Polyhedral Challenges and Tools throughout Combinatorics

The focus of this project is polytopes and polyhedra. These are jewel-like geometric structures that, because of their essential basic nature, have universal importance in many areas of pure and applied mathematics. For example, the scheduling of airplane crews and routing of airplanes requires fast computation with polytopes. Similarly, polytopes are enjoyed by school age children and teachers and have played a role in architecture, art, and philosophy for thousands of years. Despite their importance we still do not know the answer to some of the most basic mathematical questions about them. The PI plans to advance our mathematical understanding of polytopes. The PI's mathematical results and software will be directly relevant to a wide range of mathematicians but also to researchers in theoretical computer science. The PI will also continue his efforts in developing educational materials and software for training mathematicians in the use of polytopes. In fact, training of a new generation of computational mathematicians is key to his plans. The PI will attack various problems about polytopes and develop useful tools for other researchers: in enumerative and extremal combinatorics the PI and his team will look at slices and shadows of polytopes, in algebraic combinatorics he will look at lattice points and a new (weighted, colored) Ehrhart theory, and in geometric-topological combinatorics the PI and his team will look at Baues complexes and related geometric constructions of fiber polytopes, floating bodies, illumination bodies, and others. Methods used by the PI include combinatorics, discrete mathematics, algebraic geometry, convex geometry, commutative algebra, lattices, representation theory, probability, and computer-aided experimentation, search, certification, and proofs. The PI will stress the emerging connection between Artificial Intelligence and polyhedral research. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.