All Seminars
| 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: MSC W303 |
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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: 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: 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: Independent transversals in multipartite graphs |
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| Seminar: Discrete Math |
| Speaker: Yi Zhao, PhD of Georgia State University |
| Contact: Dr. Cosmin Pohoata, cosmin.pohoata@emory.edu |
| Date: 2025-02-19 at 4:00PM |
| Venue: MSC E408 |
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| Abstract: An independent transversal in a multipartite graph is an independent set that intersects each part in exactly one vertex. We show that given any positive even integer r, every r-partite graph with parts of size n and maximum degree r n / (2r-2) - t (t>0) contains c t n^{r-1}) independent transversals. This is best possible up to the constant c=c_r, confirming a conjecture of Haxell and Szabo from 2006 and partially answering a question Erdos from 1972 and a question of Bollobas, Erdos and Szemeredi from 1975. We also show that for every s\ge 2, even r\ge 2 and sufficiently large n, every r-partite graph with parts of size n and maximum degree \Delta |
| Title: Local-global principles on stacky curves and solving generalized Fermat equations |
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| Seminar: Algebra |
| Speaker: Yidi Wang, PhD of University of Western Ontario |
| Contact: Deependra Singh, deependra.singh@emory.edu |
| Date: 2025-02-18 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: The primitive solutions of certain generalized Fermat equations, i.e., Diophantine equations of the form Ax^2+By^2 = Cz^n, can be studied as integral points on certain stacky curves. In a recent paper by Bhargava and Poonen, an explicit example of such a curve of genus 1/2 violating local-global principle for integral points was given. However, a general description of stacky curves failing the local-global principle is unknown. In this talk, I will discuss our work on finding the primitive solutions to equation of the form by studying local-global principles for integral points on stacky curves constructed from such equations. The talk is based on a joint project with Juanita Duque-Rosero, Christopher Keyes, Andrew Kobin, Manami Roy, and Soumya Sankar. |
| 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. |