Algebra / Number Theory / Combinatorics Seminar

A triple of natural numbers (a,b,c) is an S-set if a+b=c. I. Schur used the S-sets to show that for n >3, there exists s(n) such that for prime p […]

Augusta Ada Byron King Lovelace (1815-1852) is today celebrated as the first computer programmer in history. This might be confusing to some because in 1852 there were no machines that […]

One of the main drivers of current research in geometry is the classification of Calabi-Yau threefolds.  Towards this effort, a particular approach in algebraic geometry is via the study of […]

The classical Frobenius problem asks for the largest integer not representable as a non-negative integer linear combination of a relatively prime integer n-tuple. This problem and its various generalizations have […]

In Euclidean geometry, the sum of  two sides of any  triangle is greater than the third side. We  introduce this idea to labeling of graphs. A (p,q)-graph G=(V,E) is said to […]

An “Adinkra” is a graphical tool to describe a branch of particle physics known as supersymmetry. Understanding the mathematics of Adinkras shines a light on the underlying physics, as well […]

The classical, one-boundary, and two-boundary Temperley-Lieb algebras arise in mathematical physics related to solving certain rectangular lattice models.They also have beautiful presentations as "diagram algebras", meaning that they have basis elements depicted as certain kinds of graphs, and multiplication rules are given by stacking diagrams and gluing of vertices. In this talk, we will explore […]

In this talk, I will give an overview of the theory of matroids. These are mathematical objects which capture the combinatorial essence of linear independence. Besides providing some basic definitions of this theory, I will discuss several examples of matroids and explain some connections with optimization. Also, in this talk, I will introduce matroid polytopes, […]

A classic and fundamental result in number theory is due to Mordell who proved that the set of points on an elliptic curve defined over a number field forms a finitely generated abelian group; in particular, it has a finite torsion subgroup. An essential tool to study elliptic curves is the modular curves which are […]

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 is to count the number of points over prime power rings, e.g., the integers modulo a prime power. However, the theory behind the latter kind of point […]

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 a power permutation of $F$. For any function $f: F \to F$, and $a, b \in F$, we define the differential multiplicity of $f$ with respect to […]

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 […]