(All times Eastern)
9.45  10.00  Welcome and house rules  
10.00  11.00 
Kris Shaw — Real phase structures on matroid fans
The tropical geometry of matroid fans has been a recent powerful tool for understanding many matroid invariants. In this talk, I will define real phase structures on fans and prove that a real phase structure on a matroid fan is cryptomorphic to providing an orientation of the underlying matroid. Therefore, we can use tropical techniques to study oriented matroids. We can also define the real part of a fan equipped with a real phase structure. In the matroid setting, this yields the topological representation of an oriented matroid in the sense of Folkman and Lawrence, and is related to results of ArdilaKlivansWilliams and Celaya. 

11.00  11.30 
Max Hlavacek — Signed poset polytopes
Posets can be viewed as subsets of the typeA 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 typeB 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. 

11.30  12.00 
Erika Roldan Roa — Shuffling Polyominoes
Do you know what algorithm is deciding which tetromino piece you get next in a Tetris game? In this talk I will start by answering this question and then I will tell you about several different ways of sampling random polyominoes (polyominoes are like tetrominoes but with any desired amount of squares). We will also analyze how the topological and geometric properties of polyominoes change depending on the distribution that we choose to sample them. 

12.00  13.00  Break  
13.00  13.30 
Holly Swisher — Generalizations of the AlderAndrews Theorem in Partition Theory
Integer partitions with parts that differ by at least $d$, called $d$distinct partitions, arise in a famous identity of Euler and the first RogersRamanujan identity. These identities state that the number of $1$distinct or $2$distinct partitions of $n$ is equal to the number of partitions of $n$ into parts that are $\pm 1$ modulo $4$ or $5$, respectively. Alder showed that for $d \geq 3$ no identity of such a type exists, and conjectured what is now the AlderAndrews Theorem, namely that there are at least as many $d$distinct partitions of $n$ as partitions of $n$ into parts that are $\pm1$ modulo $d+3$. This was proved partially by Andrews in 1971, by Yee in 2008, and was fully resolved by Alfes, Jameson and Lemke Oliver in 2011. 

13.30  14.00 
Christian Gaetz — The $1/32/3$ Conjecture for Coxeter groups
The $1/32/3$ Conjecture, originally formulated in 1968, is one of the bestknown open problems in the theory of posets, stating that the balance constant of any nontotal order is at least $1/3$. By reinterpreting balance constants of posets in terms of convex subsets of the symmetric group, we extend the study of balance constants to convex subsets $C$ of any Coxeter group. Remarkably, we conjecture that the lower bound of $1/3$ still applies in any finite Coxeter group, with new and interesting equality cases appearing. We generalize several of the main results towards the $1/32/3$ Conjecture to this new setting: we prove our conjecture when $C$ is a weak order interval below a fully commutative element in any acyclic Coxeter group (a generalization of the case of widthtwo posets), we give a uniform lower bound for balance constants in all finite Weyl groups using a new generalization of order polytopes to this context, and we introduce generalized semiorders for which we resolve the conjecture. We hope this new perspective may shed light on the proper level of generality in which to consider the $1/32/3$ Conjecture, and therefore on which methods are likely to be successful in resolving it. This is joint work with Yibo Gao. 

14.00  16.00  Fun activity  Solving Fun (Puzzle activity) 
10.00  11.00 
Renzo Cavalieri  Why a combinatorialist should like the moduli spaces of curves
Moduli spaces of curves are highly sophisticated geometric
objects of fundamental importance. Most of the success we have had in
understanding (aspects of) their geometry stems from the fact that
they have a rich combinatorial structure. First off, they are
stratified spaces, where the strata are themselves built out of other
moduli spaces of curves. Secondly, there is a discrete infinite family
of moduli spaces that are connected to each other via tautological
morphisms. 

11.00  11.30 
Juliette Bruce — The top weight cohomology of $\mathcal{A}_{g}$
I will discuss recent work calculating the top weight cohomology of the moduli space $\mathcal{A}_{g}$ of principally polarized abelian varieties of dimension $g$ for small values of $g$. The key idea is that this piece of cohomology is encoded combinatorially via the relationship between the boundary complex of a compactification of $\mathcal{A}_{g}$ and the moduli space of tropical abelian varieties. 

11.30  13.00  Break  
13.00  13.30 
Emily Eckels — Strong Set Colorings of Odd Trees
A tree $T$ on $2^n$ vertices is called setsequential if the elements in $V(T)∪E(T)$ can be labeled with distinct nonzero $(n+1)$dimensional $01$vectors such that the vector labeling each edge is the componentwise sum modulo 2 of the labels of the endpoints. It has been conjectured that all trees on $2^n$ vertices with only odd degree are setsequential (the Odd Tree Conjecture), and in this paper, we present progress toward that conjecture. We show that certain kinds of caterpillars (with restrictions on the degrees of the vertices, but no restrictions on the diameter) are setsequential. Additionally, we introduce some constructions of new setsequential graphs from smaller setsequential bipartite graphs (not necessarily odd trees). We also make a conjecture about pairings of the elements of $\mathbb{F}_2^n$ in a particular way; in the process, we provide a substantial clarification of a proof of a theorem that partitions $\mathbb{F}_2^n$ from a 2011 paper by Balister, Győri, and Schelp. Finally, we put forward a result on bipartite graphs that is a modification of a theorem in the same paper by Balister et al. 

13.30  14.00 
Alexander Wiedemann — On digraphs with polygonal restricted numerical range
In 2020, we introduced the restricted numerical range of a digraph (directed graph) as a tool for characterizing digraphs and studying their algebraic connectivity. In particular, digraphs that have a restricted numerical range of a single point, horizontal line segment, and vertical line segment were characterized as kimploding stars, directed joins of bidirectional digraphs, and regular tournaments, respectively. We now extend this work by investigating digraphs whose restricted numerical range forms a convex polygon in the complex plane. We provide computational methods for identifying these polygonal digraphs, and show that these digraphs can be broken into three disjoint classes that are closed under digraph complement: normal, restrictednormal, and pseudonormal digraphs. We prove sufficient conditions for normal digraphs and show that the directed join of two normal digraphs results in a restrictednormal digraph. Also, we prove that directed joins are the only restrictednormal digraphs when the order is squarefree or twice a squarefree number. Finally, we provide a construction for restrictednormal digraphs that are not directed joins for all orders that are neither squarefree nor twice a squarefree number. 

14.00  15.00  Community Discussion  
15.00  15.15  Closing/announcements  
15.15  17.15  Social hour(s) — Gather.town 