M@X -- Math At Xavier

Mathematics Department
Xavier University of Louisiana

Fall 2024 seminar schedule

Date	Place	Speaker	Title
Thu Oct 17, 12:15pm	NCF 574	Navvye Anand California Institute of Technology (student)	On Bounds and Diophantine Properties of Elliptic Curves
Summary Mordell equations are celebrated equations within number theory and are named after Louis Mordell, an American-born British mathematician, known for his pioneering research in number theory. In this talk, we discover all Mordell equations of the form \(y^2 = x^3 + k\), where \(k \in \mathbb Z\), with exactly \(|k|\) integral solutions. We also discover explicit bounds for Mordell equations, parameterized families of elliptic curves, and twists on elliptic curves. Using the connection between Mordell curves and binary cubic forms, we improve the lower bound for the number of integral solutions of a Mordell curve by looking at a pair of curves with unusually high rank.
Click here to access the slides from Navvye Anand's talk
	Navvye (pronounced Nav-yay!) is a freshman at Caltech, majoring in Applied and Computational Mathematics with a minor in Control and Dynamical Systems. His passion for mathematics began at an early age, leading him to win a national competition in number theory. Beyond number theory, Navvye is particularly interested in chaos theory and graph theory. Navvye's work in Computational Linguistics, aimed at reviving the endangered Kangri language, earned him the 2023 Spirit of Ramanujan Award. Apart from math and linguistics, he enjoys playing chess and watching basketball: his favorite chess player is Mikhail Tal, and his favorite team is the Lakers.

Date	Place	Speaker	Title
Tue Oct 29, 12:15pm	NCF 574	Angelos Koutsianas Aristotle University of Thessaloniki	Local-global principle of elliptic curves
Summary Let \(E\) be an elliptic curve over a number field \(K\) and \(\ell\) a prime number. Homomorphisms \(\phi:E\rightarrow E^\prime\) with finite kernel of order \(\ell\) defined over \(K\) are called \(K\)-rational \(\ell\)-isogenies and have been studied a lot because of their important arithmetic and cryptographic properties. Given an elliptic curve \(E\) over \(K\) with a \(K\)-rational \(\ell\)-isogeny we know that the reduction of \(\tilde{E}_{\mathfrak{p}}/\mathbb{F}_{\mathfrak{p}}\) also has a \(\mathbb{F}_{\mathfrak{p}}\)-rational \(\ell\)-isogeny for almost all primes \(\mathfrak{p}\). In this talk I will explain when the converse holds and how it is related to the determination of rational points on modular curves. I will also explain how we can compute rational points on curves with the Chabauty-Coleman method. No prior knowledge of Number Theory will be assumed while a brief introduction to all new objects will be given.
Click here to access the slides from Dr. Koutsianas's talk
		Angelos Koutsianas finished his PhD at the Mathematical Institute, University of Warwick, under the supervision of Prof. John Cremona (2012-2016), who is a well-known name in the field of Elliptic Curves and Modular Forms. In his thesis, he studied the relation between S-unit equations and elliptic curves with good reduction outside S. After the University of Warwick, he moved to Ulm University to work with Professors Irene Bouw and Stefan Wewers. Then, he spent one year at the University of Piraeus before moving again abroad to the University of British Columbia in Canada to work with Professor Michael Bennett. Then he spent one year at Université Clermont Auvergne et CNRS, working with Prof. Nicolas Billerey. Since Fall 2021, he has been an Assistant Professor at the Department of Mathematics at Aristotle University of Thessaloniki, Greece. Currently, he is interested in Computational Algebraic Number Theory and Arithmetic Geometry, Elliptic Curves, Rational points on Curves, Galois Representations, Diophantine Equations, Modular Forms and Cryptography. He is also a co-organizer of the Greek Algebra and Number Theory Seminar.