Jesús A. De Loera’s research while affiliated with University of California, Davis and other places

We study the question whether the affine semigroup of integer points in a convex cone can be finitely generated up to symmetries of the cone. We establish general properties of finite generation up to symmetry, and then concentrate on the case of irrational polyhedral cones.

We investigate the semigroup of integer points inside a convex cone. We extend classical results in integer linear programming to integer conic programming. We show that the semigroup associated with nonpolyhedral cones can sometimes have a notion of finite generating set with the help of a group action. We show this is true for the cone of positive semidefinite matrices (PSD) and the second-order cone (SOC). Both cones have a finite generating set of integer points, similar in spirit to Hilbert bases, under the action of a finitely generated group. We also extend notions of total dual integrality, Gomory-Chvátal closure, and Carathéodory rank to integer points in arbitrary cones.

This paper studies three different ways to assign weights to the lattice points of a convex polytope and discusses the algebraic and combinatorial properties of the resulting weighted Ehrhart functions and their generating functions and associated rings. These will be called q-weighted, r-weighted, and s-weighted Ehrhart functions, respectively. The key questions we investigate are \emph{When are the weighted Ehrhart series rational functions and which classical Ehrhart theory properties are preserved? And, when are the abstract formal power series the Hilbert series of Ehrhart rings of some polytope?} We prove generalizations about weighted Ehrhart $h^*$-coefficients of q-weighted Ehrhart series, and show q- and s-weighted Ehrhart reciprocity theorems. Then, we show the q- and r-weighted Ehrhart rings are the (classical) Ehrhart rings of weight lifting polytopes.

We study the slices or sections of a convex polytope by affine hyperplanes. We present results on two key problems: First, we provide tight bounds on the maximum number of vertices attainable by a hyperplane slice of d-polytope (a sort of upper bound theorem) and discuss a new algorithm to find all sections. Second, we investigate the sequence of numbers of vertices produced by the different slices over all possible hyperplanes and analyze the gaps that arise in that sequence. We study these sequences for three-dimensional polytopes and for hypercubes. Our results were obtained with the help of large computational experiments, and we report on new data generated for hypercubes.

A famous result in arithmetic Ramsey theory says that for many linear homogeneous equations E there is a threshold value $R_k(E)$ (the Rado number of E) such that for any k-coloring of the integers in the interval [1,n], with $n \ge R_k(E)$, there exists at least one monochromatic solution. But one can further ask, how many monochromatic solutions is the minimum possible in terms of n? Several authors have estimated this function before, here we offer new tools from integer and semidefinite optimization that help find either optimal or near optimal 2-colorings minimizing the number of monochromatic solutions of several families of 3-variable non-regular homogeneous linear equations. In the last part of the paper we further extend to three and more colors for the Schur equation, improving earlier work.

We present three new pivot rules for the Simplex method for Linear Programs over 0/1-polytopes. We show that the number of nondegenerate steps taken using these three rules is strongly polynomial, linear in the number of variables, and linear in the dimension. Our bounds on the number of steps are asymptotically optimal on several well-known combinatorial polytopes. Our analysis is based on the geometry of 0/1-polytopes and novel modifications to the classical steepest-edge and shadow-vertex pivot rules. We draw interesting connections between our pivot rules and other well-known algorithms in combinatorial optimization. Funding: A. E. Black and J. A. De Loera are grateful for the support received through the National Science Foundation [Grants DMS-1818969 and NSF GRFP]. L. Sanita is grateful for the support received from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant VI.Vidi.193.087].

In this paper, we explore affine semigroup versions of the convex geometry theorems of Helly, Tverberg, and Carathéodory. Additionally, we develop a new theory of colored affine semigroups , where the semigroup generators each receive a color and the elements of the semigroup take into account the colors used (the classical theory of affine semigroups coincides with the case in which all generators have the same color). We prove an analog of Tverberg’s theorem and colorful Helly’s theorem for semigroups, as well as a version of colorful Carathéodory’s theorem for cones. We also demonstrate that colored numerical semigroups are particularly rich by introducing a colored version of the Frobenius number.

In the early 1990s, Billera and Sturmfels introduced the monotone path polytope (MPP), an important case of the general theory of fiber polytopes, which has led to remarkable combinatorics. Given a pair (P,φ)$(P,\varphi )$ of a polytope P and a linear functional φ$\varphi$, the MPP is obtained from averaging the fibers of the projection φ(P)$\varphi (P)$. They also showed that MPPs of (regular) simplices and hyper-cubes are combinatorial cubes and permutahedra respectively. As a natural follow-up we investigate the monotone paths of cross-polytopes for a generic linear functional φ$\varphi$. We show the face lattice of the MPP of the cross-polytope is isomorphic to the lattice of intervals in the sign poset from oriented matroid theory. We also describe its f-vector, some geometric realizations, an irredundant inequality description, the 1-skeleton and we compute its diameter. In contrast to the case of simplices and hyper-cubes, monotone paths on cross-polytopes are not always coherent.