Studies in Geometry: Convexity, Polyhedra, Rigidity, and Point Lattices

University Of Massachusetts Lowell Research Foundation, Lowell MA

Investigators

Abstract

Rybnikov will study geometric and combinatorial properties of polytopes, surfaces, graphs, and point lattices. In particular, he will address connections between local and global convexity and convexity verification, topological approaches to Voronoi's conjecture on parallelohedra, rigidity and flexibility of large graphs, and perfect Delaunay polytopes in lattices and arithmetic of inhomogeneous quadratic forms. The educational activity will include supervision of undergraduate and Masters students, which will help them not only improve their mathematical skills, but also get involved in active research. Graphs, polytopes, and lattices are studied and used not only in mathematics, but also in applications such as operations research, computational molecular biology, cryptography, CAD, etc. Rybnikov's research will be motivated, among other things, by problems in software reliability, computational molecular biology, and cryptology. For example, convexity verification is an important issue, which is addressed not only in academic papers, but also in widely used software, such as e.g. LEDA; most of PI's research on rigidity and flexibility of graphs is motivated by problems in protein structure analysis and prediction; point lattices and quadratic forms in higher dimensions are important to the computational complexity theory and cryptology.