Counting points on algebraic curves over finite fields has numerous applications in communications and cryptology, and has led to some of the most beautiful results in 20th century arithmetic geometry. A natural generalization […]
We review a beautiful 17th century result by the philosopher Rene Descartes: a univariate real polynomial with t monomial terms has no more than t-1 positive roots. We then see how one can prove a generalization that counts roots of two bivariate polynomials (with few monomial terms), using nothing more than basic calculus. In other […]
TOPIC: The Mathematics of Information We are surrounded by information. Words in books, ones and zeros in computers, mathematical equations, and DNA sequences are all examples of information, but can […]
Markov chains are widely used throughout mathematics, statistics, and the sciences, often for modelling purposes or for generating random samples. In this talk I’ll discuss a different, more recent application […]
If $F$ is a finite field and $d$ is a positive integer relatively prime to $|F^\times|$, then the power map $x \mapsto x^d$ is a permutation of $F$, and so is called […]
I will discuss joint work with Jim Hoste, where we prove that a unique folded strip of paper can follow any polygonal knot with odd stick number. In the even […]
In 2007, Dr. Maria D'Orsogna learned of proposed oil activities in her home region of Abruzzo, Italy. Century-old wineries were to be uprooted to build clusters of oil wells, refineries and pipelines, turning scenic Abruzzo into an oil district. Although based in California, 6,000 miles away, Dr. D'Orsogna took it upon herself to raise awareness […]
Clustering in image analysis is a central technique that allows to classify elements of an image. We describe a simple clustering technique that uses the method of similarity matrices, and an algorithm in which a collection of image elements is treated as a dynamical system. Efficient clustering in this framework is achieved if the dynamical system admits […]
Mathematicians like to count things. Often in very complicated and fancy ways. In this talk I will explain how we can use quantum Airy structures -- an abstract formalism recently proposed by Kontsevich and Soibelman, underlying the Eynard-Orantin topological recursion -- to count various interesting geometric structures. Quantum Airy structures can be seen as a […]
This triple-header of topology talks will include three speakers: First, Hyeran Cho from The Ohio State University will speak about Derivation of Schubert normal forms of 2-bridge knots from (1,1)-diagrams. In this talk, we show that the dual (1, 1)-diagram of a (1, 1)-diagram (a.k.a. a two pointed genus one Heegaard diagram) D(a, 0, 1, […]
A fascinating fact on mathematics is that there are many interesting connections between seemingly different mathematical disciplines. In this talk, I will present a surprising formula counting integral points on polygons and sketch its proof. We will see a delightful interaction between algebra, combinatorics, and geometry. This talk aims primarily for undergraduate students. No prerequisite […]
We review some recent developments in the study of biquandle brackets and other quantum enhancements.
