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.
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.
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.