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My Research Interests

Zhongmin Shen

My primary research interests are in several areas in differential geometry: Metric geometry (Riemannian or Finslerian), Spray Geometry and Information Geometry. In Riemann geometry I study the relationship between the geometry and topology of spaces with Riemann metrics, including all submanifolds in the Euclidean space. In Finsler geometry, I study more general regular metric spaces using calculus. Metrics arising from some psychological models, geological models and physical models are non-Riemannian metrics. Spray geometry studies paths in a space. A spray structure (also called a path structure) on a space is a class of parameterized curves subject to a system of second order ordinary differential equations. A typical spray structure is the class of geodesics of a Riemannian metric or a Finsler metric. Information Geometry studies two (possibly independent) metric and spray structures arising from various models, such as statistical models (exponential family of distributions) with the f-divergence.

Riemannian geometry is used by Albert Einstein in his general relativity theory. One of the fundamental problems in Riemann geometry is to study the topology of manifolds with certain restrictions on their curvature. Since my graduate study under D. Gromoll (1986-1990), I have made great efforts to understand the topological properties of manifolds admitting complete Riemann metrics of positive (or nonnegative) kth Ricci curvature. I establish a vanishing theorem for homotopy groups of manifolds with positive kth Ricci curvature and a large injectivity radius. I also obtain some finiteness and vanishing theorems on the topological type of non-compact n-manifolds with positive or nonnegative kth-Ricci curvature. With Christina Sormani (CUNY), we successfully obtain a sharp result on the homological type of complete Riemannian manifolds of nonnegative Ricci curvature. With John Lott (Michigan University), we determine the topological structure of the ends of open Riemannian manifolds with certain asymptotic geometric properties. With Jyh-Yang Wu, we introduce a new notion of bounded Betti numbers and show how the bounded Betti numbers are controlled by the lower Ricci curvature bound and the size of the manifold.

Encouraged, advised and supported by S.S. Chern, I became interested in Finsler geometry since 1995. Finsler geometry is an old area in differential geometry, and developed very slowly in the last 90 years, due to the complexity of Finsler structures. Roughly speaking, Finsler geometry is to study metric spaces without quadratic restriction on its metric function. In the past 10 years, Finsler geometry has been greatly developed, due to work by many geometers in Canada, China, Brazil, France, Germany, Hungary, Iran, Japan, Romania, Russia and etc. I have made great efforts to introduce new geometric quantities which describe Finslerian phenomena. I discovered some new non-Riemannian Finsler metrics of constant flag curvature. I classified locally projectively flat Finsler metrics of constant curvature.  Many interesting Finsler metrics of constant curvature have been constructed through the study, which can serve as models in Finsler geometry. Randers metrics are the most simplest non-Riemannian Finsler metrics. They are computable. Many Finsler metrics arising from applications are of Randers type. The solution to Zermelo's navigation problem is a geodesic of a Randers metric.  Bao, Robles and I classified all Finsler metrics of Randers type with constant curvature.

In order to encourage more young mathematicians to devote their research efforts to Finsler geometry, D. Bao, S.S. Chern and myself have written a graduate textbook on Finsler geometry, which has be published by Springer-Verlag in 2000. I have also written two monographs (Differential Geometry of Spray and Finsler Spaces, Lectures on Finsler Geometry, 2001) for geometers who are working in the area of Finsler geometry. Since some significant progress has been made since the publication of the above books, S.S. Chern and I published another book which includes the most recent results up to 2005.  Currently, Xinyue Cheng and I are writing a book on the geometry of Randers spaces.

The study by math-physicists in Russia and other countries shows that it seems inevitable to adopt Finsler geometry as a tool in General Relativity Theory. An annual international conference on  General Relativity Theory using Finsler geometry has been held  since 2005. I personally believe that the golden era of Finsler geometry has arrived, although  so far there are still very few US-based mathematicians working in Finsler geometry. I am very optimistic about the development of Finsler geometry.

My current interest is also in Information Geometry. Information Geometry provides a new method applicable to various areas including information sciences and physical sciences. It has emerged from investigating the geometrical structures of a family of probability distributions, and has applied successfully to statistical inference problems. It has been proved that information geometry opens a new paradigm useful for elucidation of information systems, intelligent systems, imaging process, control systems, physical systems, mathematical systems, and so on. Remarkable progress has been made recently by S.I. Amari in the neurocomputing and related areas. My primary goal is to make some progress in the mathematical foundation of Information Geometry.

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This page was updated on 06/19/2008