# 2022-2023 Colloquium Schedule

## Fall 2022

**Arithmetic Progressions of Powers in Cyclotomic Polynomials**

**Speaker:**Hung Chu (WLU Alum '19)- Wednesday November 2nd at 5pm
- Colloquium will be presented via
**Zoom Meeting ID: 912 8071 2538 Password: 668723** **Abstract:**We begin by recalling the definition of complex numbers and cyclotomic polynomials as well as their properties. For each natural number n, we have a unique cyclotomic polynomial of order n, denoted by Φn(x). For example, Φ_{1}(x) = x-1, Φ_{2}(x) = x+ 1, Φ_{3}(x) =x^{2}+x+ 1, Φ_{4}(x) = x^{2}+ 1, and so on. Consider the following two cyclotomic polynomials: Φ_{21}(x) = x^{12}- x 11+ x^{9}- x^{8}+ x^{ 6}- x^{ 4}+ x^{3}- x + 1, Φ_{33}(x) = x^{20}- x^{19}+ x^{17}- x^{16}+ x^{14}- x^{13}+ . . . - x 4+ x 3- x + 1. Observe that the set of powers corresponding to positive coefficients of Φ21(x), denoted by S+n, forms the arithmetic progression 12, 9, 6, 3, 0. Similarly, the set of powers corresponding to negative coefficients of Φ_{33}(x), denoted by S - n , form the arithmetic progression 19,16, 13, . . . , 4, 1. In this talk, we give a necessary condition for when such situations occur. When n is the product of two distinct primes, our necessary condition is also sufficient. As a corollary, we observe that no cyclotomic polynomial has both S+nand S-nin arithmetic progression.

## Winter 2020

### Mathematics & Social Justice

**Speaker**: Karen Saxe (AMS, Macalester College)- March 12th at 5:00 pm in Chavis Hall 105
- Refreshments at 4:40 pm in Chavis Hall foyer
**Abstract**: Societal inequalities pose some of the biggest and most intractable challenges facing our nation today. Can the mathematical sciences help us understand and analyze social inequality? What is the relationship between various imbalances in the U.S. today such as those we see in income distribution and political polarization? This talk will explore answers to these questions, and focus on quantitative approaches that mathematicians, statisticians, and political scientists use to measure inequalities. I can also discuss my advocacy work for the American Mathematical Society, and we’ll have plenty of time for Q & A. No background in mathematics needed.

### Birds, Beer, and Big Data: Non-negative matrix factorization and the intersection of Computer Science, Mathematics, and Statistics

**Speaker**: Julian M. Dymacek (Longwood University)- March 5th at 5:00 pm in Chavis Hall 105
- Refreshments at 4:40 pm in Chavis Hall foyer
**Abstract**: Non-negative matrix factorization (NMF) is used to identify significant patterns within a set of data. This technique has a variety of uses including: data mining, image analysis, and bioinformatics. In this presentation we will explore the mathematical basis for matrix factorization, extend the NMF algorithm to incorporate prior information in the form of pattern constraints, and utilize statistics for understanding our results.

## Fall 2019

### Torsion Points of Elliptic Curves

**Speaker**: Abbey Bourdon (Wake Forest University)- November 14th at 5:00 pm in Chavis Hall 105
- Refreshments at 4:40 pm in Chavis Hall foyer
**Abstract**: An elliptic curve is a geometric object with an especially rich arithmetic structure. These curves are involved in applications that vary from Wiles' proof of Fermat's Last Theorem to secure web browsing, and the challenge of working with them has captured the interest of mathematicians from Weierstrass to Serre. Despite their complexity, elliptic curves and their basic properties may be defined using only high-school algebra, and many important questions can be stated with little additional background. For this reason, they are in a superb position to provide an accessible glimpse of modern mathematics.This talk will provide an introduction to elliptic curves and then focus on questions related to a class of points known as torsion points. No special mathematical background will be required.

