We study the Ehrhart \(h^\ast\)-polynomial of (the boundary of) a lattice polytope via regular unimodular triangulations and Gr\"obner degenerations of toric ideals. Our main result is a boundary analogue of the well-known Sturmfels correspondence. This allows us to connect the boundary \(h^\ast\)-polynomial to the \(h\)-polynomial of any regular unimodular triangulation, in analogy to the classical Betke-McMullen Theorem.

Providing a direct link between Ehrhart theory and the face enumeration of simplicial complexes, we then transfer structural results from the theory of simplicial polytopes to the setting of lattice polytopes. In particular, we derive general Dehn-Sommerville-type relations between \(h^\ast(P)\) and \(h^\ast(\partial P)\). Under the additional assumption of \(\partial P\) admitting a regular unimodular triangulation, we recover old and prove new characterization results concerning symmetry or unimodality, as well as upper and lower bounds for coefficient-wise differences within \(h^\ast(P)\).

We study combinatorial aspects of hyperplane sections of polytopes, focusing on how much the combinatorics of the sections determines the combinatorics of the original polytope. We show that, in general, combinatorial information about the sections is not enough to determine even the f-vector of the polytope. In contrast, for sufficiently generic simple polytopes, the function recording the number of vertices of each central section determines the full combinatorial type. We organize different combinatorial slicing properties into hierarchies, separately for affine and central sections, according to the level of combinatorial structure they determine on the polytope. In analogy with classical metric slicing problems, we formulate combinatorial analogues of the Busemann-Petty problem and Bourgain’s slicing problem by replacing volume with face numbers, and show that they fail in every dimension and every face dimension.

Order polytopes for generalized snake posets were recently studied by von Bell et al. (2022), and are known to be unimodularly equivalent to strength-one flow polytopes for acyclic directed graphs strongly dual to generalized snake posets. Lee, Vindas-Meléndez, and Wang (2026) conjectured that the Ehrhart h∗-polynomials of these order polytopes are real-rooted. We prove this conjecture using a connection between these h∗-polynomials and non-nesting rook polynomials, which were recently introduced by Alexandersson and Jal (2024+) in connection with P-Eulerian polynomials for width two posets.

We study monoids in an infinite-dimensional setting that are invariant under the action of the infinite symmetric group Sym. Our main result establishes a local--global principle characterizing equivariant finite generation for arbitrary Sym-invariant monoids, extending earlier results that required additional assumptions. We further analyze local--global phenomena for other fundamental properties, including positivity, normality, seminormality, and simplicity. In addition, we obtain structural results for symmetric monoids, including characterizations of positivity and non-positivity, a description of their groups of units, and explicit formulas for the ranks of local symmetric monoids and stabilizing Sym-invariant chains.

We study duality for monoids in an infinite-dimensional setting that are invariant under the action of the infinite symmetric group Sym. Our main result is an equivariant Minkowski--Weyl theorem for monoids. More precisely, we analyze the evolution of dual monoids along stabilizing Sym-invariant chains and describe the eventual behavior of their equivariant Hilbert bases. In addition, we develop a systematic study of structural properties of dual symmetric monoids, including a characterization of the duals of positive and non-positive monoids.

We describe an implementation of the biset category of finite groups as a tower of standard categorical constructions, all of which are implemented in the software project CAP for algorithmic category theory. In particular, we describe the composition of bisets as a composition in a Kleisli category of some biadjunction monad. This composition relies on the universal property of the coequalizer completion of a group viewed as a groupoid on one object. Implementing this universal property makes nontrivial use of the Schreier-Sims orbit algorithm.

We investigate symmetric edge polytopes generated by Erdős--Rényi random graphs in a high-dimensional regime. These objects provide a natural and largely unexplored model of random lattice polytopes, in which geometric properties are governed by graph-theoretic structure. Focusing on the number of polytope edges and on the number of edges in unimodular triangulations, we derive precise asymptotics for expectations and variances and establish central limit theorems with explicit rates of convergence. Our analysis combines a detailed combinatorial-geometric study of the graph configurations determining the facial structure with the discrete Malliavin--Stein method for normal approximation. In particular, we identify a distinguished parameter value at which the leading variance term cancels, producing an atypical fluctuation regime. To the best of our knowledge, the results obtained here constitute the first distributional limit theorems for random lattice polytopes

This book provides an introduction to the theory of random beta-type simplices and polytopes, exploring their connections to key research areas in stochastic and convex geometry. The random points defining the beta-type simplices, a class of random simplices introduced by Ruben and Miles, follow beta, beta-prime, or Gaussian distributions in the Euclidean space, and need not be identically distributed. A key tool in the analysis of these simplices, the so-called canonical decomposition, is presented here in a generalized form and is employed to derive explicit formulas for the moments of the volumes of beta-type simplices and to prove distributional representations for these volumes. Three independent approaches are described, including the original Ruben–Miles method. In addition, a version of the canonical decomposition for beta-type polytopes is provided, characterizing their typical faces as volume-weighted beta-type simplices. This is then applied to compute various expected functionals of beta-type polytopes, such as their volume, surface area and number of facets. The formulas for the moments of the volumes are also used to investigate several high-dimensional phenomena. Among these, a central limit theorem is established for the logarithmic volume of beta-type simplices in the high-dimensional limit. The canonical decomposition further motivates the study of beta-type distributions on affine Grassmannians, a subject to which the last chapter is dedicated.

Largely self-contained, requiring minimal prior knowledge, the book connects these topics to a broad range of past and current research, serving as an excellent resource for graduate students and researchers seeking to engage with the field of stochastic and integral geometry.

Consider the triangle \(T\) with vertices \((0,0)\), \((0,1)\), and \((1,0)\). The lower boundary of the convex hull of \((0,1)\), \((1,0)\), together with \(n\) independent uniformly distributed random points in \(T\), is called a random convex chain and denoted by \(T_n\). We study the random variable \(f_0(T_n)\), the number of vertices of this chain. Our first result gives an explicit expression for the bivariate generating function of the probabilities \(\mathbb{P}(f_0(T_n)=k+2)\) in terms of the Gaussian hypergeometric function. Building on this analytic representation, we apply a careful singularity analysis to derive a variety of limit theorems for \(f_0(T_n)\), including a quantitative central limit theorem, a large deviation principle as well as a precise asymptotics for the probabilities \(\mathbb{P}(f_0(T_n)=k+2)\). Conceptually, our results establish a novel bridge between stochastic geometry and methods from analytic combinatorics.

Simplicial arrangements are a special class of hyperplane arrangements, having the property that every chamber is a simplicial cone. It is known that the simpliciality property is preserved under taking restrictions. In this article we focus on the class of reflection arrangements and investigate two different polynomial invariants associated to them and their restrictions, the h-polynomial with its γ-vector and the Chow polynomial. We prove that all restrictions of reflection arrangements are γ-positive and give an explicit combinatorial formula of the Chow polynomial in type B. Furthermore we prove that for a special class of restrictions of arrangements of type D, called intermediate arrangements, both the h-polynomial as well as the Chow polynomial behave arithmetically, that is they interpolate linearly between the respective invariants for type B and D.