A Trip Through Knot Theory
How can we tell when two knots are truly different? Topologists approach this question by studying invariants, immutable properties of knots. One invariant is the Jones polynomial which has been conjectured to detect the unknot. In this talk we’ll explore how trip matrices provide a way to study and compute the Jones polynomial and their connection to Gauss codes. We’ll provide the necessary background of the basics of knot theory and then build up to discussing these research tools. The talk will be accessible to undergraduate students who have taken calculus.
Fair Dice and Where to Find Them
With the exploding popularity of tabletop role-playing games over the last several years, many people are familiar with the funny-shaped dice used for these and other games. Are there other shapes we can use for dice? How can we know if these shapes are fair? How can a mathematician approach these questions? We will examine these and related questions through the lens of group theory. We will also try to show how one could start to answer these questions themselves. Prior experience with proofs is helpful but not required.
Cycles in Graph-Based Codes
In this talk we will introduce the basics of graph theory and coding theory, discuss why cycles in graphs are important to the study of graph-based codes, and explore an example that applies recent results in the field.
The Resultant of Two Quadratics
Determining when multiple polynomials have a common root is a classical problem in algebra and number theory. In 1840, J.J. Sylvester introduced what is now called the Sylvester matrix, which led to the notion of the resultant, a tool for detecting when two polynomials share a solution.
Since then, this idea has been generalized to determine when several homogeneous polynomials in several variables have a common solution, with applications in areas such as geometric modeling, robotics, and computer vision. In this talk, we will introduce the notation and basic ideas behind the classical resultant, and then compute the resultant explicitly in the special case of two quadratic polynomials.
Nonlocal models: Analytical and Modeling Aspects
A variety of phenomena such as fracture, phase separation, cell migration are driven by interactions over finite neighborhoods that classical local PDEs struggle to capture. This issue is addressed by formulating the systems in a nonlocal framework. In this talk I will present a formulation of nonlocal calculus based on integral operators that use antisymmetric kernels, thus defining a nonlocal gradient, divergence, curl, and Laplacian, and showing how these operators recover their classical counterparts as the interaction radius shrinks. Furthermore, I will present some mathematical tools in nonlocal theory, such as boundedness results and nonlocal Poincaré inequalities, which are essential in theoretical studies of nonlocal systems. If time permits, I will conclude with an application of nonlocal frameworks by introducing a nonlocal model for cell migration in tumors.
Knot Theory: An Introduction
Winning Ways for Your Mathematical Plays
Let's play some games! In this talk, we'll explore the world of combinatorial game theory, which uses mathematics to analyze 2-player games of pure strategy. We'll look at the game of NIM and see how to win this game every time. Using our knowledge of NIM, we'll develop a winning strategy for a large class of games, called impartial games. This strategy will follow from a beautiful structure theorem for impartial games called the Sprague-Grundy theorem.
Winning the Lottery with Finite Geometry
Geometry is one of the oldest areas of mathematics; the Euclidean plane is the most famous and most studied geometry. But what happens as we strip away the axioms until we only need finitely many points? Affine planes are one of these finite geometries, and in this talk we will see some results centered around them, how they connect to a failed Euler conjecture, Latin squares, military displays, and winning the lottery.
To Knot or to Unknot?
Knot theory is an active area of research in low-dimensional topology, much of which centers on a deceptively simple question: how do we tell knots apart? In this talk, we will represent knots using two-dimensional diagrams and explore the tools used to study and compare them. We will introduce key ideas such as Reidemeister moves and crossing number to distinguish knot diagrams, as well as the unknotting number to measure how difficult a knot is to untangle. These seemingly simple questions lead to rich problems that mathematicians are still actively investigating today.