wave parameters in nonlinear periodic wavetrains. Although modulation equations can, in principle, be derived for a general nonlinear, evolutionary model, they often become intractable when explicit formulas for periodic solutions are unavailable. We will discuss the Whitham modulation equations for a general class of dispersive hydrodynamic model equations and examine how their properties can be studied asymptotically and numerically. Special attention will be given to how this approach can provide practical insights into the stability of periodic traveling waves. |
| Title: Picard Groups of Stacky Curves |
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| Seminar: Algebra and Number Theory |
| Speaker: Rose Lopez of UC Berkeley |
| Contact: Santiago Arango, santiago.arango@emory.edu |
| Date: 2024-11-19 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: In Mumford's famous 1965 paper Picard Groups of Moduli Problems, he computed the Picard group of the moduli stack of elliptic curves, a stacky curve over the affine j-line. While Mumford relies heavily on the moduli description, much can be said about the Picard group of a stacky curve by understanding its geometry alone. In this talk, I will give and explain an exact sequence which relates the Picard group of a stacky curve to the Picard group of its coarse space and the gerbe class over its rigidification |
| Title: On the stable birationality of Hilbert schemes of points on surfaces |
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| Seminar: Algebra and Number Theory |
| Speaker: Morena Porzio of Columbia University |
| Contact: Santiago Arango, santiago.arango@emory.edu |
| Date: 2024-11-19 at 5:00PM |
| Venue: MSC W303 |
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| Abstract: The Cassels--Swinnerton-Dyer conjecture says that cubic surfaces of index 1 have a rational point. This can be reformulated into a statement about the existence of rational maps between Hilbert schemes of points $\mathrm{Hilb}^n_X$, which in turn motivates the study of the stable birational type of $\mathrm{Hilb}^n_X$. In this talk, we will address the question for which pairs of integers $(n,n’)$ the variety $\mathrm{Hilb}^n_X$ is stably birational to $\mathrm{Hilb}^{n'}_X$, when $X$ is a surface with $H^1(X,\mathcal{O}_X)=0$. In order to do so we will relate the existence of degree $n'$ effective cycles on $X$ with the existence of degree $n$ ones using curves on $X$. We will then focus on geometrically rational surfaces, proving that there are finitely many birational classes among $\mathrm{Hilb}^n_X$'s. |
| Title: Global, robust, multi-objective optimization of stellarators |
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| Seminar: CODES@emory |
| Speaker: David Bindel of Cornell University |
| Contact: Yuanzhe Xi, yuanzhe.xi@emory.edu |
| Date: 2024-11-14 at 10:00AM |
| Venue: MSC N306 |
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| Abstract: A stellarator is a type of toroidal magnetic field geometry used to confine fusion-relevant plasmas. Unfortunately for applied mathematicians, physicists do not measure how good a stellarator is with a single number. Among other things, an “optimal” stellarator should approximately satisfy desirable field symmetries, satisfy macroscopic and local stability, minimize different types of transport, and minimize engineering complexity. We also want designs that are robust to small variations and do not require unrealistically tight tolerances for coil manufacturing and placement. These objectives and constraints are complicated, non-convex functions of the plasma boundary and coil shapes, and they may be subject to subtle tradeoffs. In this talk, we describe why state-of-the-art stellarator optimization tools do not yet fully explore the range of design tradeoffs to find robustly “optimal” designs. We discuss formulations of robust optimization and show some results achieved by members of our group in the context of stellarator optimization. We then discuss the shortcoming of the “scalarization” formulation of multi-objective optimization used in the current generation of stellarator optimizers, showing why this approach may miss parts of the Pareto frontier of best tradeoffs. Finally, we discuss how to go beyond “black box” approaches with optimization algorithms that use both derivative information and physics-based approximations to quickly explore and converge to the globally best parts of the design space. |
| Title: The Turán Density of 4-Uniform Tight Cycles |
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| Seminar: Discrete Mathematics |
| Speaker: Maya Sankar of Stanford University |
| Contact: Dr. Cosmin Pohoata, cosmin.pohoata@emory.edu |
| Date: 2024-11-14 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: For any uniformity r and residue k modulo r, we give an exact characterization of the r-uniform hypergraphs that homomorphically avoid tight cycles of length k modulo r, in terms of colorings of (r-1)-tuples of vertices. This generalizes the result that a graph avoids all odd closed walks if and only if it is bipartite, as well as a result of Kam?ev, Letzter, and Pokrovskiy in uniformity 3. In fact, our characterization applies to a much larger class of families than those of the form C_k^(r)={r-uniform tight cycles of length k modulo r}. We also outline a general strategy to prove that, if C is a family of tight-cycle-like hypergraphs (including but not limited to the families C_k^(r)) for which the above characterization applies, then all sufficiently long cycles in C will have the same Turán density. We demonstrate an application of this framework, proving that there exists an integer L_0 such that for every L>L_0 not divisible by 4, the tight cycle C^(4)_L has Turán density 1/2. |