International Conference on Riemann-Finsler Geometry

February 15-17, 2008

Organizing Committee:  Zhongmin Shen (Chair),  David Bao (Co-Chair) and Pit-Mann Wong (Co-Chair)

Contact Person: Z. Shen (zshen@math.iupui.edu)

The conference will be held at Indiana University-Purdue University Indianapolis from February 15-16, 2008. The conference will focus on recent developments in the study on Finsler spaces with emphasis on non-Riemannian geometry and applications to problems in other areas of mathematics and outside  of mathematics.

In 1854, B. Riemann introduced the notion of curvature for spaces with a family of inner products (Riemannian metrics). In his Paris address in 1900, D. Hilbert formulated 23 problems, the 4th and 23rd problems being in Finsler geometry's category. There was no significant progress in the general case until 1918, when P. Finsler studied the variation problem for spaces with a family of norms (Finsler metrics). Meanwhile, A. Einstein used Riemannian geometry to present his general relativity. By that time, however, the geometry of Finsler spaces was still at its infant stage. Until 1926, L. Berwald extended Riemann's notion of curvature to Finsler spaces and discovered a new non-Riemannian quantity using his connection..

In the past decade, many mathematicians around the world have made significant progress in solving the fundamental problems in Riemann-Finsler geometry. Several monographs and textbooks have been published. It has been shown that modern differential geometry provides the concepts and tools to effect a treatment of Riemann-Finsler geometry in a direct and elegant way. This not only gives a better understanding of the geometry but opens a vista comparable to the developments of algebraic geometry from quadrics to general algebraic varieties.

The driving force of Riemann-Finsler geometry is its applications in other branches of mathematics and science. We list few of them below.

• In navigation problem, if an object moves on a (Riemannian) space with external force acting on it, then shortest time paths are the geodesics of certain type of Finsler metrics.
• In Analysis, Louis Nirenberg and Yanyan Li have shown that the solution of Hamilton-Jacobi equations on a domain in the Euclidean space can be described as the Finsler  distance function to the boundary.
• In image processing, geodesics active contours have proven to be a very useful tool for a number of segmentation tasks. If we add directionality to the active contours which allows for the segmentation of image data in oriented domains, then we must consider Finsler metrics. Geodesic active contours in Finsler framework provide a mechanism for the minimization of energy functionals defined on oriented domains.
• A. Einstein was forced to get out of the classical Euclid geometry while creating the relativity theory. He replaced it with the Riemannian one. It is natural to assume that the future development of physics also will demand some new geometry. Most recently, a class of Finsler metrics, mth root metrics, are strongly taken into consideration by a group of physicists as a subject for a possible model of space-time.

Our goal of this conference to bring some experts in north America together with experts from other countries to report their current research results. Besides, we hope that this conference is to nurture the development of a group of researchers in North America with interests and training in this field. We feel confident that this conference will attract new generations of mathematicians to this area.

Previous Conferences in North America:

·        Joint Summer Research Conference on Finsler Geometry, Seattle, July 1995.

·        Special session on Finsler geometry in the American Mathematical Society meetings, San Diego, January 1997.

·        Finsler Laplacian Conference, University of Alberta, August 20-22, 1997.

·        International Conference on Finsler Geometry, University of Alberta, August 13-20, 1998.

·        Workshop on Finsler Geometry, MSRI, June 3-7, 2002.

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