Queer and Trans Mathematicians 2023

Joint work with Eleonore Faber and Matthew Pressland

A Conway-Coxeter frieze is an array of finitely many interlaced rows of positive integers in the plane satisfying a determinantal relation. Conway-Coxeter showed that such friezes with n rows are in bijection with triangulations of a regular polygon with n+1 sides. I will give an introduction to Conway-Coxeter friezes and a generalisation (frieze patterns with coefficients) introduced by Propp and developed by Cuntz-Holm-Jorgensen, in the context of cluster algebras.

Cuntz-Holm-Jorgensen observe that a frieze pattern with coefficients can be obtained from a Conway-Coxeter frieze pattern by cutting out a subpolygon. We show that this procedure can be modelled categorically in the context of the Grassmannian cluster category model for friezes of Jensen-King-Su. Funders: EPSRC, Isaac Newton Institute (Cambridge), Simons Foundation, Centre for Advanced Study (Oslo).

Universal Partial Cycles (or "upcycles") are a generalization of De Bruijn cycles where in addition to alphabet letters one allows a wildcard character (representing all possible letters). My research group has proven the existence of infinite classes of nontrivial such cycles (De Bruijn cycles are a subset using no wildcard characters, and the use of only wildcard characters would also be trivial), but it has yet to be shown whether it is possible to have more than one wildcard character per wordlength. My thesis research is investigating this, but a nice result from my research group that can be shared in twenty minutes would be that perfect necklaces (as defined in 2016 by Alvarez et al.) are precisely the tool required to unfold an upcycle with d wildcard characters per wordlength into an upcycle with fewer than d wildcards per wordlength (such as fully unfolding it into a De Bruijn cycle), and I can show precisely how to do so for a hypothetical d>1 upcycle.

The Ehrhart polynomial for a given integral polytope counts the lattice points inside the polytope for any positive integer dilate. A graphical zonotope is a polytope constructed from the Minkowski sum of the column vectors of a graph's incidence matrix. I will introduce the audience to Ehrhart theory in the context of graphical zonotopes and how the polytope relates to the induced forests of a graph. Then, I will discuss my current work with signed graphs and their polytopes.

We study the connected blocks polytope, which, apart from its own merits, can be seen as the generalization of certain connectivity based or Eulerian subgraph polytopes. We provide a complete facet description of this polytope, characterize its edges and show that it is Hirsch. We also show that connected blocks polytopes admit a regular unimodular triangulation by constructing a squarefree Gr\"obner basis. In addition, we prove that the polytope is Gorenstein of index 2 and that its $h^\ast$-vector is unimodal.

Classical matroid theory develops a set-theoretical foundation to the study of linear independence, and in doing so, establishes a far more general theory than what is captured in the linear algebra of vectors in Euclidean space. The theory of valuated matroids is a further generalization, where in addition to independence relations we record a weight on each matroid basis, generalizing the data captured by determinants of maximal minors in an (m x n)-matrix. Just as in classical matroid theory, the application of valuated matroids depends largely on which of its many definitions we choose to work with.

In this talk I will present my joint work with Felipe Rincon on a new, equivalent definition for valuated matroids in terms of closure operators. This definition has roots based in tropical scheme theory and relates to earlier works in $l_\infty$ nearest approximations and phylogenetics, as well as recent works in tropical principal component analysis.

Grassmannians and flag varieties are important moduli spaces in algebraic geometry. Their linear degenerations arise in representation theory as they describe quiver representations and their irreducible modules. As linear degenerations of flag varieties are difficult to analyze algebraically, we describe them in a matroidal setting and further investigate their tropical counterparts.

In this talk, I will introduce matroidal and tropical analoga of linear degenerate flags and their varieties obtained in joint work with Alessio Borzì and describe them in terms of morphisms of valuated matroids. Using techniques from matroid theory and linear tropical geometry, we use the correspondences between the different descriptions to gain insight on the structure of linear degeneration. For small examples, we relate observations on the tropical linear degenerate flag varieties to the flat irreducible locus studied in representation theory.

The famous Bernstein’s theorem tells us that the number of isolated roots over $\mathbb{C}^*$ of a square polynomial system with fixed support is at most equal to a mixed volume, and that this bound is attained for generic choices of coefficients. In practice, though, many systems that arise in applications have coefficients that depend on the parameters in such a way that they are algebraically dependent, and in such cases, the mixed volume bound might be far from sharp for generic choices of parameters.

In this talk, I will discuss a generalization of Bernstein’s theorem, based on previous work by Helminck and Ren, that for sufficiently well-behaved (so-called tropically equivariant) parametric systems makes it possible to compute the correct generic root count, by replacing the mixed volume with a tropical stable intersection. Moreover, I’ll discuss a tropical generalization of the polyhedral homotopy method of Sturmfels and Huber, that makes it possible to find all the roots with homotopy continuation without tracing superfluous paths.

