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FRINATEK-Fri prosj.st. mat.,naturv.,tek

Geometry and Analysis of Complex Shapes and Applications to Mathematical Physics

Tildelt: kr 3,4 mill.

Hovedideen i prosjektet og de ??viktigste resultatene som er oppnådd i løpet av året av kjøring er at analyse av komplekse former kan beskrives ved hjelp av verktøy som brukes i matematisk fysikk, konform feltteori. Det er ganske naturlig at en jevn kontur i bevegelse i et rom av figurene besitter typiske konforme symmetrier. Vi vurderte utviklingen beskrevet av klassisk og stokastiske Loewner ligninger samt vi vurderte geodetisk bevegelse i romet av komplekse former. Som et resultat vi utarbeidet en uendelig-dimensjonale analog av sub-Riemannsk geometri. Deltakerne i prosjektet har deltatt i flere internasjonale konferanser, publisert artikler i ledende tidsskrifter og en monografi. En Ph.D. student disputerte med suksess. 2 postdoktorer ble trent i år.

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.

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FRINATEK-Fri prosj.st. mat.,naturv.,tek