# Minicourses

### First Week

The Wonderland of Polytopes

A polytope is the convex hull of a finite set of points in a Euclidean space; equivalently, polytopes are bounded intersections of finitely many closed half-spaces. Despite this simple definition, polytopes have a very rich and continuously developing theory. In this mini-course, we will mostly concentrate on the f-vectors of polytopes: for a polytope P, its f-vector records the number of faces of P of various dimensions. While the f-vectors of 3-dimensional polytopes and those of simplicial (or simple) d-dimensional polytopes are well understood, very little is known about f-vectors of general polytopes or even about f-vectors of simplicial polytopes with some additional structure (e.g., symmetry). We will discuss several classical as well as several very recent results about f-vectors of polytopes.

Ehrhart Polynomials

Ehrhart theory measures a polytope P discretely by counting the integer lattice points inside its dilates P, 2P, 3P, ... , and the resulting counting functions have a beautiful polynomial structure. We will outline a few fundamental theorems on Ehrhart polynomials, give a sample of recent results in Ehrhart theory, and exhibit applications and open problems.

### Second Week

Combina-Torics

An algebraic variety is the set of solutions to a system of polynomial equations, and algebraic geometry is the study of such objects. Algebraic geometry is often regarded as a very abstract field, but a certain class of algebraic varieties known as toric varieties has beautiful connections to combinatorics and discrete geometry. In particular, the study of toric varieties blends algebra, geometry, and combinatorics to produce wonderful and surprising bridges between these areas.

We will start from ground zero, do lots of examples, and wrap up the lectures with a sketch of the famous g-theorem, which describes the possible number of faces of a simplicial polytope (a fundamental object in combinatorics and discrete geometry). For example, the octahedron has 6 zero-dimensional faces (the vertices), 12 one-dimensional faces (the edges), and 8 two-dimensional faces (the triangles). The g-theorem describes the possible variants of this sequence (6,12,8) of numbers, and toric varieties are the key tool.

Real Rootedness, Log-concavity, and Matroids

Matroids are combinatorial structures that model independence, such as among edges in a graph and vectors in a linear space. I will introduce the theory of matroids along with classes of real polynomials capturing many of their important features. Real rooted univariate polynomials are ubiquitous in combinatorics and there are several interesting multivariate generalizations. In increasing order of generality, we will discuss determinantal, stable, and log-concave polynomials, their real and combinatorial properties, and their applications to matroids and the mixing times of certain random walks.