All Seminars
| Title: Rigidity of Extremal Point-Line Arrangements |
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| Seminar: Combinatorics |
| Speaker: Hung-Hsun Hans Yu of Princeton University |
| Contact: Dr. Cosmin Pohoata, cosmin.pohoata@emory.edu |
| Date: 2024-10-25 at 10:00AM |
| Venue: MSC N306 |
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| Abstract: It is a classical theorem by Szemerédi and Trotter that n points and m lines in the Euclidean plane form at most C(mn)^{2/3} + m + n incidences, and the bound is optimal up to the constant C. However, there is still no satisfactory description of configurations maximizing the number of incidences. As a small step toward such description, in this talk, I will show how to prove that the extremizers are rigid in some sense. This is based on joint work with Gabriel Currier and Jozsef Solymosi. |
| Title: Erdos-Pósa property of tripods in directed graphs |
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| Seminar: Discrete Mathematics |
| Speaker: Meike Hatzel of Institute for Basic Science |
| Contact: Liana Yepremyan, lyeprem@emory.edu |
| Date: 2024-10-23 at 4:00PM |
| Venue: MSC E406 |
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| Abstract: Let $D$ be a directed graphs with distinguished sets of sources $S\subseteq V(D)$ and sinks $T\subseteq V(D)$. A \emph{tripod} in $D$ is a subgraph consisting of the union of two $S$-$T$-paths that have distinct start-vertices and the same end-vertex, and are disjoint apart from sharing a suffix. We prove that tripods in directed graphs exhibit the Erd?s-Pósa property. More precisely, there is a function $f\colon $N$ \rightarrow{N}$ such that for every digraph $D$ with sources $S$ and sinks $T$, if $D$ does not contain $k$ vertex-disjoint tripods, then there is a set of at most $f(k)$ vertices that meets all the tripods in $D$. |
| Title: Steinitz classes of number fields and Tschirnhausen bundles of covers of the projective line |
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| Seminar: Algebra and Number Theory |
| Speaker: Sameera Vemulapalli of Harvard University |
| Contact: Santiago Arango, santiago.arango@emory.edu |
| Date: 2024-10-22 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Given a number field extension $L/K$ of fixed degree, one may consider $\mathcal{O}_L$ as an $\mathcal{O}_K$-module. Which modules arise this way? Analogously, in the geometric setting, a cover of the complex projective line by a smooth curve yields a vector bundle on the projective line by pushforward of the structure sheaf; which bundles arise this way? In this talk, I'll describe recent work with Vakil in which we use tools in arithmetic statistics (in particular, binary forms) to completely answer the first question and make progress towards the second. |
| Title: Combining Bayesian inference with data-consistent inversion: Leveraging population-level information to construct informative priors |
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| Seminar: CODES@emory |
| Speaker: Rebekah (Bekah) White of Sandia National Labs |
| Contact: Levon Nurbekyan, lnurbek@emory.edu |
| Date: 2024-10-18 at 10:00AM |
| Venue: Atwood 240, Atwood Chemistry Building |
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| Abstract: Computational models underpin many important applications at Sandia, such as additive manufacturing, nuclear waste repository management, and structural dynamics. However, such models often contain uncertain or unknown parameters that must be estimated from observational data by solving an inverse problem. Bayesian inference is a popular approach to inverse problems, but when data is limited (as is often the case for Sandia applications), a highly informative prior is needed; in practice, the physical knowledge required to construct such priors may not be available. Consequently, this work presents an alternative, novel approach for leveraging data from “population” of related experiments or physical assets to construct highly informative Bayesian priors. Specifically, we use a more recently developed inversion technique, known as data-consistent inversion (DCI), to estimate properties of a given population. Combining DCI with Bayesian inference in this way is shown to improve the inference process overall, further reducing model parameter uncertainty. This talk will provide an overview of both Bayesian inference and data-consistent inversion, illustrate how such inversion techniques can be combined, and demonstrate the combined approach for a computational mechanics exemplar governed by partial differential equations (PDEs). |
| Title: Inverse Problem for Hyperbolic Partial Differential Operators on Riemannian Manifolds Without Boundary |
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| Seminar: Analysis and PDE |
| Speaker: Teemu Saksala of North Carolina State University |
| Contact: Yiran Wang, yiran.wang@emory.edu |
| Date: 2024-10-18 at 11:00AM |
| Venue: MSC W301 |
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| Abstract: In this talk we consider an inverse problem for a hyperbolic partial differential operator on a Riemannian manifold without boundary. Such a manifold can be either compact, like a sphere or torus, or unbounded, like Euclidean or Hyperbolic spaces The hyperbolic partial differential operator we are studying is a self-adjoint first order perturbation of the Riemannian wave operator. In particular, this operator has time-independent lower order terms which can be written in the form of a vector field and a function which models the magnetic and electric potentials respectively. Our goal is to recover the speed of sound as well as these lower order terms by sending lots of waves from some open set of the manifold and measuring these waves on the same open set. This is called the local source-to-solution map. In this talk I will introduce the natural obstruction for recovering the aforementioned quantities from such measurements. This is the gauge of the problem. I will outline a proof that shows that modulo this gauge we can recover the wave speed, together with the magnetic and electric potentials from the local source-to-solution map. Our proof is based on a variation of the celebrated boundary control method (BC-method) which was developed by Belishev and Kurylev, and used to solve Gel'fand's inverse boundary spectral problem: "Can you recover a Riemannian manifold with boundary from the spectral data of its Dirichlet-Laplacian?" The BC-method reduces the PDE-based inverse problem to a geometric inverse problem of recovering a Riemannian manifold from a family of distance functions. We will also outline the proof of this problem.\\ \\ This talk is based on my earlier work with Tapio Helin, Matti Lassas and Lauri Oksanen and on an ongoing project with my PhD student Andrew Shedlock. |
