For any questions about the workshop please contact the organizers by email.
Real rootedness, log-concavity, and matroids
Cynthia Vinzant (North Carolina State University)
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 completely log-concave polynomials, their real and combinatorial properties, and their applications to matroids.
Entropy and counting
Wojciech Samotij (Tel Aviv University)
Entropy is a notion coming from information theory; it was introduced by Claude Shannon in the 1940s. Entropy quantifies the expected amount of information contained in a realisation of a discrete random variable. In particular, if X is a uniformly chosen random element of a finite set S, then the entropy of X is the logarithm of |S|. This allows one to rephrase counting problems in the language of entropy. Such rephrasing opens up the possibility of applying several tools from information theory to studying counting problems in combinatorics; in the recent years, such approach was applied very successfully to a range of enumeration problems. In this course, we will introduce the notion of entropy and derive several useful identities and inequalities relating entropies. We will then demonstrate the power of these inequalities by presenting several applications of the `entropy method' in combinatorics.