Principal Investigator: Jean-Francois R. Lafont, Michael W. Davis, Pedro Ontaneda
This is a proposal to provide funding for the conference Geometry, Topology and Dynamics in Negative Curvature to be held in Bangalore, India, during the week of August 2-7, 2010. The conference is an official satellite conference to the 2010 International Conference of Mathematicians (held in Hyderabad, India). The goal of this conference is to bring together mathematicians working on different aspects of negatively curved spaces. It will bring together three distinct communities of mathematicians: geometers, topologists, and dynamicists. It will provide a forum for sharing the most recent developments in the study of spaces of negative (and more generally, non-positive) curvature. A selection of topics that will be touched upon in the conference include: (1) Gromov hyperbolic groups, CAT(0)-groups, and geometric group theory, (2) spaces of metrics (Teichmuller space, moduli space, representation varieties), (3) links with 3-dimensional topology (Geometrization conjecture, mapping class groups), (4) links with high-dimensional topology (Novikov conjecture, Borel conjecture), (5) analysis on boundaries of groups and spaces, (6) Anosov flows and dynamics, (7) flows on homogeneous spaces and applications to number theory. We expect the conference to provide a "snapshot" of the current state of our knowledge in matters related to nonpositively curved spaces, as well as to provide a roadmap for future research in this interdisciplinary field.
Curvature is a fundamental notion in geometry. Zero curvature is the most physically familiar, and corresponds to an ordinary flat surface. Positive and negative curvature can be described relative to the familiar zero curvature setting as follows: one tries to "flatten" the surface near a point, and looks to see if there is too little or too much fabric to achieve the flattening. For instance, the surface of a cone or of a cylinder can be unfolded to lie flat, so these also have zero curvature. But if one tries to flatten the surface of a sphere (think of the skin of an orange), there is not enough fabric to do this (i.e. the skin of the orange splits), which corresponds to positive curvature. Alternatively, if one tries to flatten a saddle shaped surface, folds appear. There is too much fabric, which corresponds to negative curvature. A space of nonpositive curvature can now be described as one having the property that, near every point and for every pair of directions, it looks either flat or like a saddle shaped surface. Surprisingly, such spaces are much more prevalent than one would initially guess. Aside from the field of geometry, they naturally appear in numerous other fields of mathematics: topology, dynamics, number theory, representation theory, etc. Aspects of nonpositive curvature have also made an increasing appearance in more applied fields, wherever there has been an interest in understanding the "shape" of objects (special relativity theory, crystalline structures in organic chemistry, configuration spaces in robotics, etc). This conference will bring together international experts working on various aspects of nonpositively curved spaces. The conference web site is www.icts.res.in/program/gtdnc.
The International Congress of Mathematicians (ICM) is held every four years, and is the single biggest event in the mathematical community. In 2010, the ICM was held in Hyderabad (India). Surrounding the ICM, there are typically a large number of satellite conferences, focusing on more specialized topics. The present grant provided funding for the ICM satellite conference "Geometry, topology, and dynamics in negative curvature" (GTDNC), held in Bangalore (India) during the week of August 2nd-7th, 2010. The conference focused on spaces of non-positive curvature. These are spaces where, on the small scale, metric invariants (length, area, volume, etc) grow faster than in the corresponding Euclidean space. Such spaces are prevalent in nature, and moreover, in a suitable sense, most spaces will have large regions where non-positive curvature is present. When studying such spaces, one can adopt several distinct viewpoints. The geometric viewpoint is concerned with quantitative information about these spaces: for example, how do the length, area, volume (and other more complex metric invariants) behave? The topological viewpoint is concerned with qualitative information: for instance, is the space "large" or "small", does it have holes in it, etc. Finally, the dynamical viewpoint is concerned with understanding motion inside these spaces. In recent years, the dynamical viewpoint has had significant applications, particularly to resolving several famous problems from number theory. These three viewpoints often interact, and many of the deepest results exploit methods inspired from all three viewpoints. The conference in Bangalore brought together many of the leading experts in the field. The talks provided insight into the most recent developments in the area. The US was particularly well represented, with close to half of the invited speakers being based in US institutions.
