Queer and Trans Mathematicians 2021

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 Ardila-Klivans-Williams and Celaya.
This is joint work with Johannes Rau and Arthur Renaudineau.

Then using real structures I will propose a definition of the first Stiefel-Whitney class of a matroid. This is a homological class which is zero if and only if the matroid is orientable. Thus it is the linguistic analog of the first Stiefel-Whitney class of a manifold in the matroid setting. However, determining whether a matroid is orientable is an NP-complete problem, so determining whether or not the class is non-zero can be expected to be very difficult in general!

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.

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.

Integer partitions with parts that differ by at least $d$, called $d$-distinct partitions, arise in a famous identity of Euler and the first Rogers-Ramanujan 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 Alder-Andrews 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.
In 2020, Kang and Park constructed an extension of Alder's conjecture which relates to the second Rogers-Ramanujan identity. Namely, that there are at least as many $d$-distinct partitions of $n$ with parts at least $2$ as partitions of $n$ into parts that are $\pm 2$ modulo $d+3$, excluding the part $d+1$. Kang and Park proved their conjecture when $d = 2^r - 2$ and $n$ is even.
Here, we prove Kang and Park's conjecture for all $d\geq 62$. Toward proving the remaining cases, we adapt work of Alfes, Jameson and Lemke Oliver to generate asymptotics for the related functions. Additionally, we present a more generalized infinite family of conjectures and provide proofs for infinite classes of $n$ and $d$.
This work is joint with REU students Adriana Duncan (Tulane), Simran Khunger (Carnegie Mellon), and Ryan Tamura (Berkeley) and was partially supported by the National Science Foundation REU Site Grant DMS-1757995, and Oregon State University.

The $1/3-2/3$ Conjecture, originally formulated in 1968, is one of the best-known open problems in the theory of posets, stating that the balance constant of any non-total 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 $1/3-2/3$ 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 width-two 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 $1/3-2/3$ Conjecture, and therefore on which methods are likely to be successful in resolving it. This is joint work with Yibo Gao.

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.
In this talk I will describe in detail this combinatorial structure, and present some results and some open questions that I hope will intrigue a crowd of combinatorialists.

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.
This is joint work with Madeline Brandt, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.

A tree $T$ on $2^n$ vertices is called set-sequential 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 component-wise 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 set-sequential (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 set-sequential. Additionally, we introduce some constructions of new set-sequential graphs from smaller set-sequential 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.
This is joint work with Ervin Győri (supported by the National Research, Development and Innovation Office under Grant K132696), Junsheng Liu, and Sohaib Nasir.

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 k-imploding 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, restricted-normal, and pseudo-normal digraphs. We prove sufficient conditions for normal digraphs and show that the directed join of two normal digraphs results in a restricted-normal digraph. Also, we prove that directed joins are the only restricted-normal digraphs when the order is square-free or twice a square-free number. Finally, we provide a construction for restricted-normal digraphs that are not directed joins for all orders that are neither square-free nor twice a square-free number.