Geometry, topology, and dynamics in negative curvature

负曲率中的几何、拓扑和动力学

• 批准号: 1016098
• 负责人: Jean-Francois Lafont
• 金额: --
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1016098&HistoricalAwards=false
• 关键词: Geometry; topology; dynamics; negative; curvature

AbstractAward: DMS-1016098Principal Investigator: Jean-Francois R. Lafont, MichaelW. Davis, Pedro OntanedaThis is a proposal to provide funding for the conferenceGeometry, Topology and Dynamics in Negative Curvature to be heldin Bangalore, India, during the week of August 2-7, 2010. Theconference is an official satellite conference to the 2010International Conference of Mathematicians (held in Hyderabad,India). The goal of this conference is to bring togethermathematicians working on different aspects of negatively curvedspaces. It will bring together three distinct communities ofmathematicians: geometers, topologists, and dynamicists. It willprovide a forum for sharing the most recent developments in thestudy of spaces of negative (and more generally, non-positive)curvature. A selection of topics that will be touched upon in theconference include: (1) Gromov hyperbolic groups, CAT(0)-groups,and geometric group theory, (2) spaces of metrics (Teichmullerspace, moduli space, representation varieties), (3) links with3-dimensional topology (Geometrization conjecture, mapping classgroups), (4) links with high-dimensional topology (Novikovconjecture, Borel conjecture), (5) analysis on boundaries ofgroups and spaces, (6) Anosov flows and dynamics, (7) flows onhomogeneous spaces and applications to number theory. We expectthe conference to provide a "snapshot" of the current state ofour knowledge in matters related to nonpositively curved spaces,as well as to provide a roadmap for future research in thisinterdisciplinary field.Curvature is a fundamental notion in geometry. Zero curvature isthe most physically familiar, and corresponds to an ordinary flatsurface. Positive and negative curvature can be describedrelative to the familiar zero curvature setting as follows: onetries to "flatten" the surface near a point, and looks to see ifthere is too little or too much fabric to achieve theflattening. For instance, the surface of a cone or of a cylindercan be unfolded to lie flat, so these also have zero curvature.But if one tries to flatten the surface of a sphere (think of theskin of an orange), there is not enough fabric to do this(i.e. the skin of the orange splits), which corresponds topositive curvature. Alternatively, if one tries to flatten asaddle shaped surface, folds appear. There is too much fabric,which corresponds to negative curvature. A space of nonpositivecurvature can now be described as one having the property that,near every point and for every pair of directions, it lookseither flat or like a saddle shaped surface. Surprisingly, suchspaces are much more prevalent than one would initiallyguess. Aside from the field of geometry, they naturally appear innumerous other fields of mathematics: topology, dynamics, numbertheory, representation theory, etc. Aspects of nonpositivecurvature have also made an increasing appearance in more appliedfields, wherever there has been an interest in understanding the"shape" of objects (special relativity theory, crystallinestructures in organic chemistry, configuration spaces inrobotics, etc). This conference will bring together internationalexperts working on various aspects of nonpositively curvedspaces. The conference web site is http://www.icts.res.in/program/gtdnc.

摘要奖：DMS-1016098首席研究员：Jean-Francois R.Lafont，MichaelW。戴维斯，Pedro Ontaneda这是一项为负曲率几何、拓扑和动力学会议提供资金的提案，会议将于2010年8月2日至7日在印度班加罗尔举行。该会议是2010年国际数学家会议(在印度海得拉巴举行)的官方卫星会议。这次会议的目的是让研究负曲率空间不同方面的数学家们聚集在一起。它将把三个不同的数学家群体聚集在一起：几何学家、拓扑学家和动态学家。它将提供一个论坛，分享负(更广泛地说，非正)曲率空间研究的最新发展。会议将涉及的主题精选包括：(1)Gromov双曲群，CAT(0)-群和几何群论，(2)度量空间(TeichMuller空间，模空间，表示簇)，(3)与三维拓扑的链接(几何化猜想，映射类群)，(4)与高维拓扑的链接(Novikov猜想，Borel猜想)，(5)群和空间的边界分析，(6)Anosov流和动力学，(7)齐次空间上的流和数论的应用。我们期望这次会议能为我们在与非正曲线空间相关的问题上的知识现状提供一个“快照”，并为这一交叉学科领域的未来研究提供一个路线图。零曲率是最常见的物理概念，它对应于一个普通的平面。相对于熟悉的零曲率设置，正曲率和负曲率可以描述为：一种尝试将某点附近的曲面“展平”，并查看是否有太少或太多的布料无法实现展平。例如，圆锥体或圆柱体的表面可以展开以平放，因此它们也具有零曲率。但是，如果试图展平球体的表面(想想橙子的皮)，就没有足够的布料来做到这一点(即橙子的皮分裂)，这与拓扑正曲率相对应。或者，如果试图将马鞍形曲面展平，则会出现褶皱。有太多的布料，这对应于负曲率。非正曲率空间现在可以描述为这样一个空间，它具有这样的性质：在每个点附近，对于每一对方向，它看起来要么是平的，要么像马鞍形的曲面。令人惊讶的是，这样的空间比人们最初想象的要普遍得多。除了几何领域，它们自然出现在许多其他数学领域：拓扑学、动力学、数论、表示论等。非正向曲率的方面也越来越多地出现在更多的应用领域，无论是在哪里，只要人们有兴趣了解物体的“形状”(狭义相对论、有机化学中的晶体结构、机器人学中的位形空间等)。这次会议将汇聚研究非正曲线空间各个方面的国际专家。会议的网址是http://www.icts.res.in/program/gtdnc.。