This project aims at systematically analyzing and studying the combinatorial statistics database FindStat with machine learning techniques. The two guiding open problems are longstanding questions in enumerative, bijective and algebraic combinatorics. They concern the famous q,t-Catalan numbers. The first is to find a combinatorial proof of their symmetry and the second is to find a combinatorial definition for general reflection groups. These two longstanding open problems are perfect candidates to be approached using machine learning techniques. A lot of research has been devoted to solutions and we expect machine learning combinatorial statistics and maps to provide genuinely new combinatorial insight.

Projects

An arrangement of finitely many linear hyperplanes is simplicial if all its regions are simplicial cones. From the geometric perspective, simpliciality imposes strong restrictions and it is widely believed that simplicial arrangements are rare. In contrast to the geometric side, computer experiments suggest an abundance of simplicial matroids! This project initiates a coherent study of simpliciality in arrangements and matroids with an emphasis on algebra, combinatorics, and geometry. The three main directions of research, Generation and Realization, Algebra and Convexity, and Matroidal and Simplicial Combinatorics are pursued in parallel.

The project concerns studying sums of nonnegative circuit polynomials (SONC) and their use for sparse polynomial optimization. These concepts regard key ideas in the rapidly growing interplay of combinatorics, convexity and and non-linear optimization. The goal is to develop new combinatorial methods for the construction of certificates of nonnegativity in sparse contexts and the interaction of the SONC cone with a variety of combinatorial concepts, including convex cones, matroids, and lattice points. Specific working directions concern exactness results, convex-algebraic questions, generalized combinatorial models for the optimization and nonnegativity of sparse polynomials as well as selected applications.

We study a probabilistic real intersection theory in compact homogeneous spaces M. The examples of prime interest are the real Grassmannians. The intersection of randomly moved submanifolds of M is captured by a notion of multiplication of certain convex bodies, that we call Grassmann zonoids. They live in the exterior power of a Euclidean vector space V, chosen as the cotangent space V of M. Alternatively, these zonoids can be viewed as positive measures on the Grassmannians of V. (They should be invariant under the action of the isotropy group.) There are close connections to integral theory and the theory of valuations.

One goal of the project is to prove a generalization of the Alexandrov-Fenchel inequality for higher order Grassmann zonoids. This would imply that certain expected intersection numbers arise as coefficients of Lorentzian polynomials and are therefore define logconcave sequences. In the same direction, we want to show that the Grassmann zonoid algebra has a homomorphic image that satisfies the properties of a Kähler package.

Another goal is to investigate the Grassmann zonoid algebra with the tools of harmonic analysis and represention theory. In particular, we would like to compute the volume of the Schubert zonoids and to understand their multiplication.

A lattice polytope is the convex hull of finitely many lattice points. One possibility to investigate the generic behaviour of lattice polytopes is to study random lattice polytopes, more precisely the randomized lattice convex hull which is the convex hull of the lattice points inside a random convex set.

We consider the integer lattice and choose a random convex set whose shape is given but its position is chosen at random. We are interested in the limiting structure of the random lattice polytope as the size of the convex set tends to infinity. In particular, we would like to analyse the combinatorial behaviour (expected f-vector) and metric behaviour (expected intrinsic volume) of the random lattice polytopes and how these depend on the shape of the underlying convex set.

Further investigations will deal with higher moments and questions concerning distributional properties of functionals of the random lattice polytopes.

This proposal aims at a deeper understanding of several recently discovered mechanisms from the theory of Lorentzian polynomials that preserve log-concavity. Beyond the intrinsic interest, our main motivation for this investigation is to obtain precise information about the equality cases in log-concave inequalities. Primary objectives of this project are the characterization of the equality cases in the generalized Alexandrov-Fenchel inequality and Kahn-Saks inequality for convex polytopes and applications to combinatorial poset inequalities. A key feature of our proposed research is the utilization of connections and synergies between combinatorics, convexity, and algebraic geometry.

