RUI: FOUR PROBLEMS IN POLYTOPAL ALGEBRAIC COMBINATORICS

San Francisco State University, San Francisco CA

Investigators

Abstract

The proposed research focuses on problems at the crossroads of discrete geometry, convex polytopes, combinatorial commutative algebra, and algebraic K-theory. The topics involved are: integral Caratheodory property of normal polytopes and rational cones, homological and K-theoretical properties of affine monoid rings, and cofibrations in the category of convex polytopes. The first research direction proposes a new dynamical approach to normal polytopes, as opposed to the traditional study of the static picture of a single polytope. This is done via encoding the interactions of normal polytopes in a certain global poset. The associated topology has a potential of shedding much light to some central open questions on Hilbert bases. The second research topic is an algorithmic attempt at disproving the conjecture that the affine cones over smooth projective toric varieties are Koszul. This includes algorithmic analysis of several closely related properties of independent interest: normality, quadratic generation, resolutions of toric singularities etc. The third research topic is higher K-theory of affine monoid rings. An explicit description of higher K-theory of a singular ring is a rare phenomenon. Here a finer multigraded structure of the involved K-groups is conjectured, as opposed to the weaker graded structures known so far. The fourth research topic concerns the category of convex polytopes and their affine maps. Concrete suggestions are made on how to apply universal categorial concepts to such hypothetical objects as quotient polytopes. Combinatorics is the science of organizing, arranging and analyzing discrete data. An illustrative example is the set of integer points in a convex polygon in the plane or in a convex polytope in the space. Algebraic combinatorics of lattice polytopes studies such point configurations to encode important constructions in algebra, geometry, and topology, while combinatorial methods are well suited for related computations. The interaction of combinatorics and abstract mathematical techniques, which is the leitmotif of this research, over the last two decades has resulted in a number of fundamental theorems in a variety of disciplines. Applications range from algebraic geometry (the science of solution sets to systems of multidimensional polynomial equations) to integer programming, computer science, probability theory, physics, cryptography etc. The progress would have been unimaginable without computer assisted investigation and experimentation, the increasing importance of which is related to the demand for explicit or algorithmic understanding of discrete structures. The latter aspect makes the project especially well suited for engaging beginning graduate students in the research.