The proposed research lies at the crossroads of discrete geometry, combinatorial commutative algebra, toric geometry, higher algebraic K-theory of rings, and the theory of convex polytopes. The expected results will have applications to other fields, such as geometric number theory, integer programming (via Groebner bases of binomial ideals) and, potentially, physical sciences (algebraic structure of physical units). The first group of problems concerns the Hilbert bases of polyhedral cones - distinguished lattice point configurations in discrete geometry for which no satisfactory geometric characterization is known to date. Very concrete working conjectures are made on Caratheodory ranks in higher dimensions, an effective uniform version of Knudsen-Mumford's result on unimodular triangulations, and point configurations with extremal arithmetic properties. Despite much effort in recent years the current state of the art is that positive results are very difficult to obtain as the dimension of the space increases, and in higher dimensions only a few striking counterexamples are known. Verification of the proposed conjectures would shed much light on the global picture. The second group of problems is more related to algebraic geometry. The problems concern sophisticated homological and high K-theoretical aspects of lattice point configurations, such as Koszul property of projective embeddings of smooth toric varieties and K-homotopy invariance of toric singularities. Related algorithmic issues include analysis of smooth polytopes and factorization of invertible matrices over monomial algebras. The project also aims at a deeper understanding of the category of polytopes (mapping, tensor and cofiber objects).
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 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.
