Project 1643

Geometry and Topology of Manifolds and Integrable Dynamical Systems

Leader: dr Vladimir Dragovic


Subject of research

Integrable hierarchies (IH) - Frobenius manifolds; Hamiltonian, Poisson, symplectic structures; reductions of IH in geometry and mechanics; mechanisms of integration (with quantization); analytical and combinatorial properties of Abel-Jacobi map; Floer homology for Lagrangian submanifolds (after Koncevich mirror partner of some Frobenius manifolds); Hamiltonian diffeomorphisms; important metrics; Geometric resolution of singular spaces (after V. Vassilev), homotopy colimits and topological order complexes; discriminants and configuration spaces (divisor spaces); group actions and equivariant maps.

Description of the work

Geometric resolutions of stratified spaces (V. Vassilev), homotopy colimits of diagrams, order complexes of (topological) complexes etc. are treated as objects of the same nature and they are studied from the same point of view. R. Živaljević and G. Ziegler (TU-Berlin) have developed, in foundational papers (Math. Ann. 1993, (with Welker) J. Reine Angew. Math. 1999), so called Ziegler-Živaljević methodology for combinatorial analysis of singular spaces. This approach will be applied on confuguration spaces, especially on "particle and divisor spaces". The problem of existence of equivariant maps is studied in relation to some immediate applications, say in geometric combinatorics (S. Vrećica, R. Živaljević, Discrete Comput. Geom. 2001). The research is focused on the evaluation of relevant opstructions in the group of normal (equivariant) bordisms. New integrable rigid-body system and L-A pair for the Hess-Apelrot case constructed by Dragović and Gajić (Proc. Roy. Soc. Edinburgh (2001)) motivate further development of methods of finite-zone integration (Dubrovin, Krichever, Novikov, Springer Encyclopaedie of Mathematical Science 4 (1990)). Using Krichever vacuum vector techincs (Krichever Funct. An. Appl. (1981), Dragović: Russ. Acad. Sci. Math. (1993)) representations of Sklyanin algebra will be studied. Structures close to Frobenius and isomonodromy on Lie algebras; exact bihamiltonian; connection between separability and hypergeometry (from Dragović J. Physics A (1996)). Milinković ideas (Proc. AMS (2001), Trans. AMS (1999)) on Floer homology and Hofer geometry, will be applied on metric properties of group of Hamiltonian diffeomorphisms, Lagrangian intersections and general symplectic manifolds. Based on Jovanović results (joint with Bolsinov and Mischenko (MGU): Russ. Acad. Sci. Sbornik (2001)) relationship between commutative and noncommutative integrability will be studied (Mischenko-Fomenko Conjecture); construction on manifolds with integrable geodesic flows.

Originality of research

Applications of homotopy colimits (Ziegler-Živaljević methodology) for desingularization of spaces. The technique for evaluation of obstructions for the for the existence of equivariant maps.

Research Goal

Isospectrality of skew symmetric matrices (Euler-Poison equations) and isomonodromy; construction of integrable geodesics, subRiemannian and nonholonommic flows; the relation between separability, bihamiltonian and alg.-geom, integrability; the hypothesis of Mischenko-Fomenko about non-commutative integration; construction of representations of Sklyanin algebras; hypothesis about nonempty intersection of the exact Lagrange embedding and a conormal bundle of a closed submanifold etc.

State of the Art in Scientific Field

World-wide Situation

Integrable dinamical systems are one of the strongest unifying disciplines in contemprorary mathematics. One of the highlights is Dubrovin's invention of Frobenius manifolds, motivated by Witten and Koncevich solution of the old problem of topology of moduli spaces of algebraic curves in terms od KdV hierarchies, which now connects classification of integrable hierarchies, twodimensional topological field theories, Gromov-Witten invariants, mirror simetries, singularity theory, reflection groups, isomonodromy deformations, Floer homology (FH), generalisation of Witten's constructions of Morse homology, are rich and more complicated in the case of Lagrangian submanifolds then in the case of Hamiltonian diffeomorphisms. Results of Fukaya, Oh, Ohta, Ono connect FH with mirror symetry program of Koncevich. FH is connected also with Hofer's geometry on the group of Hamiltonian diffeomorphisms (McDuff, Lalonde, Eliashberg, Chekanov). V. Vassilev is the founder of the topological theory of complements of discriminant spaces. His "geometric resolution method" has produced breakthroughs like Vassilev's knot invariants, evaluation of the complexity of algorithms (S. Smale problems), singularity theory (problems of V. Arnold). Related is the study of the complements of arrangements (Orlik, Terao, Goresky, MacPherson, Stanley, Deligne, Ziegler).

Domestic Situation

Deep interaction among mathematical disciplines is fundamental for the contemporary mathematics ("Mathematics: Frontiers and Perspectives'', AMS 2000). Several Belgrade groups have accepted this multidisciplinary approach, e.g. in mid eighties the so called G-T-A seminar (G-T-A stands for Geometry, Topology and Algebra) was founded. The conference "Geometric Combinatorics'', satellite to ICM '98, gathered together leading world mathematicians in Yugoslavia and fully justified this orientation. The collaboration of G-T-A seminar with other seminars (Diff. Geometry, Analysis, M-M-M (Mathematical methods in Mechanics)) eventually led to a more organized research in the areas of topology, differential geometry, integrable systems and field theory and recently mathematical physics. Among the consultants and active collaborators of the participants in the project are well known experts in this areas, Oh, Salamon, Dubrovin, Jimbo, Fomenko, Kozlov, Mischenko, Vasilliev, Bjoerner.

Planned Project Results

The project "Geometry and topology of manifolds and integrable dynamical systems" has a goal to be a leading project in Serbia for interaction of the central mathematical disciplines, Geometry and Topology on one side, and Theoretical Mechanics and Mathematical Physics, on the other. The project will provide the expertize in these fields and encourage younger associates to work on multidisciplinary themes. A particular attention will be given to spreading of information and acquiring and developing of the technique used in existing applications both in natural sciences (e.g. quantum physics, simplectic methods in physics and mechanics) and more applied areas. For example, integrable dynamical systems have found important applications, say in optics and fluid dynamics (KdV equation etc.). A main guiding principle, accepted by the project, illustrating the relationship with the applications of mathematics is the following apstract of an invited lecture of V. I. Arnold on the "Third International Congress on Industrial and Applied Mathematics": "The difference between the pure and applied mathematics is social rather than scientific. Examples from matrix theory, optics, Hamiltonian dynamics, quantum physics and simplectic geometry will be presented, illustrating the danger of the divorce between the pure and applied mathematics communities for both groups."