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.
The research focused on problems in algebraic combinatorics, at the crossroads of discrete geometry, convex polytopes, combinatorial commutative algebra, and algebraic K-theory. 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. This project outcomes include (1) three publications and two preprints, submitted for publication - all in peer reviewed journals, (2) presentations at six national and international confereces, and a series of colloquium talks at various universities in several countries, (3) supervision of six masters theses, (4) creating an online course and short video recordings for attracting prospective students, (5) initiating collaboration with a group of researchers and students from Georgia (country). The two preprints, mentioned in (1) above, are coauthored by two of PI's recent masters students. In one of the publications a complete resolution of a well known problem in the theory of normal polytopes is given. Normal polytopes serve as discrete versions of continuous geometric figures and are the building blocks for toric varieties - popular objects of study in algebraic geometry. The result was already quoted and used a number of times by other authors. In two other publications a novel pure algebraic study of general convex polytopes is initiated, providing a new unified context for classical geometric constructions and leading to new challenges in the theory of polytopes. Other results include (i) a deeper insight into the structure of higher K-theory of toric varieties - a sophisticated topology (the science of continuous deformations of shapes) inspired study of geometric objects of discrete type, and (ii) discovery of new classes of lattice polytopes, exhibiting extremal properties, not known to be possible before. The interaction of combinatorics and abstract mathematical techniques, which was 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. Algorithmic aspects of many of the challenges in the field offered excellent opportunities for involving beginning graduate students in the research. This opportunity was fully used by the PI, resulting in many masters theses over the whole period of the grant and creating conditions for future extensive work with students, further faciliated by several video recordings - short 20 min. "promos" on research problems and an online course, all posted on PI's web page. Six months before the end of the grant period, the PI started working with Georgian (country) team of reserachers and students on some of the project related problems with strong computational element. The goal - already partly achieved - is to create international American-Georgian team of investigators and students.
