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

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