The first part of the project is about the geometric and analytic properties of the Ricci flow equation and their applications to the study of geometry. Namely to study the classifications of self-similar solutions, e.g gradient solitons and ancient solutions, as well as the convexity type estimates in general. Such results have far-reaching consequences in the singularity analysis, and applications of Ricci flow in the study of geometric-topological structure of the manifolds. The second theme of the project is on the sharp gradient estimates of Li-Yau-Hamilton type, related monotonicity formulae and applications in geometric nonlinear PDEs. Their relations to physics, statistical mechanics will be studied too. The aim is to discover a fundamental physical/geometric principle to unify various sharp estimates and monotonicity formulae. It will also provide the guideline for further discovery of the new monotonicity formulae in other geometric PDEs.

Since all physical event takes place in a space, the subject of differential geometry which studies the geometric properties of the space has important consequence in every physical event. This project mainly involves the study of partial differential equations of parabolic type which arise from differential geometry, and their applications to the understanding of various geometric/topological properties of manifolds. This area lies in the center of the current mathematics. It naturally connects various area of mathematics, such as topology, Riemannian geometry, partial differential geometry, Lie groups, as well as mathematical physics. The techniques developed can be useful in understanding problems in economics, material sciences and bio-sciences.

Project Report

This projects involves the development of new method/techinques of studying the evolution equations on Riemannian and complex manifolds, and the applications of such study to understand the geometric and topological structure of the underlying spaces. Several new method/development have been accompolished. In terms of the broader impact, the asymptotical method developed in ths project is intimated related to the study of theoretic physics. It is also originated from the study of dynamic system. The method can be of significance to these subjects. The entropy method employed in the study is relating the study of the nonlinear PDEs to the subject of convex geometry, bring the ideas from the statistical mechanics. The result wil definitely feed back to these subjects. During the duration of the projects, several new methods have been developed resulting in several resarch articles published on highly reuptable mathematical journals. These discoveries advance the theory of nonlinear PDEs on manifolds and the geometry of complex manifolds.The PI is also in the middle of process of finishing a research mongraph project which overlaps with the project. The projects also involves the training of graduate students in recent Ph.Ds in mathematics. The PI wrote several articles to disseminate the results to the general public. The PI contributes to the society by helping local middle/high schools in math counts (as a coach) and other math competitions, giving lectures to high school students in the local math circles. The PI contributes to the mathematical community by organizing conferences aiming for the traning of graduate students and recent Ph.Ds as well as the group of under-represented groups, and teaching graduate courses in geometry and analysis.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
Standard Grant (Standard)
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Joanna Kania-Bartoszynska
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University of California San Diego
La Jolla
United States
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