We study Voronoi diagrams of manifolds and varieties with respect to polyhedral norms. We provide upper and lower bounds on the dimensions of Voronoi cells. For algebraic varieties, we count their full-dimensional Voronoi cells. As an application, we consider the polyhedral Wasserstein distance between discrete probability distributions.

We study-y-vectors associated with h\ast-vectors of symmetric edge polytopes both from a deterministic and a probabilistic point of view. On the deterministic side, we prove nonnegativity of-y2 for any graph and completely characterize the case when-y2 = 0. The latter also confirms a conjecture by Lutz and Nevo in the realm of symmetric edge polytopes. On the probabilistic side, we show that the-y-vectors of symmetric edge polytopes of most ErdoH \s--Re'\nyi random graphs are asymptotically almost surely nonnegative up to any fixed entry. This proves that Gal's conjecture holds asymptotically almost surely for arbitrary unimodular triangulations in this setting.

We associate with every pure flag simplicial complex Δ a standard graded Gorenstein F-RΔ whose homological features are largely dictated by the combinatorics and topology of Δ . As our main result, we prove that the residue field F has a k-step linear RΔ-resolution if and only if Δ satisfies Serre's condition (S k) over F and that RΔ is Koszul if and only if Δ is Cohen-Macaulay over F. Moreover, we show that RΔ has a quadratic Gröbner basis if and only if Δ is shellable. We give two applications: first, we construct quadratic Gorenstein F-s that are Koszul if and only if the characteristic of F is not in any prescribed set of primes. Finally, we prove that whenever RΔ is Koszul the coefficients of its γ-vector alternate in sign, settling in the negative an ic generalization of a conjecture by Charney and Davis.

We showcase applications of nonlinear algebra in the sciences and engineering. Our review is organized into eight themes: polynomial optimiza-tion, partial differential equations, algebraic statistics, integrable systems, configuration spaces of frameworks, biochemical reaction networks, algebraic vi-sion, and tensor decompositions. Conversely, developments on these topics inspire new questions and algorithms for algebraic geometry.

Lascoux polynomials have been recently introduced to prove polynomiality of the maximum-likelihood degree of linear concentration models. We find the leading coefficient of the Lascoux polynomials (type C) and their generalizations to the case of general matrices (type A) and skew symmetric matrices (type D). In particular, we determine the degrees of such polynomials. As an application, we find the degree of the polynomial 8(m, n, n - s) of the algebraic degree of semidefinite programming, and when s =1 we find its leading coefficient for types C, A and D.

One way to study a hypergraph is to attach to it a tensor. Tensors are a generalization of matrices, and they are an efficient way to encode information in a compact form. In this paper, we study how properties of weighted hypergraphs are reflected on eigenvalues and eigenvectors of their associated tensors. We also show how to efficiently compute eigenvalues with some techniques from numerical algebraic geometry.

We study projective schemes arising from eigenvectors of tensors, called eigenschemes. After some general results, we give a birational description of the variety parametrizing eigenschemes of general ternary symmetric tensors, and we compute its dimension. Moreover, we characterize the locus of triples of homogeneous polynomials defining the eigenscheme of a ternary symmetric tensor. Our results allow us to implement algorithms to check whether a given set of points is the eigenscheme of a symmetric tensor and to reconstruct the tensor. Finally, we give a geometric characterization of all reduced zero-dimensional eigenschemes. The techniques we use rely on both classical and modern complex projective algebraic geometry.

An independence model for discrete random variables is a Segre Veronese variety in a probability simplex. Any metric on the set of joint states of the random variables induces a Wasserstein metric on the probability simplex. The unit ball of this polyhedral norm is dual to the Lipschitz polytope. Given any data distribution, we seek to minimize its Wasserstein distance to a fixed independence model. The solution to this optimization problem is a piecewise algebraic function of the data. We compute this function explicitly in small instances, we study its combinatorial structure and algebraic degrees in general, and we present some experimental casestudies.