Geometry, topology, and dynamics in negative curvature

[摘要]项目编号：dms -1016098。Davis， Pedro ontaneda：这是为将于2010年8月2-7日在印度班加罗尔举行的“负曲率几何、拓扑和动力学”会议提供资金的提案。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。这次会议的目的是把研究负曲线空间不同方面的数学家聚集在一起。它将汇集三个不同的数学家群体：几何学家、拓扑学家和动力学家。它将提供一个论坛，分享负（更一般地说，非正）曲率空间研究的最新进展。本次会议将涉及的主题包括：(1) Gromov双曲群、CAT(0)群、几何群论；(2)度量空间（teichmuller空间、模空间、表示变体）；(3)与三维拓扑的联系（几何化猜想、映射类群）；(4)与高维拓扑的联系（novikov猜想、Borel猜想）；(5)群与空间的边界分析；(6)Anosov流与动力学；(7)齐次空间上的流及其在数论中的应用。我们希望这次会议能够提供我们在非正弯曲空间相关问题上的知识现状的“快照”，并为这一跨学科领域的未来研究提供路线图。曲率是几何学中的一个基本概念。零曲率在物理上是最常见的，它对应于一个普通的平面。相对于我们熟悉的零曲率设置，正曲率和负曲率可以这样描述：一个人试图在一个点附近“拉平”表面，并观察是否有太少或太多的织物来实现拉平。例如，锥体或圆柱体的表面可以展开平躺，因此它们也具有零曲率。但是，如果一个人试图把一个球体的表面弄平（想象一下橘子的表皮），就没有足够的织物来做到这一点。橙色的表皮裂开，这对应于正曲率。另外，如果试图将鞍形表面拉平，则会出现褶皱。面料太多，对应负曲率。一个非正曲率的空间现在可以被描述为具有这样的性质，即在每一点附近，在每一对方向上，它要么看起来平坦，要么看起来像一个鞍形表面。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。非正曲率的各个方面也越来越多地出现在更多的应用领域，无论人们对理解物体的“形状”有兴趣（狭义相对论，有机化学中的晶体结构，机器人技术中的构型空间等）。本次会议将汇集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。

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.