Graduate Student Meeting on
Applied Algebra and Combinatorics

Preliminary course material: Slides part 1. Slides part 2. Exercises.

The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals, introduced in the 1950s by Kuo Tsai Chen. They are central to the theory of rough paths, a revolutionary view on stochastic analysis. They have also found applications in areas like time series analysis and machine learning. We will examine path signatures through an algebraic and combinatorial perspective, revealing some of the rich theory and fascinating geometry behind them. 

Useful references and background reading:

• Varieties of Signature Tensors.

• Learning Paths from Signature Tensors.

• Rough Paths and Signatures.

• Inversion of Signature.

• The rough Veronese Variety.

• Toric geometry of path signature varieties.

• Convex Hulls of Curves: Volumes and Signatures.

Preliminary course material: Notes part 1. Notes part 2. Exercises.

These two lectures will focus on topics in the overlap of combinatorics and real algebraic geometry. 

The first lecture will focus on the topology of real algebraic hypersurfaces and construction methods. I will present Viro’s patchworking method, which uses combinatorics to construct spaces having the same topologies as some real algebraic hypersurfaces. I will also give a contemporary formulation using the language of tropical geometry. 

Lecture two will focus on generalising the patchworking construction in higher codimensions and to objects outside of algebraic geometry. More specifically I will focus on oriented matroids, which are matroids with extra structure inspired by real vector spaces or directed graphs. The patchworking construction applied here provides a strong link between oriented matroids and real toric varieties.

The problem sessions will consist of a mix of theoretical and computational questions with the possibility of using polymake and other software.

Useful references and background reading:

• A brief introduction to tropical geometry (Sections 1, 2, 3).

• Patchworking algebraic varieties (original source on patchworking).

• Bounding the Betti numbers of real tropical hypersurfaces near the tropical limit (tropical source on patchworking in all dimensions, Sections 1, 2, 3, 7). 

• Oriented matroids.

• Real phase structures on matroid fans and matroid orientations.

• Patchworking oriented matroids.

The matroid of a point configuration in space encodes how special the positions of these points are with respect to each other, in terms of linear dependence. In this talk I show how matroids arise naturally in algebraic statistics and information theory where "points in space" are replaced by random variables and "special position" by conditional independence.

The focus of this talk is on one classical aspect of matroid theory which has received considerable attention from information theorists: the Ingleton inequality. This inequality holds for linear matroids but not in the probabilistic setting. Linear conditions under which a probabilistically representable matroid does satisfy Ingleton lead to many celebrated information inequalities — and these conditions have recently been classified with the help of computer algebra.

The main thesis of this talk is that optimization and algebra can be applied to solve problems in information theory. On the way, I will point out some open problems and computational challenges for the non-linear algebra community. 

We compute the reach and the the second fundamental form of the (spherical) real Segre-Veronese variety; i.e., the manifold of rank-one partially symmetric tensors in $\mathbb R^{n_1\times \cdots \times n_d}$ of norm one. In particular, we show that the reach only depends on the dimension $d$ of the tensor, but not on the size of its modes $n_1,\ldots, n_d$. We also express the volume of the tubular neighborhood of the Segre variety in terms of the number of perfect matchings of a multipartite graph. This is joint work with Paul Breiding.

To each lattice simplex one can associate a linear code over a ring of integers modulo some integer N. This allows us  to translate some questions arising in Ehrhart theory to those of linear codes. In particular, the vanishing of local h*-polynomial translates to the corresponding code having no words of maximal weight.  I will report on some progress made in classifying the spanning simplices with vanishing local h*-polynomial. We will focus on the four dimensional simplices and the simplices of prime volume. 

Point-line configurations are combinatorial objects that generate an important family of matroids and matroid varieties. In particular, we are interested in providing a minimal set of polynomial equations defining matroid varieties associated with point-line configurations. Recent literature shows that this can be deduced by the incidence structure of the configuration itself when embedded in a projective space. We will apply this idea to quadrilateral sets i.e. plane configurations consisting of four lines and their six intersection points. First, we will provide a new characterization of quadrilateral sets in incidence geometry. Second, we will use this characterization to provide a set of polynomial generators for the variety itself which is geometrically meaningful. We will also see how this set of generators can be extended to more complicated configurations. 

Originally interested in presenting combinatorial views on the adjunction theory of toric varieties, the adjoint polytope $P^{(s)}$ was defined by Di Rocco et al. in 2012 as the set of points in $P$ with lattice distance to every facet at least $s$. On the other hand, introduced by Fine and Reid in the study of plurigenera of toric hypersurfaces, the Fine interior of a lattice polytope got recently into the focus of research. Based on the Fine interior, we propose a modification of the original adjoint polytopes by defining the Fine adjoint polytope $P^{F(s)}$ of $P$ as consisting of the points in $P$ that have lattice distance at least $s$ to all valid inequalities for $P$. We obtain a Fine Polyhedral Adjunction Theory that is, in many respects, better behaved than its original analogue. Many existing results in Polyhedral Adjunction Theory carry over to the Fine case, some with stronger conclusions, as decomposing polytopes into Cayley sums, and most with simpler, more natural proofs as in the case of the finiteness of the Fine spectrum. 

We study statistical models defined by reflexive polytopes in terms of maximum likelihood estimation problems. A measure of complexity of the estimation problem is the focus of our investigations. We present our computational results for the 4319 three-dimensional reflexive polytopes obtained using homotopy continuation methods. The results are examined in more detail with regard to occurring singularities using algebraic tools such as A-discriminants. We also give a closed formula for the complexity measure for the n-dimensional cube and its dual. 

Matroids are objects that translate into the combinatorial world the notion of hyperplane arrangements.

During the last decade, matroid theory has been revolutionized by the introduction of algebro-geometric techniques: long-standing combinatorial conjectures were solved by first studying a variety associated to realizable matroids, and then by constructing an abstract combinatorial analogue in the non-realizable case. Nowadays, the Chow ring and the Kazhdan-Lusztig invariants can be rightfully considered an essential tool for anyone interested in matroids. All the new polynomials that arose in this context are conjectured to have nice combinatorial properties, for example being real-rooted.

The plan of the talk is to introduce matroids and their invariants, and then show recent developments in tackling some open combinatorial problems.

This is based on a joint work with Luis Ferroni, Jacob Matherne, and Matthew Stevens. 

In this talk, we generalize discrete Morse theory to the context of symmetric Delta-complexes. First introduced by Forman, discrete Morse theory is an adaptation of Morse theory for regular CW-complexes. It gives a schematic for collapsing cells in a CW-complex while maintaining the same homotopy type. On the other hand, symmetric Delta-complexes are a generalization of cell-complexes. They are topological spaces built from quotients of standard simplices, and they have played important roles in recent developments of tropical geometry. We generalize various concepts from discrete Morse theory, including discrete Morse functions and acyclic matchings on face posets, and prove parallel theorems for symmetric Delta-complexes. We also apply this new theory to the moduli space of tropical abelian varieties.