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We were delighted to have three amazing keynote speakers joining us for the 2021 GSCC! You can watch recordings of their talks at our conference YouTube channel.
San Francisco State University
Federico Ardila received his Ph.D. from MIT in 2003 with Richard Stanley. Since 2005 he is Professor of Mathematics at San Francisco State University and the Universidad de Los Andes. He is a Fellow of the American Mathematical Society and a recipient of the MAA National Teaching Award of the Mathematical Association of America in the US, and the winner of the Premio Nacional de Ciencias and Premio Nacional de Matemáticas in his native Colombia. He is also one of the founding editors of Combinatorial Theory, the new mathematician-owned-and-run, doubly anonymous refereed journal in combinatorics. Federico's research is in combinatorics and its connections to geometry, algebra, topology, and applications.
Federico is constantly working towards fostering an increasingly diverse, equitable, and welcoming community of mathematicians that serves the needs of all communities. With that goal, he founded the SFSU-Colombia Combinatorics Initiative, he co-directs the MSRI-UP undergraduate research program, he hosts more than 200 hours of combinatorics lectures on YouTube, and he reads and writes about the role of mathematics in education and society. Federico has advised more than 50 thesis students; the majority of his US advisees are members of underrepresented ethnic groups and the majority are women. These days, when he is not at work, he is probably reading, playing records, or playing marimba de chonta.
Recording of Federico's GSCC talk
Federico's Talk - "A tale of two polytopes: The bipermutahedron and the harmonic polytope"
The harmonic polytope and the bipermutahedron are two related polytopes which arose in our work with Graham Denham and June Huh on the Lagrangian geometry of matroids. This talk will discuss their algebraic and geometric combinatorics and explain their geometric origin.
The bipermutahedron is a (2n−2)-dimensional polytope with (2n!)/2^n vertices and 3^n−3 facets. Its f-polynomial, which counts the faces of each dimension, is given by a simple evaluation of the three variable Rogers-Ramanujan function. Its h-polynomial, which gives the dimensions of the intersection cohomology of the associated topic variety, is real-rooted, so its coefficients are log-concave.
The harmonic polytope is a (2n−2)-dimensional polytope with (n!)^2(1+1/2+...+1/n) vertices and 3^n−3 facets. Its volume is a weighted sum of the degrees of the projective varieties of all the toric ideals of connected bipartite graphs with n edges; or equivalently, a weighted sum of the lattice point counts of all the corresponding trimmed generalized permutahedra.
These two polytopes are related by a surprising fact: in any dimension, the Minkowski quotient of the bipermutahedron and the harmonic polytope equals 2.
The talk will be as self-contained as possible, and will feature joint work with Graham Denham, Laura Escobar, and June Huh.
Iowa State University
Steve Butler is the Barbara J Janson Professor of Mathematics at Iowa State University. He earned his degree from UC San Diego studying under Fan Chung and was an extensive collaborator with Ron Graham (which included one paper with Paul Erdos, making him the last person to attain an Erdos number of one). His research interests include spectral graph theory and the mathematics of juggling. More information can be found on his website: SteveButler.org
Steve's Talk - "A short course in Spectral Graph Theory"
Graphs can have their structure efficiently stored in an array format. We can go one step further and replace arrays by matrices (which are arrays with benefits!), the eigenvalues of these matrices tell us some information about the graph. The extent to which we can understand the structure of the graph from these eigenvalues is spectral graph theory. We will do a quick walk through of some of the flavors of matrices that are studied in spectral graph theory, and some results that arise from their study.
Northeastern University
Gordana Todorov is a Professor of Mathematics at Northeastern University in Boston. Gordana received her PhD from Brandeis university under the guidance of Maurice Auslander. Her research interests were originally in representation theory of artin algebras. After the introduction of cluster algebras, she used her representation theory knowledge in order to work on categorification of cluster algebras: she coauthored the paper in which the original cluster categories were created. This lead to her works in several other related fields, e.g. combinatorics, triangulations of surfaces, semi-invariants of quiver representations - mostly related to cluster theory via representation theory.
Gordana's Talk - "Friezes, Triangulations, and Representation Theory"
The well known combinatorial notion of friezes was introduced and studied by Conway and Coxeter. In their celebrated theorem, they show that the finite friezes are characterized by triangulations of polygons. That theorem gave a nice connection between friezes, representation theory and cluster theory. In a joint work with Karin Baur, Ilke Canakci, Karin Jacobsen, Maitreyee Kulkarni we study infinite periodic friezes their relation to triangulations of annulus, using both quiver representation and cluster algebras.
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