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

雷布尼科夫将研究多面体、曲面、图和点格的几何和组合性质。特别是，他将讨论局部和全局凸性与凸性验证之间的联系，Voronoi关于平行四面体猜想的拓扑方法，大型图的刚性和灵活性，格中的完美Delaunay多面体和非齐次二次型的算法。教育活动将包括指导本科生和硕士研究生，这将帮助他们不仅提高他们的数学技能，而且还参与到积极的研究中。图形、多面体和格子不仅在数学中被学习和使用，而且在运筹学、计算分子生物学、密码学、计算机辅助设计等应用中也被学习和使用。雷布尼科夫的研究将受到可软件性、计算分子生物学和密码学等问题的推动。例如，凸性验证是一个重要的问题，不仅在学术论文中得到解决，而且在广泛使用的软件中也得到了解决，例如。Pi对图的刚性和灵活性的研究大多是受蛋白质结构分析和预测问题的启发；高维点阵和二次型是计算复杂性理论和密码学的重要内容。

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.