Geometry and Analysis of Complex Shapes and Applications to Mathematical Physics

The main idea of the project and the main results obtained during the the year of its execution are that the analysis of complex shapes may be described by means of tools used in mathematical physics, conformal field theory. It is rather natural that a smooth contour moving in the space of shapes possesses typical conformal symmetries. We considered evolution described by classical and stochastic Loewner equations as well we considered geodesic motion in the space of complex shapes. As a result we elaborated an infinite-dimensional analog of sub-Riemannian geometry. The participants of the project participated in several international conferences, published papers in leading journals and 1 monograph. 1 Ph.D. student successfully defended his thesis. 2 postdocs were trained this year.

An important feature of the development in natural sciences during last decades is the increasing degree of cross-fertilization between mathematics and physics with great benefits to both subjects. Complex analysis and geometry recently became most releva nt parts of mathematics for the study of classical and quantum field theories. An elegant description of 2D conformally invariant statistical physical systems in terms of stochastic (Schramm) Loewner contour evolution implied revolutionary progress in thi s direction at the beginning of XXI century. Deterministic and random complex shapes will be encoded using circle diffeomorphisms and the Virasoro-Bott group ('fingerprints'). The evolution in the space of shapes will be studied. Two sample contour evolut ion models are the Laplacian growth, Loewner and Loewner-Kufarev evolution and their generalizations. The ground idea of the project is that these evolutions are projections of a general evolution in the universal Sato's Grassmannian to corresponding phas e spaces. This will lead to relations to classical KdV type integrable hierarchies, to investigation of infinite dimensional constrained systems and to analysis on Lie-Fréchet groups, which proved to have an important role in classical/quantum mechanics a nd classical/quantum field theories. Geometric control will be developed on infinite dimensional groups. These relations will have far-reaching effect on the progress both in integrable systems and in classical complex analysis, geometry and control. Appl ications to the field of vision and pattern recognition will follow. More generally, topics of study are motivated by nonlinear physics and will include areas such as integrable systems, optimal control, Virasoro and Krichever-Novikov algebras, Loewner eq uations, and sub-Riemannian/Lorentzian geometric analysis. In other words, we shall work on a theory of constrained motion in difficult, mostly infinite dimensional, spaces.