This project concerns the number of terms of polynomials as a complexity measure. This is an area of commutative algebra that is much less explored than degree-based complexity measures like Castelnuovo--Mumford regularity. As the finiteness results that drive the Gröbner machinery are based on induction on the degree, they often need to be replaced by more synergetic tools to make progress here. We envision that combinatorial data structures like Newton polyhedra and matroids will help us to solve the fundamental problem of this project:

-- Is it algorithmically decidable if an ideal in a polynomial ring contains a short polynomial? --

The context of this project are recent applications of combinatorics in particle physics motivating new research questions and directions in mathematics that the project is going to explore. The area is Positive Geometry and the focus in this project is mainly on questions in discrete geometry inspired by physics. In terms of the themes of the Schwerpunktprogramm, the project is mainly an the intersection of Convexity and Mathematical Physics. There are possible connections to Matroids as well as Commutative Algebra.

This project has two intertwined goals: Compute the graded parts of Orlik-Solomon algebras of geometric lattices together with its algebra structure and generalise the Lehrer-Solomon conjecture, possibly altering its statement, to other classes of arrangements and nonrepresentable matroids.

For the first goal, a major strategy will be the exploitation of equivariant structures under (subgroups of) the automorphism group G of the underlying lattice of the Orlik-Solomon algebra.

For the second goal, reasonable classes of arrangements that allow both for an inductive process and have nontrivial automorphism groups are the class of inductively free arrangements and the larger class of free arrangements. Among these classes are the class of Coxeter arrangements and the larger classes of ideal subarrangements of Weyl arrangements and of crystallographic arrangements, all of which include arrangements with relatively large automorphism groups.

The need to combine both strategies naturally leads us to the use of the category-theoretic language, both as a theoretical framework and as an algorithmic machinery.

The results will be made accessible via an online database.

Tropical geometry is a dequantized version of classical algebraic geometry, which studies a combinatorial piecewise linear shadow associated to compactifications and degenerations of algebraic varieties.

One of its most successful and active parts is tropical linear algebra, a term that stands for the many different incarnations of linearity in tropical geometry. This area ranges from the study of matrices over the min-plus algebra coming to us from optimization to the geometric study of (valuated) matroids. It includes the recent breakthrough results in the Hodge theory of matroids, but also the study of the tropical Grassmannians and Dressians with its numerous applications, e.g. to phylogenetics.

The central objectives of this project are:

(1) to develop new foundations for tropical linear algebra using techniques from the geometry of affine buildings, the combinatorics of (valuated) bimatroids, and the algebra of hyperstructures;

(2) to expand tropical linear algebra beyond the A_n case, integrating valuated Coxeter matroids and affine buildings; and

(3) to explore applications of tropical linear algebra, focussing on the combinatorial Hodge theory of bimatroids and Coxeter matroids, the yet-to-be-fully-developed geometry of tropical vector bundles, and the dequantization process in categorical quantum theory.

Wachspress coordinates were initially conceived as a particular generalization of barycentric coordinates for general polytopes, but have since re-emerged in a wide variety of seemingly unrelated contexts in mathematics (adjoint hypersurfaces, cone volumes), statistics (Wachspress models) and physics (positive geometries). They and their related "Wachspress objects" have now been linked to a number of phenomena in algebraic geometry, polyhedral combinatorics, convex geometry, spectral graph theory and rigidity theory.

This project aims to explore, explain and exploit the ubiquity of the Wachspress objects, also including the Wachspress variety, the adjoint polynomial and the Izmestiev matrix. Particular focus will be on a study of the Wachspress variety, which is expected to inform problems in polytope rigidity, geometric modelling and polyhedral combinatorics; as well as generalizations of the Wachspress objects to non-convex and non-polytopal contexts.

further details: https://martinwintermath.github.io/wachspress-geometry.html

In algebra, geometry, combinatorics and areas of applications of these disciplines there exist in many interesting situations the phenomena that objects of interest occur in ascending chains A(n) and there are suitable and compatible group actions on these objects. Example are generic determinantal ideals in polynomial rings with an increasing number of variables or symmetric polytopes as well as cones in real vector spaces of increasing dimensions. Often there exist a limit object B of the chain, which can be useful to study properties of the elements of the chain and vice versa. In the two examples, the union of the elements of the chain is such a limit object (defined in a suitable way).

