Upcoming Seminars

Title: Local-global principles on stacky curves and solving generalized Fermat equations
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: Independent transversals in multipartite graphs
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: Minimal and nilpotent images of Galois for elliptic curves
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: Crossing and Color
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.