### It's all fun and games until someone becomes a mathematician.

**Speaker**: Allison Henrich (Seattle University)- November 7th at 5:00 pm in Chavis Hall 105
- Refreshments at 4:40 pm in Chavis Hall foyer
**Abstract**: As former MAA President Francis Su recently reminded us, PLAY is essential for human flourishing. Whether you are a poet or a scientist, a grandparent or a child, play can powerfully enrich your life. For mathematicians, play is essential for building intuition. For undergraduates, play can inspire a desire to get involved in mathematical research. The world of knots provides fertile ground for understanding these connections. Playing games on knot diagrams can give us intuition about knotty structures, while learning about the theory of knots can reveal the “magic” behind rope tricks and excite us to learn more.

### Some aspects of Ramsey theory in Banach spaces.

**Speaker**: Valentin Ferenczi (Universidade de Sao Paulo)- October 29th at 5:00 pm in Chavis Hall 105
- Refreshments at 4:40 pm in Chavis Hall foyer
**Abstract**: This talk will begin by introducing the classical Ramsey Theorem, as well as Banach spaces and Lp spaces. I will then present some recent and classical Ramsey type theorems in Banach space theory, in particular, a Ramsey type property for finite dimensional Lp spaces. Then I will explain in which respect this result may be seen as a multidimensional Borsuk-Ulam antipodal point theorem. Based on joint work with J. Lopez-Abad, B. Mbombo, S. Todorcevic.

### Easy as ABC: A Marginal Proof of Fermat's Last Theorem

**Speaker**: Alexander Barrios (Carleton Coillege)- October 24th at 5:00 pm in Chavis Hall 105
- Refreshments at 4:40 pm in Chavis Hall foyer
**Abstract**: Elliptic curves have provided the mathematical bridge for solving intractable problems in number theory such as Fermat’s Last Theorem and possibly the ABC Conjecture. Fermat's Last Theorem, first conjectured by Fermat in 1637 states that the only integer solutions to the equation*x*for^{n }+ y^{n}=z^{n}*n>=3*satisfy*xyz=0*. This is a stark contrast to the*n=2*case which has infinitely many integer solutions, for instance,*3*. While Fermat did not provide a proof of this assertion, he famously wrote in the margin of a book that he had a proof which the margin could not contain. In this talk, we discuss the history of Fermat's Last Theorem and introduce the ABC Conjecture, an incredibly powerful statement which as we will see, can provide a marginal proof of Fermat's Last Theorem in its explicit form. We will also introduce elliptic curves which are implicit functions of the form^{2}+4^{2}=5^{2}*y*with^{3}=x^{2}+Ax+B*A*and*B*integers and the property that the derivative exists at every point on the graph of this implicit function. From here we show how Fermat's Last Theorem and the ABC Conjecture can be rephrased in the language of elliptic curves and conclude the talk with some of the surprising applications of elliptic curves to cybersecurity.

### The Parallel Postulate and Geometries without Rigidity

**Speaker**: Edwin O'Shea (James Madison University)- October 15th at 5:00 pm in Chavis Hall 105
- Refreshments at 4:40 pm in Chavis Hall foyer
**Abstract**: The discovery of non-Euclidean geometry in the 19th century elicited two distinct reactions from mathematicians wishing to preserve the ethos of classical geometry: address the crises of foundations (led by David Hilbert and others) and recast geometry vis-a-vis transformation groups (led by Felix Klein and others). This talk aims to build a bridge between these paradigms by exploring axiomatic models of geometry without Hilbertian Side-Angle-Side congruence / Kleinian rigidity. Using these models, we show that the classical equivalence of Euclid’s parallel postulate and Playfair’s axiom collapses in the absence of SAS and that a perfectly standard sum of angles property is equivalent to SAS. The talk is introductory: I won’t have assumed you have studied geometry since high school.This is joint work with my colleague, Elizabeth T. Brown and our four REU students, Emily Castner, Stephen Davis, Edouard Seryozhenkov, and AJ Vargas.

### What We Did Last Summer (Parents and Family Weekend Event)

**Speakers**: W&L Math students: Maia Baldridge, Katie Cones, Jacob Kintzing, Emily Matthews, John Coleman Ward and Yoseph Wolde- October 4th at 4:00 pm in Chavis Hall 105
- Refreshments at 3:40 pm in Chavis Hall foyer
**Abstract**: The students will talk about their summer research on as part of the RANCID GROUP studying Sierpinski time, spirolateral graphs and bounding boxes.