Three questions are then of importance: (1) Determine the asymptotic behavior of properties of interest of the A(n); (2) Study corresponding properties of B; (3) Show a connection between the properties of A(n) and the ones of B.

An answer to (1) can be seen as interesting local information and one to (2) as some kind of global information. (3) can then be considered as a local-global principle which one would like to understand. In the recent years, this approach was extremely successfully applied to various situation in the areas mentioned above with applications, for example, also in machine learning, optimization, and representation theory. The overall goal beyond the project is to develop foundations of commutative algebra up to symmetry and polyhedral geometry up to symmetry in a systematic way.

The key objectives of this project are to:

- Develop algorithms and algebraic structures to efficiently count permutation and higher-dimensional chirotope patterns. Identify subclasses of patterns that can be counted efficiently.
- Develop a notion of entropy based on chirotope patterns to analyze the complexity of multidimensional time series.
- Introduce cumulants of permutation and chirotope patterns and study their properties and applications in statistics.
- Establish connections between counting patterns, dynamic programming, multiparameter integrals and (crossed modules of) Hopf algebras.

The main focus of classical Ehrhart theory is the h*-polynomial of a lattice polytope: a linear transformation of the Ehrhart polynomial, the fundamental invariant that counts lattice points in dilates of lattice polytopes. The h*-polynomial can be further decomposed into nonnegative local contributions of faces using toric g-polynomials. The main local contribution from the polytope itself is called its local h*-polynomial.

We advertize local Ehrhart theory as the rich field of study of local h*-polynomials of lattice polytopes. We plan to probe the space of local h*-vectors, to investigate interesting classes and polytopal constructions using machine-learning algorithms, and to explore new and existing synergies such as those with dual defective varieties, hypergeometric motives and mirror symmetry.

The project is devoted to the study of the flatness constant and other geometric constants associated to lattices and defined as extremal problems along with the respective extremal bodies. Such studies are relevant in a number of contexts, including algebraic geometry and integer optimization. In our studies we will combine structural methods of polyhedral combinatorics, the theory of lattice with computational computer-assisted approaches.

Directed acyclic graphical models, sometimes called Bayesian networks, are of critical importance in modern data science and statistics through their applications to causality and probabilistic inference. These statistical models use directed acyclic graphs (DAGs) to represent causal relationships between random variables and are often specified by a system of structural equations which is defined using the associated DAG. This project studies a relatively new family of directed graphical models which are called max-linear Bayesian networks. Unlike many other classical families of graphical models, they are not faithful to the well-known d-separation criterion and thus they exhibit very different conditional independence structures. This means that many standard algorithms for learning graphical models from data can not be applied to data which is generated by a max-linear Bayesian network. A main goal of this project is to develop new combinatorial algorithms which are able to reconstruct a graph that best represents given data based on which conditional independencies are observed.

The classes of free arrangements and free multiarrangments play pivotal roles in the theories of hyperplane arrangements and multiarrangements, respectively. Ziegler showed in his seminal work that a free arrangement A gives rise to a canonical free multiarrangement on any restiction of A with exponents only depending on the exponents of A. We refer to this construction as a Ziegler restriction. In 2010, Yoshinaga asked for a reverse procedure, which he coins as extension: given a free multiarrangement is this the Ziegler restriction of a free arrangement A? In general this is not the case. We call this a Ziegler extension.

The ultimate goal of Yoshinaga's work in this context is a complete description of all free intermediate arrangements between the extended Shi and extended Catalan arrangements for simply laced underlying root systems. There is a long history and continued interest in the literature regarding questions of freeness of the families of extended Shi and extended Catalan arrangements. Among other aspects, this proposal is a contribution to this theory.

