All Seminars
| Title: An optimal Bayesian estimator for a stochastic problem in Diffuse Optical Tomography |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Anuj Abhishek of University of North Carolina at Charlotte |
| Contact: Elizabeth Newman, elizabeth.newman@emory.edu |
| Date: 2021-11-19 at 12:30PM |
| Venue: MSC W201 |
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| Abstract: Studying coefficient inverse problems in a stochastic setting has increasingly gained in prominence in the past couple of decades. In this talk, we will present some results that were obtained for a Bayesian estimator built from the noisy data obtained in a simplified one-parameter Diffuse Optical Tomography (DOT) Model. We establish the rate of convergence of such an estimator in the supremum norm loss and show that it is optimal. This work extends the approach proposed by Abraham and Nickl in a recent article (On Statistical Calderon problems) and applies it to the problem in DOT setting. We also present some preliminary numerical simulations in support of our theoretical findings. This is joint work with Dr. Taufiquar Khan (UNCC) and Dr. Thilo Strauss (Robert Bosch GmbH). |
| Title: Generation and propagation of moments for the binary-ternary Boltzmann equation |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Ioakeim Ampatzoglou of Courant Institute |
| Contact: Maja Taskovic, maja.taskovic@emory.edu |
| Date: 2021-11-18 at 3:00PM |
| Venue: ATWOOD 360 |
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| Abstract: The binary-ternary Boltzmann equation is a recently derived kinetic equation describing the evolution of the probability density a non-ideal gas in non-equilibrium. In this talk we focus on the homogeneous (space invariant) equation and discuss the generation and propagation of polynomial and exponential moments properties of a solution. We will then employ these properties to discuss global in time existence and uniqueness. This is a joint work with Maja Taskovic. |
| Title: Non-orientable enumerative problems in $\mathbf{A}^{1}$-homotopy theory |
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| Seminar: Number Theory |
| Speaker: Andrew Kobin of Emory University |
| Contact: David Zureick-Brown, dzureic@emory.edu |
| Date: 2021-11-16 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Many enumerative problems in classical algebraic geometry, such as counting lines on a smooth cubic surface, admit a solution over an arbitrary ground field $k$ (of characteristic $\not = 2$) using Morel and Voevodsky's $\mathbf{A}^{1}$-homotopy theory. Recently, several authors have formulated and solved such ``enriched'' enumerative problems using Kass and Wickelgren's ``enriched'' Euler class, which takes values in the Grothendieck--Witt group of $k$ and is only defined when the associated vector bundle is orientable. In joint work with Libby Taylor, we extend Kass--Wickelgren's construction to non-orientable vector bundles using a stacky construction. This allows us to enrich a larger class of enumerative problems, including the count of lines meeting $6$ planes in $\mathbf{P}^{4}$. |
| Title: Geometric and Statistical Approaches to Shallow and Deep Clustering |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: James M. Murphy of Tufts University |
| Contact: Elizabeth Newman, elizabeth.newman@emory.edu |
| Date: 2021-11-05 at 12:30PM |
| Venue: MSC W201 |
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| Abstract: We propose approaches to unsupervised clustering based on data-dependent distances and dictionary learning. By considering metrics derived from data-driven graphs, robustness to noise and ambient dimensionality is achieved. Connections to geometric analysis, stochastic processes, and deep learning are emphasized. The proposed algorithms enjoy theoretical performance guarantees on flexible data models and in some cases guarantees ensuring quasilinear scaling in the number of data points. Applications to image processing and computational chemistry will be shown, demonstrating state-of-the-art empirical performance. |
| Title: Hamiltonian cycles in uniformly dense hypergraphs |
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| Seminar: Combinatorics |
| Speaker: Mathias Schacht of The University of Hamburg and Yale University |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2021-11-05 at 3:00PM |
| Venue: MSC E408 |
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| Abstract: We are studying the minimum density d such that every uniformly dense hypergraph with density bigger than d, combined with a mild minimum degree restriction, contains a Hamiltonian cycle. |
| Title: Some Galois cohomology classes arising from the fundamental group of a curve |
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| Seminar: Number Theory |
| Speaker: Padmavathi Srinivasan of University of Georgia |
| Contact: David Zureick-Brown, dzureic@emory.edu |
| Date: 2021-11-02 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: We will first talk about the Ceresa class, which is the image under a cycle class map of a canonical algebraic cycle associated to a curve in its Jacobian. This class vanishes for all hyperelliptic curves and was expected to be nonvanishing for non-hyperelliptic curves. In joint work with Dean Bisogno, Wanlin Li and Daniel Litt, we construct a non-hyperelliptic genus 3 quotient of the Fricke--Macbeath curve with vanishing Ceresa class, using the character theory of the automorphism group of the curve, namely, $\mathrm{PSL}_2(\mathbf{F}_8)$. This will also include the tale of another genus 3 curve by Schoen that was lost and then found again! \\ Time permitting, we will also talk about some Galois cohomology classes that obstruct the existence of rational points on curves, by obstructing splittings to natural exact sequences coming from the fundamental group of a curve. In joint work with Wanlin Li, Daniel Litt and Nick Salter, we use these obstruction classes to give a new proof of Grothendieck’s section conjecture for the generic curve of genus $g > 2$. An analysis of the degeneration of these classes at the boundary of the moduli space of curves, combined with a specialization argument lets us prove the existence of infinitely many curves of each genus over $p$-adic fields and number fields that satisfy the section conjecture. |
