Y20-21

Sep 23

Speaker: Dr. Ali Uncu,

Title: Where do the maximum absolute q-series coefficients of $(1-q)(1-q^2)...(1-q^n)$ occur?

Abstract: A generalization of factorials, $n! := 1.2...(n-1).n, is the products $(1-q)(1-q^2)...(1-q^{(n-1)})(1-q^n)$; they are conveniently named as $q$-rising factorials. These polynomials are a source of many interesting properties. They are especially widely used in combinatorics and theory of partitions as the main constructive ingredient of generating functions.

In a recent study, joint with Alexander Berkovich (University of Florida), we found a way to classify the $q$-rising factorials with respect to their maximum absolute coefficient. This lead us to a two-year-long experimental search for the location of these maximum absolute coefficients. We used the MACH2 supercomputer to study coefficients in the q-series expansion of $(1−q)(1−q^2)…(1−q^n)$, for all $n \leq 75,000$. As a result, we were able to conjecture some periodic properties associated with the before unknown location of the maximum coefficient of these polynomials with odd n. Remarkably the observed period is $62,624$.

In this talk, we would like to present how this research evolved over time and what our conjectures might mean for the future.

Oct 7

Speaker: Dr. Kyle Bradford

Title: Straus-Erdös Conjecture

Abstract: In this talk I describe the history of the Erdös-Straus Conjecture and outline some recent advances. This Number Theory problem has been unsolved for 75 years and is easily accessible to all mathematicians. In a short statement for any integer n greater than 1 there exist positive integers x,y and z so that the following diophantine equation holds: 4/n = 1/x + 1/y + 1/z. Please join the conversation and lend us your voice as we try to solve the problem as a community.

Oct 21

Speaker: Dr. Jim Brawner

Title: Wise Women Wearing Hats

Abstract: There is a long history of recreational mathematics problems involving hat wearers trying to guess the color or the number associated with the hat they are wearing. Typically, others can see their hat, but they cannot see their own. Occasionally, the hat wearers are prisoners of a diabolical warden who can free them or execute them depending on whether they guess correctly. Invariably, the hat wearers use logic to determine an optimal strategy. In this month's American Mathematical Monthly, a problem in memory of the late John Horton Conway asks a number of intriguing questions about wise women wearing hats with numbers. We will explore these questions and give a brief survey of problems about hats.

Nov 4

Speaker: Dr. Jeremy Rouse

Title: Provably finding rational points on curves.

Abstract: The problem of finding all of the integer or rational solutions to an equation is one of the oldest problems in number theory. In this talk, we state Faltings's theorem which guarantees that many curves only have finitely many rational points on them. We then discuss five methods for provably finding all of the rational points on a given curve. We will focus on those methods that are connected with elliptic curves, and many examples will be included.

Nov 18

Speaker: Dr. Saeed Nasseh

Title: INTERACTIONS BETWEEN COMMUTATIVE ALGEBRA AND OTHER AREAS: SOLUTIONS TO LONG-STANDING CONJECTURES

Abstract: Commutative algebra and algebraic geometry are inseparable fields and have a long history. In 1950's, homological algebra (that originates from algebraic topology) also made its way into the commutative algebra toolbox in the works of Auslander-Buchbaum and Serre. Tight connections between commutative algebra, representation theory, and deformation theory appeared in the past century in numerous places including the works of Auslander as well. In this talk, I introduce two long-standing major conjectures in commutative algebra { namely, Vasconcelos' Conjecture (1974) and the Auslander-Reiten Conjecture (1975) { that have been solved in my recent works (one of them completely and the other one partially) using techniques from the above-mentioned areas of mathematics plus linear algebra, group theory, invariant theory, and rational homotopy theory. I will describe, in down-to-earth terms, the cast of characters that play fundamental roles in the solutions of these conjectures. Finally, I will sketch a new method that uses partial differential equations and I will show evidences that it might be used in solving the Auslander-Reiten Conjecture in its full generality.

Feb 3

Speaker: Dr. Jim Brawner

Title: Turn Dials, Triangular Numbers, and an Unproperty of the Powers of Two

Abstract: In this talk we will investigate integers that can be expressed as the sum of two or more consecutive positive integers and discuss an interesting correspondence with odd divisors. We will then use this correspondence to examine the distribution of triangular numbers modulo a positive integer n. This talk is suitable for a general audience; all mathematical sciences students are especially welcome to attend!

Feb 17

Speaker: Dr. Daniel Gray

Title: Introduction to Permutation Patterns

Abstract: Central to the formal study of any mathematical object is the concept of a substructure which inherits the properties of that object, i.e. subsets of sets or subspaces of vector spaces. In the case of permutations, these `subpermutations' are called patterns. In this talk, we will formally define what a pattern is, show how the definition arises naturally out of an application, discuss the history, and briefly outline some of the ``classical results'' in permutation patterns.

Mar 3

Speaker: Dr. Ha Nguyen

Title: Incorporating Social Justice into Mathematics Instruction to Impact Preservice Teachers

Abstract: In this talk, we will share how we planned and implemented a teaching mathematics for social justice lesson in a mathematics content course for K-8 preservice teachers (PSTs). We will then present findings on changes in beliefs of our K-8 PSTs about the importance of cultural, social, or political knowledge as they learned probability and sampling through a social justice lens. This talk is accessible to a general audience, including all mathematics and mathematics education students.

Mar 24

Speaker: Dr. Sungkon Chang

Title: The weak converse of Zeckendorf Theorem

Abstract: Zeckendorf's Theorem states that each positive integer is uniquely written as a sum of nonadjacent Fibonacci terms. This neat result has a striking feature, called the converse of Zeckendorf's theorem, which states that the Fibonacci sequence is the only sequence of positive integers that uniquely represents each positive integer in such a fashion. If we further impose a monotone property on the sequences, we call the result the weak converse of Zeckendorf's theorem. In our recent work, we introduced a general approach to conditions that yield a result such as Zeckendorf's theorem not only for integers, but also for real numbers and p-adic integers. In this talk, we shall introduce our results for integers and real numbers, and various open problems.

Apr 28

Speaker: Dr. Paul Sobaje

Title: Dimensions and Characters of Representations

Abstract: In the representation theory of a finite group, Lie group, or algebraic group, one hopes to first classify all of the irreducible representations, and then give further information regarding their dimensions and their "characters." A famous theorem by Hermann Weyl completes this latter step in the case of compact Lie groups, and also applies to representations of certain algebraic groups over the complex numbers. The corresponding problem over fields of prime characteristic is still unresolved. We will give special attention as to why this is true in the case of special linear groups, and will work through several concrete examples.