Mathematical Sciences: Symplectic and Contact Geometry in the Interaction with Topology and Complex Analysis

点击翻译按钮获取中文摘要

This project envisions the continuing development of symplectic geometry and topology in its interaction with other fields of mathematics. The main progress under prior support was achieved in 3-dimensional contact geometry, construction of new invariants of high-dimensional closed and open contact manifolds, in the description of invariant properties of boundaries of symplectic manifolds, and in the application of symplectic methods to the theory of 2-knots in real 4-space. In the present continuation, several new areas of research are suggested. In particular, there is the study of newly discovered connections between symplectic geometry, pseudoisotopy theory and algebraic K-theory, applications of Lagrangian intersection theory to Morse theory for plurisubharmonic functions on complex manifolds, and the development and application of contact homology theory. Other projected directions of research are: 3-dimensional contact topology and further development of symplectic methods in 4-dimensional topology. It seems that in modern mathematics interdisciplinary research plays a very special and important role. Many recent discoveries were made in areas which do not fit into any traditional classification. For instance, a breakthrough in 4-dimensional topology by Donaldson came from gauge theory, a topic in quantum physics, and many other recent topological discoveries were motivated by physics. Symplectic topology at present is a beautiful blend of different mathematical sciences: topology, Hamiltonian dynamics, complex analysis, differential geometry, etc. It attracts more and more ideas from different fields and repays them with unexpected new applications, which makes research in this direction especially promising.