The goal of this project is to continue Yoshinaga's investigation about extensions of free multiarrangements. Firstly, we aim to answer some of the questions and conjectures raised in Yoshinaga's paper. Secondly, we intend to extend the classification from this work to all intermediate free arrangements between extended Shi and extended Catalan arrangements for arbitrary Coxeter types, i.e. to also include non-simply laced ones. Thirdly, we plan to investigate extensions of free multiarrangements over finite fields, a topic which has not been studied in the literature as of yet. Finally, in a fourth research stand we aim at studying free extensions of various natural free complex multireflection arrangements.

In a recent paper, Cuntz and Kühne introduce a special class of hyperplane arrangements stemming from a given (connected) graph G, so called connected subgraph arrangements A(G). They include such prominent classes such as the braid arrangements, Shi arrangements, and resonance arrangements among countless many others.

The aims of this project are fourfold. Firstly, we intend to answer some of the questions raised in the work of Cuntz and Kühne. Secondly we aim to strengthen several of the results from their paper. Thirdly we hope to prove some new results for this class of arrangements, e.g. over finite fields. In their work, Cuntz and Kühne classified all connected subgraph arrangements over the rationals which are free, factored, simplicial or supersolvable.

In this project we aim to extend and strengthen these results as follows. We aim to show that a connected subgraph arrangement A(G) over the rationals is free if and only if it is inductively free; that it is factored if and only if it is inductively factored.

Cuntz and Kühne specifically raise the question of classifying members among the arrangements A(G) over the rationals that are aspherical. This is probably a hopeless task, as determining asphericity for a given set of arrangements is a notoriously difficult undertaking. However, our further aim is to compile a comprehensive list of graphs G for which A(G) is not aspherical. As asphericity is a local property, this then allows us to conclude that large classes of connected subgraph arrangements are not aspherical. These in particular encompass the aforementioned resonance arrangements of rank at least 5.

Undoubtedly, polytopes do not only belong to the central objects studied in contemporary research in geometry and combinatorics, but they are also important for applications in a multitude of areas, including optimization, theoretical physics, mathematical statistics, computational geometry, and machine learning, among others. Two prominent classes of lattice polytopes that are interesting not only for their intrinsic combinatorial and geometric properties but also due to their multiple applications to e.g., metric space theory and in particular physics, are symmetric edge polytopes and cosmological polytopes that are associated to graphs.

In the proposed research program we study these two classes of polytopes (and further generalizations) from a deterministic and a probabilistic perspective. The objectives can be summarized as follows:

(a) We propose several ways to extend the definition of classical symmetric edge polytopes by replacing the underlying graph by a higher-dimensional simplicial complex. We investigate fundamental geometric and combinatorial properties of the resulting lattice polytopes, such as the dimension, the f-, h-, h^*- and the gamma-vector. To achieve this we are interested in triangulations of these polytopes and ask for the existence of a regular unimodular triangulation (equivalently a squarefree Gröbner basis of the toric ideals).

(b) We investigate combinatorial, geometric and probabilistic properties of randomly generated lattice polytopes such as random symmetric edge polytopes, as well as random cosmological polytopes. We study expectation and variance asymptotics of the f-vector as the size of the underlying graph or the simplicial complex tends to infinity. The considered models of random graphs encompass the classical Erdös-Rényi model, but also r-regular random graphs and random geometric graphs.

(c) We study asymptotic distributional properties of randomly generated lattice polytopes in high dimensions. In particular, we establish quantitative central limit theorems for the number of facets and the logarithmic volume for random symmetric edge and cosmological polytopes using variants of Stein's method for normal approximation such as the discrete Malliavin-Stein method.

(d) We contribute to a deeper understanding about the typical or expected behaviour of lattice polytopes by studying such polytopes from a probabilistic angle. This complements the more traditional approach in which properties of specific classes of lattice polytopes are usually investigated.

This project also opens up several new directions for future research, including the study of generalized symmetric edge polytopes for random simplicial complexes and matroids as well as the quest for more refined central limit theorems, such as moderate or large deviation principles and concentration bounds.