## Discrete Mathematics 2-Day at Albany## Virtually Sat/Sun April 25-26, 2020## Northeast Combinatorics Network |

Sat. 8:15am- 8:45 | Pre-meeting, technology dealing |

Sat. 8:45am | Welcome |

Sat. 9:00am- 10:00 |
Megan Owen (CUNY, Lehman College) Representations of partial leaf sets in phylogenetic tree space
Abstract: A phylogenetic tree depicts evolutionary relationships
between sets of organisms. Billera, Holmes, and Vogtmann defined a
metric space of phylogenetic trees (BHV treespace) to provide a
natural geometric setting for describing collections of trees with the
same set of leaves (ie. for the same set of organisms). However,
sometimes biologists want to analyze collections of trees on
overlapping, but non-identical, leaf sets. We refine and adapt a
combinatorial algorithm of Ren et al. to work for metric trees to give
a full characterization of the subspace of extensions of a subtree.
We describe how to apply our algorithm to define and search a space of
possible trees on all leaves and, for a collection of trees with
different leaf sets, to measure their compatibility. I will end this
talk with some open problems. |

BREAK | |

Sat. 10:30- 11:30 |
Francois Bergeron (Univ. of Quebec at Montreal) Recent advances in the study of multivariate diagonal harmonics
Abstract: Over the last 25 years, the study of \( GL_2\times S_n\)-modules of diagonal harmonic polynomials has seen many interesting developments; on top of having been shown to relate to several fields of mathematics (algebraic combinatorics, representation theory, algebraic geometry, knot theory) and theoretical physics (conformal field theory, statistical physics). Its extension to several sets of variables makes it possible to give a new understanding to links between many questions of the subject. We will present recent results and conjectures for the corresponding \( GL_\infty\times S_n\)-modules, including ties with the Delta-conjecture, representation theoretic models for the effect of the nabla "Macdonald eigenoperator" on hook indexed Schur functions, as well as explicit descriptions of some components of the \( GL_\infty\times S_n\)-modules. We will also discuss some of the ties with closely related combinatorial questions, which include rectangular Catalan combinatorics, enumeration of chains in the Tamari poset, and the sweep map. |

LUNCH BREAK meeting will remain available | |

Sat. 1:00pm- 2:00 |
Alejandro Morales (UMass Amherst) Volumes and triangulations of flow polytopes of graphs
Abstract: A flow polytope of a directed acyclic graph is the set of flows on the edges of the graph with prescribed netflows on vertices. Flow polytopes of graphs are a rich family of polytopes that includes polytopes of interest in probability, optimization, representation theory, and algebraic combinatorics. These polytopes are related to partially ordered sets when the graphs are planar and special cases have remarkable formulas for their volumes and lattice points due to Baldoni-Vergne and Postnikov-Stanley. I will talk about recent results on these polytopes including a relation between seemingly different triangulations by Postnkov-Stanley and Danilov-Karzanov-Koshevoy. This talk is based on joint work with Mészáros and Striker. |

BREAK | |

Sat. 2:30- 3:50 | |

Virtual hangout, dinner etc. Meeting will remain available. |

Sun. 8:45am- 9:15 | Pre-meeting, technology dealing |

Sun. 9:15am | Welcome |

Sun. 9:30am- 10:30 |
Jessica Sidman (Mount Holyoke) Geometric Equations for Matroid Varieties
Let \( x \) denote a \( k \)-dimensional subspace of \( \mathbb{C}^n \) and let \( A_x \) be a \( k\times n \) matrix whose rows are a basis for \( x \). The matroid \( M_x \) on the columns of \( A_x \) is invariant under a change of basis for \( x \). What can we say about the set \( \Gamma_x \) of all \( k \)-dimensional subspaces \( y \) such that \( M_y=M_x \)? We will explore this question algebraically, showing that for some matroids that arise geometrically, many non-trivial equations vanishing on \( \Gamma_x \) can be derived using geometry. This work is joint with Will Traves and Ashley Wheeler. |

BREAK | |

Sun. 11:00- 12:00 |
Sophie Spirkl (Princeton) Pure Pairs (Excluding induced subgraphs from sparse graphs)
Abstract: Let us call two sets \( A \) and \( B \) of vertices in a graph \( G \) a "pure pair" if either all or no edges between the two sets are present in \( G \). Random graphs are unlikely to have pure pairs with both \( A \) and \( B \) large. This changes when excluding induced subgraphs. I'll present recent progress on the question of how large a pure pair we can guarantee in different settings. Joint work with Maria Chudnovsky, Jacob Fox, Alex Scott, and Paul Seymour. |

LUNCH BREAK meeting will remain available | |

Sun. 1:00pm- 2:40 | |

Virtual hangout, dinner etc. Meeting will remain available. |

Will be emailed to the network's mailing list and additional information may be emailed to meeting registrants.

Discrete Math Days in the Northeast is a series of one-day research meetings that seeks to bring together a community of combinatorists in the northeast. We seek to provide a relaxed atmosphere, a friendly environment conducive to fostering collaboration across institutions and disciplines.

We hold three meetings per year, Fall, Spring and Summer Combo. The meetings take place at different colleges and universities in the northeast. Most participants drive to and from the conference on the same day. We hope that by holding meetings at different universities it will make it possible for researchers with higher teaching loads and those with limited institutional support to attend, and helped them to keep up with research in their field.

We also seek to provide a non-intimidating entry into the mathematics community for graduate students and very strong undergraduates in the region.

We now also hold several Virtual Combinatorics Colloquia throughout the academic year.

Discrete Math Days in the Northeast is organized by the Northeast Combinatorics Network and funded by the US National Science Foundation.

National Science Foundation

Discrete Math Day at Albany 2020 is also supported by Univ. at Albany NSF grants DMS-1855592 (PI: Cristian Lenart) and CCF-1850052 (PI: Justin Curry), and by the University at Albany Department of Mathematics and Statistics.