# Discrete Math Day of the Northeast

April 30, 2016

### Northampton, Massachusetts

### All talks in McConnell 103

### (Tentative) Schedule and Abstracts:

- 9:15am -- Coffee and Light Breakfast
- 9:45am -- Welcome and Announcements
- 10:00am -- Jennifer Morse, Drexel University and University of Virginia
**Title: Combinatorics of Gromov-Witten invariants and Macdonald polynomials****Abstract:**We will discuss several long-standing open problems involving the search for descriptions of intriguing non-negative constants that arise in geometry, representation theory, and physics. In joint work with Lapointe and Lascoux studying one of these problems, we discovered a family of symmetric, multivariate polynomials that refine the Schur basis for the symmetric function algebra. We will see how pursuant developments with this family led to a universal combinatorial framework for all the constants. - 11:00-11:15am -- Break
- 11:15am -- Paola Vera-Licona, University of Connecticut Health Center
**Title: On the minimal hitting set generation problem and its applications****Abstract:**Fix a family S of sets S1, S2, ..., Sn. A hitting set T of S is a set which intersects each of the sets Si; T is a minimal hitting set (MHS) if no proper subset of T is a hitting set of S. The problem of generating the collection of MHSes for a given set family is of interest in a wide variety of domains, and it has been explicitly studied (under a variety of names) in the contexts of combinatorics, Boolean algebra, data mining and computational biology, to name a few. While some interesting results have been obtained for the associated decision problem, the computational complexity of this problem is currently unknown. Nevertheless, there is an extensive literature of algorithms to generate minimal hitting sets. In this talk, we will survey the state of the art of algorithms for enumerating MHSes and their application and performance on problems derived from various applied domains with an emphasis on those on computational systems biology. - 12:15-2:00pm -- Lunch break
- 2:00pm -- Henry Cohn, Microsoft Research New England
**Title: The sphere packing problem in dimensions 8 and 24****Abstract:**What is the densest packing of congruent spheres in Euclidean space? This problem arises naturally in geometry, number theory, and information theory, but it is notoriously difficult to solve, and until recently no sharp bounds were known above three dimensions. Last month Maryna Viazovska found a remarkable solution of this problem in eight dimensions. In this talk I'll describe how her breakthrough works and where it comes from, as well as follow-up work extending it to twenty-four dimensions (joint work with Kumar, Miller, Radchenko, and Viazovska). - 3:00-3:15pm -- Break
- 3:15pm -- Melody Chan, Brown University
**Title:**Brill Noether theory, Young tableaux, and the jaggedness statistic on order ideals**Abstract:**I'll tell you the story of how a combinatorial proof of a theorem in Brill-Noether theory led us to some surprising results on staircase paths in Young tableaux and the jaggedness of order ideals. Joint work with Lopez, Pflueger, and Teixidor; and with Haddadan, Hopkins, and Moci. - 4:15-4:30pm --Break
- 4:30pm -- Maria Chudnovsky, Princeton University
**Title:**Even Pairs in Perfect Graphs**Abstract:**Perfect graphs are a class of graphs that behave particularly well with respect to coloring, and, as it turns out, have many other beautiful properties. In the early 1960s Claude Berge made two conjectures about perfect graphs, that motivated a great deal of research over the years. Both these conjectures have by now been solved, but our understanding of the structure of perfect graphs remains incomplete.An even pair is a pair of vertices u,v in a graph such that every induced path between u and v has an even number of edges. For several reasons even pairs are of interest in the study of the structure of perfect graphs, and a number of conjectured have been made (by Everett and Reed; Hougardy; and a related conjecture by Maffray), attempting to describe when they must be present. In this talk we will describe recent progress on these conjectures.

Joint work with Frederic Maffray, Paul Seymour and Sophie Spirkl