Title: Signed Poset Polytopes

Abstract: Posets can be viewed as subsets of the type-A root system that satisfy certain properties. Geometric objects arising from posets, such as order cones, order polytopes, and chain polytopes, have been widely studied. In 1993, Vic Reiner introduced signed posets, which are subsets of the type-B root system that satisfy the same properties. In this talk, we will explore the analogue of order and chain polytopes in this setting, focusing on the Ehrhart theory of these objects.

Bio: Max (they/them) is a nonbinary grad student at UC Berkeley studying geometric combinatorics, focusing on enumerative questions regarding polytopes. In their spare time, they enjoy spending time with their 20 lb cat, Squid!

Title: Tropicalization of graph profiles

Abstract: The number of homomorphisms from a graph H to a graph G, denoted by hom(H;G), is the number of maps from V(H) to V(G) that yield a graph homomorphism, i.e., that map every edge of H to an edge of G. Given a fixed collection of finite simple graphs {H_1, ..., H_s}, the graph profile is the set of all vectors (hom(H_1; G), ..., hom(H_s; G)) as G varies over all graphs. Graph profiles essentially allow us to understand all polynomial inequalities in homomorphism numbers that are valid on all graphs. Profiles can be extremely complicated; for instance the full profile of any triple of connected graphs is not known. To simplify these objects, we introduce their tropicalization which we show is a closed convex cone that still captures interesting combinatorial information. We explicitly compute these tropicalizations for some sets of graphs, and relate the results to some questions in extremal graph theory.

Bio: Annie Raymond (she/her/elle) is an assistant professor in the department of Mathematics and Statistics at the University of Massachusetts. Originally from Montréal (Tiohtià:ke), she studied math and music at MIT as an undergrad before pursuing a Ph.D. in mathematics at the Technische Universitaet in Berlin and a postdoc at the University of Washington. At any given moment, you will most likely find her thinking about extremal graph theory and sums of squares---perhaps while riding her bicycle or playing the piano---or reflecting on diversity in STEM and on education in prisons. She runs the instagram feed _forall (currently on hiatus during the pandemic) which features gender minorities and people of color in math.

Title: Permutree sorting

Abstract: Permutrees define combinatorial families interpolating between permutations, binary trees and binary sequences. They also correspond to certain congruence classes of the weak order lattice on permutations. In this talk, we present the Permutree sorting algorithm which attempts to sort permutations following certain constraints, succeeding only when the permutation is minimal inside its permutree congruence class. In this sense, it is a generalization of the well known stack sorting from Knuth and the c-sorting related to Cambrian lattices defined by Reading. (joint work with D. Tamayo and V. Pilaud)

Bio: Viviane Pons (she/her) is an associate professor at the Computer Science department of University Paris-Saclay. Her research focuses on algebraic combinatorics with an approach based on computer exploration and an algorithmic mindset. She is a contributor of open-source software such as Sagemath and a defensor of open science in general. She is also a feminist and is committed to work towards better inclusivity and diversity in STEM. You can read her thoughts on her blog http://openpyviv.com/ or follow her on Twitter @PyViv.

Title: Topology of random 2-dimensional cubical complexes

Abstract: We study a natural model of random 2-dimensional cubical complexes which are subcomplexes of an n-dimensional cube, and where every possible square (2-face) is included independently with probability p. Our main result exhibits a sharp threshold p=1/2 for homology vanishing as the dimension n goes to infinity. This is a 2-dimensional analogue of the Burtin and Erdős-Spencer theorems characterizing the connectivity threshold for random graphs on the 1-skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial-Meshulam theorem for random 2-dimensional simplicial complexes. However, the models exhibit strikingly different behaviors. We show that if p > 1 - sqrt(1/2) (approx 0.2929), then with high probability the fundamental group is a free group with one generator for every maximal 1-dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold. This is joint work with Matthew Kahle and Elliot Paquette.

Bio: Érika Roldán (they/them or she/her) is currently a Marie Skłodowska-Curie Fellow within the EuroTechPostdoc Programme at the Technische Universität München (TUM) and EPFL Lausanne. Their research interests include biomathematics, stochastic topology, topological and geometric data analysis, extremal topological combinatorics, discrete configuration spaces, recreational mathematics, learning analytics, and educational technology. They use digital technologies, including gamification and CAD visualization, in their teaching and research projects. Recently, they founded the outreach initiative BAMM at The Ohio State University (2019), co-founded Hypothesis in NYC (2020), Matemorfosis at CIMAT Mexico (2011), and Music-Math in Mexico (2013). All of these initiatives promote and increase public awareness and enjoyment of mathematics and its applications with special emphasis on bringing underrepresented minorities of all ages and backgrounds into STEAM.