Course 1, Week 1: Introduction to Graph Theory

Lecturer: Emily Heath (California State Polytechnic University, Pomona)

Short Description:

Course 2, Week 1:

Lecturer: Anastasia Chavez (Saint Mary’s College of California)

Short Description: Independently introduced by Whitney and Nakasawa in the 1930's, matroids are a fundamental combinatorial object that miraculously link together several areas of mathematics: graph theory, linear algebra, abstract algebra, and many more! This means that results about matroids can have far reaching implications. Additionally, results from one mathematical field can be translated into matroid language and then applied to another field. The connections seem limitless! Indeed, Dr. Gian-Carlo Rota, a prolific researcher and proponent of matroid theory, wrote about the potential of matroids to unify all of mathematics.

In this mini-course, we will explore the humble graphical beginnings of matroids and build towards a linear algebraic definition. This will lead to several equivalent definitions of matroids that act as a springboard to other mathematical areas. We will get to know important matroids and their properties, apply matroid operations inspired by graphs, and see their geometric and algebraic interpretations. Our goal is to have you leave with a set of mathematical tools that will allow you to begin finding matroids everywhere you go.

Course 3, Week 2: Introduction to Polyhedral Combinatorics

Lecturer: Federico Ardila (San Francisco State University)

Short Description: This course will show how we can model combinatorial situations with polyhedra, and how the geometry of those polyhedral models can teach us about the combinatorial objects we are interested in.

Course 4, Week 2: Introduction to Symmetric Functions

Lecturer: Rosa Orellana (Dartmouth College)

Short Description: Symmetric functions play a fundamental role in algebra, combinatorics, and geometry due to their invariance under variable permutation, which captures essential symmetries in mathematical structures. A notable example is the chromatic symmetric function, which generalizes the chromatic polynomial of a graph. This function assigns a symmetric polynomial to a graph based on its colorings, capturing richer combinatorial information than the chromatic polynomial alone. This course will offer an introduction to symmetric functions and the chromatic symmetric function.

SageMath Tutorials

Lecturer: Adrián Barquero Sánchez

Short Description: Sage is free, open-source math software that supports research combinatorics, among many other areas. We will do demonstrations of software with examples relevant from the lectures and also show how to contribute to SAGE.