Geometry, topology, and dynamics in negative curvature

摘要奖：DMS-1016098主要研究者：Jean-Francois R. Lafont，Michael W.这是一个为2010年8月2日至7日在印度班加罗尔举行的“负曲率中的几何、拓扑和动力学”会议提供资金的建议。这次会议是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。这次会议的目标是把数学家们聚集在一起，研究负弯曲空间的不同方面。它将把三个不同的数学家群体聚集在一起：几何学家、拓扑学家和动力学家。它将提供一个论坛，分享在负（更一般地说，非正）曲率空间研究的最新进展。会议将涉及的主题包括：（1）Gromov双曲群，CAT（0）-群和几何群论，（2）度量空间（Teichmullerspace，moduli space，representation varieties），（3）具有三维拓扑的链（几何化猜想，映射类群），（4）高维拓扑的链接（Novikov猜想，Borel猜想），（5）群与空间的边界分析，（6）Anosov流与动力学，（7）齐性空间上的流及其在数论中的应用。我们希望这次会议能提供一个关于非正弯曲空间的知识现状的“快照”，并为这个跨学科领域的未来研究提供一个路线图。零曲率是物理上最熟悉的，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如下：一个人试图“展平”一点附近的表面，并查看是否有太少或太多的织物来实现展平。例如，圆锥体或圆柱体的表面可以展开成平的，因此它们也具有零曲率。但如果我们试图展平球体的表面（想想橙子的皮），就没有足够的材料（比如橙子的皮）来做这件事，这对应于正曲率。或者，如果你试图压平一个马鞍形的表面，褶皱就会出现。有太多的织物，这对应于负曲率。非正曲率空间现在可以被描述为这样一个空间，它在每一点附近和每一对方向上都具有这样的性质，它看起来是平坦的，还是像一个鞍形曲面。令人惊讶的是，这样的空间比人们最初猜测的要普遍得多。除了几何学领域，它们自然地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在更多的应用领域中出现，在任何对理解物体的“形状”感兴趣的地方（狭义相对论、有机化学中的晶体结构、机器人学中的构型空间等）。这次会议将汇集国际专家在非积极弯曲空间的各个方面的工作。 会议网址是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.