Upcoming Seminars
| Title: Inner-Product Free Krylov Methods for Inverse Problems |
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| Defense: Dissertation |
| Speaker: Ariana Brown of Emory University |
| Contact: Ariana Brown, ariana.brown@emory.edu |
| Date: 2025-02-21 at 1:00PM |
| Venue: MSC W201 |
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| Abstract: Iterative Krylov projection methods have become widely used for solving largescale linear inverse problems. Certain methods that rely on orthogonality require inner-products, which create a bottleneck for parallelization and causes the algorithms to fail in low-precision. As a result, there is a need for more effective iterative methods to alleviate this computational burden. This study presents new Krylov projection methods that do not require inner products to solve large-scale linear inverse problems.\\ \\ The first iterative solver is known as the Changing Minimal Residual Hessenberg method (CMRH). The second is a new extension of CMRH to rectangular systems which we call the least squares LU method (LSLU). We further adapt both approaches to efficiently incorporate Tikhonov regularization. These methods are labeled as Hybrid CMRH and Hybrid LSLU. Each of these techniques are known as quasi-minimal residual methods rather than minimal residual methods. Still, these methods do not offer a way to control how closely the quasi-norm approximates the desired norm. In this work, we also propose a new Krylov method that is both inner-product free and minimizes a functional that is theoretically closer to the residual norm. The new scheme combines the conventional CMRH method and the newly proposed LSLU method with a randomized sketch-and-solve technique to solve the strongly overdetermined projected least-squares problem. Extensive numerical examples illustrate the effectiveness of all methods in this dissertation. |
| Title: Minimal and nilpotent images of Galois for elliptic curves |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Jeremy Rouse of Wake Forest University |
| Contact: Santiago Arango, santiago.arango@emory.edu |
| Date: 2025-02-25 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: If $K$ is a number field and $E/K$ an elliptic curve, then for every positive integer $n$, there is a Galois representation $\rho_{E,n} : G_{K} \to {\rm GL}_{2}(\mathbb{Z}/n\mathbb{Z})$. If $K = \mathbb{Q}$, $\det \circ \rho_{E,n} : G_{\mathbb{Q}} \to (\mathbb{Z}/n\mathbb{Z})^{\times}$ is surjective. We say that a subgroup $H$ of ${\rm GL}_{2}(\mathbb{Z}/n\mathbb{Z})$ is \emph{minimal} if $\det : H \to (\mathbb{Z}/n\mathbb{Z})^{\times}$ is surjective. We show that essentially the only way for the image of $\rho_{E,n}$ to be minimal is for $n$ to be a power of $2$, and that minimal subgroups of ${\rm GL}_{2}(\mathbb{Z}/2^{k} \mathbb{Z})$ are plentiful.\\ \\ The question of minimality is connected with the question of when the Galois group of $\mathbb{Q}(E[n])/\mathbb{Q}$ is a nilpotent group. In 2016, Lozano-Robledo and Gonz\'alez-Jim\'enez showed that if $E/\mathbb{Q}$ is an elliptic curve and ${\rm Gal}(\mathbb{Q}(E[n])/\mathbb{Q})$ is abelian, then $n \in \{ 2,3,4,6, 8\}$. We show that, subject to a positive answer to Serre's uniformity question, if $E/\mathbb{Q}$ is a non-CM elliptic curve and ${\rm Gal}(\mathbb{Q}(E[n])/\mathbb{Q})$ is nilpotent, then $n \in \{ 2^{k}, 3, 5, 6, 7, 15, 21 \}$.\\ \\ All of the work in this talk is joint with Harris Daniels. |
| Title: Dispersive estimates for the discrete Schrödinger equation on a honeycomb lattice |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Younghun Hong of Chung-Ang University |
| Contact: Maja Taskovic, maja.taskovic@emory.edu |
| Date: 2025-02-28 at 11:00AM |
| Venue: TBA |
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| Abstract: The discrete Schrödinger equation on a two-dimensional honeycomb lattice is a fundamental tight-binding approximation model that describes the propagation of electrons on graphene. By the Fourier transform on the honeycomb lattice, the free Schrödinger flow can be represented by a certain oscillatory integral whose phase function has conical singularities at Dirac points as well as degeneracy at some other frequencies. We show that the degenerate frequencies are completely characterized by three symmetric periodic curves, and that the three curves meet at Dirac points. Based on this observation, we prove the dispersion estimates for the free flow estimating the oscillatory integral. Our proof is direct and uses only elementary m |
| Title: Crossing and Color |
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| Seminar: Combinatorics |
| Speaker: János Pach, PhD of Rényi Institute of Mathematics, Budapest |
| Contact: Liana Yepremyan, liana.yepremyan@EMORY.EDU |
| Date: 2025-02-28 at 4:30PM |
| Venue: MSC W301 |
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| Abstract: Turán defined cr(G), the crossing number of a graph G, as the smallest number of edge crossings in a proper drawing of G in the plane. This notion has turned out to play an important role in combinatorial geometry, additive number theory, chip design, and elsewhere. The computation of cr(G) is a classical NP-hard problem, so it is not surprising that there are very few graphs whose crossing numbers are known. In particular, we do not even know the asymptotic value of the crossing number of the complete graph K_r on r vertices, as r tends to infinity. Nevertheless, Albertson made the conjecture that cr(G) is at least cr(K_r), for any graph G whose chromatic number is at least r. After giving a short and biased survey of some important results on crossing numbers, we explain the relationship between crossings and coloring, and settle Albertson's conjecture for graphs whose number of vertices is not much larger than their chromatic number. Joint work with Jacob Fox and Andrew Suk. |
| Title: From Uncertainty Aware to Decision Ready: Specialized UQ Methods for High-Stakes Predictive Modeling |
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| Defense: Dissertation |
| Speaker: Shifan Zhao of |
| Contact: Shifan Zhao, shifan.zhao@emory.edu |
| Date: 2025-03-26 at 11:30AM |
| Venue: MSC E406 |
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| Abstract: Uncertainty quantification (UQ) is essential for reliable decision-making in predictive modeling, particularly those with high-stakes outcomes. This thesis develops a unified framework that tailors uncertainty quantification methods to AI foundation models with distinct application domains. For stationary foundation models, we enhance traditional Gaussian Process regression—through kernel preconditioning and a two-stage modeling approach—to address computational inefficiencies, approximation bias, and model misspecification, thereby improving uncertainty estimates. For nonstationary foundation models, we integrate conformal prediction techniques to exploit theoretical guarantees of data coverage. We apply our methods to medical and climate foundations models, and numerical experiments demonstrate that our targeted approaches produce reliable and actionable estimates of uncertainty. This work shows the potential to substantially advance the state of predictive modeling for both healthcare and extreme weather applications. |