| Title: Approximation of differential operators on unknown manifolds and applications |
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| Seminar: CODES@emory |
| Speaker: John Harlim of Pennsylvania State University |
| Contact: Yuanzhe Xi, yuanzhe.xi@emory.edu |
| Date: 2024-10-17 at 10:00AM |
| Venue: MSC N306 |
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| Abstract: I will discuss the numerical approximation of differential operators on unknown manifolds where the manifolds are identified by a finite sample of point cloud data. While our formulation is general, we will focus on Laplacian operators whose spectral properties are relevant to manifold learning. I will report the spectral convergence results of these formulations with Radial Basis Functions approximation and their strengths/weaknesses in practice. Supporting numerical examples, involving the spectral estimation of various vector Laplacians will be demonstrated. Applications to solve elliptic PDEs will be discussed. To address the practical issue with the RBF approximation, I will discuss a weak approximation with a higher-order local mesh method that not only promotes sparsity but also allows for an estimation of differential operators with nontrivial Cristoffel symbols such as Bochner and Hodge Laplacians. |
| Title: Deep Learning for Missing Physics in Dynamical Systems |
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| Seminar: |
| Speaker: Andrea Arnold of Worcester Polytechnic Institute |
| Contact: Matthias Chung, matthias.chung@emory.edu |
| Date: 2024-10-10 at 10:00AM |
| Venue: MSC N306 |
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| Abstract: Incorporating a priori physics knowledge into machine learning leads to more robust and interpretable algorithms. In this talk, I will describe an approach combining deep learning and classic numerical methods for differential equations to solve two challenging problems in dynamical systems theory: dynamics discovery and parameter estimation, where the missing information prevents standard use of a numerical method or neural network independently for the problem. Assuming corrupt system observations, we observe promising results in predicting the system dynamics and estimating physical parameters, given appropriate choices of spatial and temporal discretization schemes and numerical method orders.\\ \\ https://emory.zoom.us/j/94678278895?pwd=bDFxK2RaOTZRMjA5bzQ4UUtxNWJsZz09 |
| Title: Non-measurable colourings avoiding large distances |
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| Seminar: Discrete Mathematics |
| Speaker: James Davies of Junior Research Fellow at Trinity Hall, Cambridge University |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2024-10-10 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: In 1983, Furstenberg, Katznelson, and Weiss proved that for every finite measurable colouring of the plane, there exists a $d_0$ such that for all $d\geq d_0$ there is a monochromatic pair of points at distance $d$. In contrast to this, we show that there is a finite colouring avoiding arbitrarily large distances. This is joint work with Rutger Campbell. |
| Title: Finite Field Fourier Transforms in Arithmetic Statistics |
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| Algebra and Number Theory: Algebra |
| Speaker: Frank Thorne of University of South Carolina |
| Contact: Santiago Arango, santiago.arango@emory.edu |
| Date: 2024-10-08 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: In many arithmetic statistics problems, it is useful to evaluate or bound certain Fourier transforms over finite fields. I will give an overview of (1) how these Fourier transforms arise, (2) some strategies that my collaborators and I (and others!) have developed to analyze them, and (3) some surprising structures one finds.\\ \\ Some of this work is older, but I will focus on forthcoming work with Anderson and Bhargava in Bhargava's averaging method, and recently finished work with Ishitsuka, Taniguchi, and Xiao on binary quartic forms. |
| Title: Strong $u$-invariant and Period-Index bound |
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| Seminar: Algebra and Number Theory |
| Speaker: Shilpi Mandal of Emory University |
| Contact: Santiago Arango, santiago.arango@emory.edu |
| Date: 2024-10-01 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: For a central simple algebra $A$ over $K$, there are two major invariants, viz., \textit{period} and \textit{index}.\\ \\ For a field $K$, define the \emph{Brauer $l$-dimension of $K$} for a prime number $l$, denoted by $\mathrm{Br}_l\mathrm{dim}(K)$, as the smallest $d \in \mathbb{N} \cup \{\infty\}$ such that for every finite field extension $L/K$ and every central simple $L$-algebra $A$ of period a power of $l$, we have that $\mathrm{ind}(A)$ divides $\mathrm{per}(A)^d$.\\ \\ If $K$ is a number field or a local field (a finite extension of the field of $p$-adic numbers $\mathbb{Q}_p$, for some prime number $p$), then classical results from class field theory tell us that $\mathrm{Br}_l\mathrm{dim}(K) = 1$. This invariant is expected to grow under a field extension, bounded by the transcendence degree. Some recent works in this area include that of Lieblich, Harbater-Hartmann-Krashen for $K$ a complete discretely valued field, in the good characteristic case. In the bad characteristic case, for such fields $K$, Parimala-Suresh have given some bounds.\\ \\ Also, the $u$-invariant of $K$, denoted by $u(K)$, is the maximal dimension of anisotropic quadratic forms over $K$. For example, $u(\mathbb{C}) = 1$; for $F$ a non-real global or local field, we have $u(F) = 1, 2, 4,$ or $8$, etc.\\ \\ In this talk, I will present similar bounds for the $\mathrm{Br}_l\mathrm{dim}$ and the strong $u$-invariant of a complete non-Archimedean valued field $K$ with residue field $\kappa$. |