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

几何研究：凸性、多面体、刚性和点格

• 批准号: 0406956
• 负责人: Konstantin Rybnikov
• 金额: --
• 依托单位: University of Massachusetts Lowell Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-11-15 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0406956&HistoricalAwards=false
• 关键词: Studies; Geometry; Convexity; Polyhedra; Rigidity

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.

Rybnikov将研究多面体，曲面，图形和点阵的几何和组合性质。 特别是，他将解决局部和全局凸性和凸性验证，拓扑方法Voronoi的猜想上parallelohedra，rigidityand灵活性的大型图形，并完善Delaunay多面体的lattices和算术的非齐次二次型之间的连接。教育活动将包括本科生和硕士生的监督，这将帮助他们不仅提高他们的数学技能，但也参与积极的研究。图，多面体和格的研究和使用不仅在数学，而且在应用程序，如运筹学，计算分子生物学，密码学，CAD等。软件可靠性、计算分子生物学和密码学的问题。 例如，凸性验证是一个重要的问题，它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。LEDA; PI对图的刚性和柔性的研究大多是由蛋白质结构分析和预测中的问题激发的; 2高维的点格和二次型对计算复杂性理论和密码学很重要。