Consider a stationary Poisson process \(\eta\) in the d-dimensional Euclidean or hyperbolic space and construct a random graph with vertex set \(\eta\) as follows. First, each point \(x\in\eta\) is connected by an edge to its nearest neighbour, then to its second nearest neighbour and so on, until \(x\) is contained in the convex hull of the points already connected to . The resulting random graph is the so-called nearest neighbour embracing graph. The main result of this paper is a quantitative description of the Gaussian fluctuations of geometric functionals associated with the nearest neighbour embracing graph. More precisely, the total edge length, more general length-power functionals and the number of vertices with given outdegree are considered.

The classes of sums of arithmetic-geometric exponentials (SAGE) and of sums of nonnegative circuit polynomials (SONC) provide nonnegativity certificates which are based on the inequality of the arithmetic and geometric means. We study the cones of symmetric SAGE and SONC forms and their relations to the underlying symmetric nonnegative cone. As main results, we provide several symmetric cases where the SAGE or SONC property coincides with nonnegativity and we present quantitative results on the differences in various situations. The results rely on characterizations of the zeroes and the minimizers for symmetric SAGE and SONC forms, which we develop. Finally, we also study symmetric monomial mean inequalities and apply SONC certificates to establish a generalized version of Muirhead's inequality.

We introduce a combinatorial model for the Milnor fibration of a complexified real arrangement using oriented matroids. It is a poset quasi-fibration, a notion recently introduced by the first author, whose domain is a subdivision of the Salvetti complex stemming from a natural subdivision of the dual oriented matroid complex. This yields a concrete finite regular CW complex which is homotopy equivalent to the Milnor fiber of the complexified real arrangement and implies that the homotopy type of the Milnor fiber of a complexified real arrangement only depends on the underlying combinatorial structure given by its oriented matroid. Moreover, our construction works for any oriented matroid, disregarding realizability, so we obtain a notion of a combinatorial Milnor fibration for any oriented matroid.

Motivated by the Gray code interpretation of Hamiltonian cycles in Cayley graphs, we investigate the existence of Hamiltonian cycles in tope graphs of hyperplane arrangements, with a focus on simplicial, reflection, and supersolvable arrangements. We confirm Hamiltonicity for all 3-dimensional simplicial arrangements listed in the Grünbaum--Cuntz catalogue. Extending earlier results by Conway, Sloane, and Wilks, we prove that all restrictions of finite reflection arrangements, including all Weyl groupoids and crystallographic arrangements, admit Hamiltonian cycles. Finally, we further establish that all supersolvable hyperplane arrangements and supersolvable oriented matroids have Hamiltonian cycles, offering a constructive proof based on their inductive structure.

Max-linear Bayesian networks (MLBNs) are a relatively recent class of structural equation models which arise when the random variables involved have heavy-tailed distributions. Unlike most directed graphical models, MLBNs are typically not faithful to d-separation and thus classical causal discovery algorithms such as the PC algorithm or greedy equivalence search can not be used to accurately recover the true graph structure. In this paper, we begin the study of constraint-based discovery algorithms for MLBNs given an oracle for testing conditional independence in the true, unknown graph. We show that if the oracle is given by the ∗-separation criteria in the true graph, then the PC algorithm remains consistent despite the presence of additional CI statements implied by ∗-separation. We also introduce a new causal discovery algorithm named "PCstar" which assumes faithfulness to C∗-separation and is able to orient additional edges which cannot be oriented with only d- or ∗-separation.

We study the canonical form \(\Omega\) as a valuation in the context of scissors congruence for polytopes. We identify the degree of its numerator - the adjoint polynomial \(\operatorname{adj}_P\) - as an important invariant in this context. More precisely, for a polytope \(P\) we define the degree drop that measures how much smaller than expected the degree of the adjoint polynomial of \(P\) is. We show that this drop behaves well under various operations, such as decompositions, restrictions to faces, projections, products and Minkowski sums. Next we define the reduced canonical form \(\Omega_0\) and show that it is a translation-invariant 1-homogeneous valuation on polytopes that vanishes if and only if \(P\) has positive degree drop. Using it we can prove that zonotopes can be characterized as the \(d\)-polytopes that have maximal possible degree drop \(d-1\). We obtain a decomposition formula for \(\Omega_0\) that expresses it as a sum of edge-local quantities of \(P\). Finally, we discuss valuations \(\Omega_s\) that can distinguish higher values of the degree drop.

We introduce a new class of matroids, called graph curve matroids. A graph curve matroid is associated to a graph and defined on the vertices of the graph as a ground set. We prove that these matroids provide a combinatorial description of hyperplane sections of degenerate canonical curves in algebraic geometry. Our focus lies on graphs that are 2-connected and trivalent, which define identically self-dual graph curve matroids, but we also develop generalizations. Finally, we provide an algorithm to compute the graph curve matroid associated to a given graph, as well as an implementation and data of examples that can be used in Macaulay2.---

Barnette's conjecture states that every cubic, bipartite, planar and 3-connected graph is Hamiltonian. Goodey verified Barnette's conjecture for all graphs with faces of size up to 6. We substantially strengthen Goodey's result by proving Hamiltonicity for cubic, bipartite, planar and (2-)connected graphs with faces of size up to 8. Parts of the proof are computational, including a distinction of 339.068.624 cases.

Volume polynomials measure the growth of Minkowski sums of convex bodies and of tensor powers of positive line bundles on projective varieties. We show that Aluffi's covolume polynomials are precisely the polynomial differential operators that preserve volume polynomials, reflecting a duality between homology and cohomology. We then present several applications to matroid theory.

In ???, a particular family of real hyperplane arrangements stemming from hyperpolygonal spaces associated with certain quiver varieties was introduced which we thus call ???. In this note we study thesearrangements and investigate their properties systematically. Remarkably the arrangements ??? discriminate between essentially all local properties of arrangements. In addition we show that hyperpolygonal arrangements are projectively unique and combinatorially formal. We note that the arrangement ??? is the famous counterexample of Edelman and Reiner ??? of Orlik's conjecture that the restriction of a free arrangement is again free.