M@X -- Math At Xavier

Spring 2024 seminar
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Mathematics Department
Xavier University of Louisiana

Spring 2024 seminar schedule

Date	Place	Speaker	Title
Tue Jan 16, 12:15pm	NCF 574


Canceled due to inclement weather

Date	Place	Speaker	Title
Tue Jan 30, 12:15pm	NCF 574	Olivia Beckwith (Tulane U.)	The Arithmetic of Quadratic Integers


Summary Since the beginning of number theory, mathematicians have been fascinated by the patterns in the integers and the elusive nature of the distribution of primes. Beyond the integers, algebraic number theory provides an infinite supply of number systems whose arithmetic possess similarities with that of the integers as well as intriguing differences. The simplest examples involve square roots of integers, and aside from being interesting in their own right, these gadgets are powerful tools in the study of modular forms, elliptic curves and L-functions. We will discuss some open questions and a recent result in this area. Originally from Columbus, Ohio, I first became interested in number theory when I attended the Ross Program at the Ohio State University as a teenager. I completed my bachelor's degree at Harvey Mudd College in 2013 and my PhD at Emory University under the supervision of Ken Ono in 2018. Then I held postdoc positions at the University of Bristol and the University of Illinois at Urbana-Champaign before coming to Tulane University in 2021. I study modular forms and number theory.
Click here to access the slides from Dr. Beckwith's talk

Date	Place	Speaker	Title
Tue Feb 6, 12:15pm	NCF 574	Christophe Vignat (Tulane U. and U. of Paris)	Dubious Identities: A Visit to the Borwein Zoo


Summary This talk will show some contributions to the zoo of dubious identities established by J.M. and P.B. Borwein in their legendary 1992 article, "Strange Series and High Precision Fraud" with five new entries, each of a different variety than the last. Some of these identities are again a high precision fraud and picking out the true from the bogus can be a challenging task with many unexpected twists along the way. This is joint work with Zachary Bradshaw. Christophe Vignat earned his PhD in physics from Université Paris-Sud 11, Orsay, now known as Université Paris-Saclay. He is now Professor at the physics department of this same university and a member of the Laboratoire des Signaux et Systèmes at CentraleSupelec. In the past, he has benefited from multiple invitations to the Tulane University department of Mathematics. His research interests are centered around experimental mathematics, special functions and symbolic computation. Department of Physics, Université Paris Saclay, L.S.S, CentraleSupélec, Orsay, 91190, France christophe.vignat@universite-paris-saclay.fr Department of Mathematics, Tulane University, New Orleans LA 70118 cvignat@tulane.edu
Click here to access the slides from Dr. Vignat's talk

Date	Place	Speaker	Title
Tue Feb 27, 12:15pm	NCF 574	Christophe Vignat (Tulane U. and U. of Paris)	Borwein integrals and Bernoulli Sophomore's Dream identity, from continuous to discrete


Summary The aim of this talk is to illustrate how integrals and sums can be related, based on two famous examples: Borwein integrals of products of functions of the form \(\sin(x)/x\) (called sine cardinal functions), and Bernoulli's Sophomore's dream identity that involves the function \(x^x\). The surprising behavior of Borwein integrals will be explained using basic techniques of Fourier analysis and generalized to other functions. The sum/integral duality of Bernoulli's identity will be illustrated in higher dimensions.
Click here to access the slides from Dr. Vignat's talk

Date	Place	Speaker	Title
Tue Mar 12, 12:15pm	NCF 574	Charles Burnette	The distribution of factor trees for random integers


Summary Given a natural number \(n \gt 2,\) a factor tree for \(n\) is a tree diagram where \(n\) is at the top of the tree, the terminal nodes are the prime factors of \(n\), and every non-terminal node is the product of the two nodes below it. Factor trees are often taught to children in elementary school as a visual aid for finding the prime factorization of an integer. However, the complexity of the tree is dependent on quite a few factors (pun very much intended). In this talk, we will see how to count the number of factor trees for a given \(n\) and study the distribution of various structural parameters of the random factor tree, such as its height and size. Along the way, we will learn about some important combinatorial objects, namely binary trees, the Catalan numbers, Dirichlet series and other related generating functions.