| Title: A Multilevel Subgraph Preconditioner for Linear Equations in Graph Laplacians |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Junyuan Lin of Loyola Marymount University |
| Contact: Elizabeth Newman, elizabeth.newman@emory.edu |
| Date: 2021-10-29 at 12:30PM |
| Venue: https://emory.zoom.us/j/94914933211 |
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| Abstract: We propose a Multilevel Subgraph Preconditioner (MSP) to efficiently solve linear equations in graph Laplacians corresponding to general weighted graphs. The MSP preconditioner combines the ideas of expanded multilevel structure from multigrid (MG) methods and spanning subgraph preconditioners (SSP) from Computational Graph Theory. To start, we expand the original graph based on a multilevel structure to obtain an equivalent expanded graph. Although the expanded graph has a low diameter, which is a favorable property for the SSP, it has negatively weighted edges, which is an unfavorable property for the SSP. We design an algorithm to properly eliminate the negatively weighted edges and theoretically show that the resulting subgraph with positively weighted edges is spectrally equivalent to the expanded graph. Then, we adopt algorithms to find SSP, such as augmented low stretch spanning trees, for the positively weighted expanded graph and, therefore, provide an MSP for solving the original graph Laplacian. MSP is practical to find thanks to the multilevel property and has provable theoretical convergence bounds based on the support theory for preconditioning graphs. |
| Title: Approximating dominant eigenpairs of a matrix valued linear operator. |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: GUGLIELMI Nicola of Gran Sasso Science Institute |
| Contact: Manuela Manetta, manuela.manetta@emory.edu |
| Date: 2021-10-22 at 12:30PM |
| Venue: https://emory.zoom.us/j/94914933211 |
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| Abstract: In this talk I will propose a new method to approximate the rightmost eigenpair of certain matrix-valued linear operators, arising e.g. from discretization of PDEs, in a low-rank setting.This is done by means of a suitable gradient system projected onto a low rank manifold. The advantage consists of a reduced memory and computationally convenient procedure able to provide good approximations of the leading eigenpair. Although the results are quite promising, the theory still needs substantial improvements to completely understand the behavior of the method in the more general setting. The talk is inspired by a joint collaboration with D. Kressner (EPFL) and C. Scalone (Univ. L'Aquila). |
| Title: Variations on a theme of Shinzel and Wójcik |
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| Seminar: Algebra and Number Theory |
| Speaker: Matthew Just of Emory University |
| Contact: David Zureick-Brown, dzureic@emory.edu |
| Date: 2021-10-19 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Let $\alpha$ and $\beta$ be rational numbers not equal to 0 or $\pm 1$. How does the order of $\alpha$ (mod $p$) compare to the order of $\beta$ (mod $p$) as $p$ varies? A result of Shinzel and W\'ojcik states that there are infinitely many primes $p$ for which the order of $\alpha$ (mod $p$) is equal to the order of $\beta$ (mod $p$). In this talk, we discuss the problem of determining whether there are infinitely many primes $p$ for which the order of $\alpha$ (mod $p$) is strictly greater than the order of $\beta$ (mod $p$). This is joint work with Paul Pollack. |
| Title: Galerkin Transformer |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Shuhao Cao of Washington University in St. Louis |
| Contact: Yuanzhe Xi, yuanzhe.xi@emory.edu |
| Date: 2021-10-15 at 12:30PM |
| Venue: https://emory.zoom.us/j/94914933211 |
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| Abstract: Transformer in "Attention Is All You Need" is now THE ubiquitous architecture in every state-of-the-art model in Natural Language Processing (NLP), such as BERT. At its heart and soul is the "attention mechanism". We apply the attention mechanism the first time to a data-driven operator learning problem related to partial differential equations. Inspired by Fourier Neural Operator which showed a state-of-the-art performance in parametric PDE evaluation, an effort is put together to explain the heuristics of, and to improve the efficacy of the attention mechanism. It is demonstrated that the widely-accepted "indispensable" softmax normalization in the scaled dot-product attention is sufficient but not necessary. Without the softmax normalization, the approximation capacity of a linearized Transformer variant can be proved rigorously for the first time to be on par with a Petrov-Galerkin projection layer-wise. Some simple changes mimicking projections in Hilbert spaces are applied to the attention mechanism, and it helps the final model achieve remarkable accuracy in operator learning tasks with unnormalized data. The newly proposed simple attention-based operator learner, Galerkin Transformer, surpasses the evaluation accuracy of the classical Transformer applied directly by 100 times, and betters all other models in concurrent research. In all experiments including the viscid Burgers' equation, an interface Darcy flow, an inverse interface coefficient identification problem, and Navier-Stokes flow in the turbulent regime, Galerkin Transformer shows significant improvements in both speed and evaluation accuracy over its softmax-normalized counterparts and other linearizing variants such as Random Feature Attention (Deepmind) or FAVOR+ in Performer (Google Brain). In traditional NLP benchmark problems such as IWSLT 14 De-En, the Galerkin projection-inspired tweaks in the attention-based encoder layers help the classic Transformer reach the baseline BLEU score much faster. |