A particular family of parametric systems that satisfies the tropical equivariance conditions are the steady state equations from chemical reaction network theory, and as a case study, we compute root counts and complete solution sets for several classes of large networks with OSCAR/Polymake, where methods based on Bernstein’s theorem fail. Basic profiling of our algorithms identifies the tropicalization of the linear part of the systems as the computational bottleneck, and I will end the talk by discussing ideas of improvements based on the theory of transversal matroids.

This is based on joint work with Elisenda Feliu, Paul Helminck, Yue Ren, Benjamin Schröter and Máté Telek.

A Career Path in Mathematics: How I met Convex Polytopes, why books areimportant, why examples are important, what my students contributed,and whether there are still interesting problems to work on…

Table of Contents:
1. On Polytopes, Ten Problems left
2. How I became "The Expert on Convex Polytopes": Teaching
3. Lectures on Polytopes, getting the book published: preprint, manuscript, Winnie The Pooh (adult material); 
4. Lectures on Polytopes, Email to Bernd Sturmfels
5. Publication List: "true publist" identify papers that were first rejected; history of my best paper
6. Publication List: identify papers with students
7. Publication List: identify papers with examples/counter-examples
8. Books are important. Getting things finished. Convex Polytopes (Grünbaum 2nd ed.), Panorama der Mathematik
9. CV "true CV": Applications for professorships
10. CV: version with "ten special things"

The skew Schur functions admit many determinantal expressions. Chief among them are the (dual) Jacobi–Trudi formula and the Lascoux–Pragacz formula, which is a skew analogue of the Giambelli identity. Comparatively, the skew characters of the symplectic and orthogonal groups, also known as the skew symplectic and orthogonal Schur functions, have received very little attention in this direction.

In this talk, we establish analogues of the dual Jacobi–Trudi and Lascoux–Pragacz formulae for these characters. Our approach is entirely combinatorial, being based on lattice path descriptions of the tableaux models of Koike and Terada.

This is joint work with Seamus Albion, Ilse Fischer and Florian Schreier-Aigner.

Every group G can be represented as a faithful transitive permutation representation of degree n. Moreover, the stabilizer of a point in this permutation representation is always a core-free subgroup of G. Conversely, the action of a group G on a core-free subgroup H ≤ G is always transitive and faithful, giving a faithful transitive permutation representation on the set of cosets G/H, with degree |G : H|. These permutation representations are powerful tools in the classification of abstract regular/chiral polytopes and hypertopes. In this talk we list all possible degrees of faithful transitive permutation representations of the toroidal regular/chiral maps {4, 4}, {3, 6} and hypermaps (3, 3, 3). This is a joint work with M. Elisa Fernandes.

The polytope algebra is well-studied and features several interesting properties. One of these is the nilpotency $([P]-1)^{n}$. We seek to define an analogue to the polytope algebra for normal complexes. Normal complexes are polytopal complexes obtained by truncating fans, but unlike the polytope case, the fan is not required to be complete. In the first step towards creating a bijection, we prove that this nilpotency holds in the one-dimensional case in the algebra for normal complexes.

In the long tradition of graph-theoretical problems related to chess, we study domination problems on graphs coming from the multidimensional version of chessboards to polycubes. We study the computational complexity of the problem of finding a maximal coclique inside queen and rook graphs on polycubes. This is a generalisation of the famous n-queens problem. We prove that the problem is NP-complete for rooks on polycubes of dimension 3 and higher and that it is in P for rooks on polyominoes. We will discuss the problem for queens and also survey the related minimal domination problems for queens and rooks and their history. Our final result is the translation of these problems to an integer linear programming problem and its use in numerical investigations improving known values of domination number of square chessboards.

The participants are invited to play beforehand the game we developed to get some hands-on experience with the minimal domination problem at the following link: https://www.erikaroldan.net/queensrooksdomination

This is joint work with M. Müßig and É. Roldán-Roa

Given a group $\Gamma$ and a generating set $S\subseteq\Gamma$, the \emph{Cayley graph} $G=\text{Cay}(\Gamma,S)$ is the graph with vertex set $\Gamma$ and, for every $\alpha\in\Gamma$ and $s\in S$, an edge $(\alpha,s\alpha)$. That is, the Cayley graph maps out the entire group and how the generating elements move the group elements around. Stallings' Theorem (1971) states that a finitely-generated group splits over a finite subgroup if its Cayley graphs have more than one end. In 2010, Bernhard Kr\""on provided a proof of Stallings' Theorem using separators. In this project, we move towards a finite version of Stallings' Theorem, using so-called \lambdaocal separators'. We show that a finitely-generated group $\Gamma$ (with bounded nilpotency), having a certain local separator, is necessarily a cycle or decomposes as a direct product of a cycle and an involution. In particular, we assume the existence of an $r$-local cutvertex or the existence of an $r$-local 2-separator, for $r$ bounded below. An \emph{$r$-local cutvertex} $x$ is a vertex in a graph that disconnects the subgraph induced by the ball of radius $\frac{r}{2}$ centred at $x$. An $r$-local 2-separator is defined so that it works in a similarly intuitive way.