Date	Place	Speaker	Title
Tue Mar 19, 12:15pm	NCF 574	Peter Bierhorst (UNO) plbierho@uno.edu	Quantum Nonlocality - a Mystery and its Applications


Quantum Nonlocality - sometimes referred to as "spooky action at a distance" - is a fascinating phenomenon in which two entangled particles, separated at great distances, can display strange and counterintuitve correlations with each other. While Einstein believed that the nonlocal effects in quantum mechanics could be eliminated with a more complete local theory, John Bell proved in the 1960s that this would be impossible for certain experiments. These "Bell test experiments" are technically challenging, however, and were only definitively performed in 2015. In my talk, I will review the history of Bell experiments, some of which were recognized with the 2022 Nobel Prize in Physics, and explain why they are so challenging while discussing my own contribution to the 2015 experimental effort. I will also talk about my current research in the field, including applications to random number generation, secure communication, and the study of quantum networks. Peter Bierhorst earned a PhD in Mathematics from Tulane University in 2014, then worked as a postdoctoral associate at the National Institute of Standards and Technology in Boulder, CO from 2015-2018, where he contributed to the design and analysis of experiments in quantum nonlocality and its applications. Since 2018 he has been an assistant professor in the mathematics department at the University of New Orleans, continuing research into quantum nonlocality, information theory, and quantum networks.
Click here to access the slides from Dr. Bierhorst's talk

Date	Place	Speaker	Title
Tue Apr 9, 12:15pm	NCF 574	John Perry	A former academic learns he can "use this stuff"


Summary Students! Do you ever wonder what it might be like to work as a mathematics major in industry? After 15 years as a mathematics professor, I decided in 2020 to try my hand at "software development". For various reasons, I worried about the shift; for example, would the hyper-specialization of a PhD in mathematics and 15 years of following my own muse in research make me a bad fit? Turns out it didn't - and I don't even need my graduate school mathematics to be a valuable mathematician! - Sort of. This talk describes similarities and differences of work in academia and industry, offers advice for finding a job in either, and concludes with one or two problems inspired by the work I personally do. While much of it is anecdotal, I'll describe the experience of some former students and advisees, as well as point to news items and industry research. - John Perry earned a PhD in mathematics from North Carolina State University. - Before he moved to industry, he taught at a wide gamut of academic institutions, moving to Mississippi for the opportunity to pursue his research at the University of Southern Mississippi, where he published 11 refereed papers in the Journal of Symbolic Computation, Mathematics Magazine, The College Math Journal, and others. - He successfully directed 1 PhD dissertation, 2 Master's theses, and somewhere from 10-15 undergraduate research projects. - He now works for Peraton Engineering as an "Analyst" and works on a team of mathematicians, computer scientists, and other STEM specialists.
Click here to access the slides from Dr. Perry's talk

Date	Place	Speaker	Title
Tue Apr 16, 12:15pm	NCF 574	Mahir Can (Tulane U.)	Harmony in Algebraic Geometry: Exploring Schubert Varieties through Fibonacci and other Numbers


Summary In this talk, we will discuss our notion of a nearly toric variety, focusing specifically on the set of Schubert varieties. We will show how Fibonacci numbers arise in this context. Our focus then shifts to Schubert varieties indexed by the 312-avoiding permutations. After discussing their combinatorial and geometric significance, we identify among these Schubert varieties the nearly toric ones. This is a joint work with Nestor Diaz from Tulane University. Dr. Mahir Bilen Can is a Full Professor of Mathematics at Tulane University. He received his B.Sc. degree from Middle East Technical University in 2001 and completed his Ph.D. at the University of Pennsylvania in 2006. Dr. Can has published 50+ research papers in esteemed mathematics journals and edited two books. He serves on the editorial boards of Semigroup Forum and the Journal of G"okova Geometry Topology. Dr. Can has mentored several Ph.D. and Master's students. He has organized numerous conferences and delivered lectures at prestigious institutions worldwide. Dr. Can has also received multiple awards for his research, including a recent grant from the Louisiana Board of Regents (2023-2025) for his project on Representation Theory and Quantum Error Correction. Recently, Dr. Can has been awarded the prestigious Fulbright US Scholar award.
Click here to access the slides from Dr. Can's talk