The subject of the project as a whole is a study of deep interactions of Geometry and Topology with other fields (mathematical physics, discrete/computational mathematics etc.). The project is envisaged as one of the key projects in fundamental research in Serbia with strong international character and capacity for integration into international (European) networks of projects with similar orientation. The focus of research is on integrable dynamical systems and on geometric and topological combinatorics. However, the real emphasis is on deep interactions of these fields with other areas and on constructions of fundamental importance as illustrated by the following list of them(key words and phrases). Matrix Lax polynomials, rigid-body systems and geometry of Prym varieties, isoholomorphic systems, systems of Hess-Appel'rot type, isospectral and isomonodromic deformations, connections with Frobenius manifolds, theta functions and applications etc. Integrable systems on Lie algebras and homogeneous spaces, integrable geodesic flows. The Poncelet theorem, Cayley's condition and separability, geometry of Stackel spaces, Appel hypergeometric functions, hyperelliptic curves and reductions of theta functions etc. Nonholonomic mechanics: geometry, integrable models and Hamiltonization. Analytical and combinatorial properties of the Abel-Jacobi map. Affine subspace arrangements and their complements (Goresky-MacPherson and Ziegler-Zivaljevic formulas), compactifications of associated configuration spaces (after Kontsevich, Fullton-MacPherson DeConcini-Procesi, Feichther-Yuzvinsky etc.) General particle/configuration spaces on manifolds (after Segal, Milgram, Kallel, etc.), including the divisor spaces on surfaces (Zivaljevic at al.).Applicastions of combinatorial methods including the discrete Morse theory (after Forman, Kozlov, et al.), enumerative combinatorics (after Bjorner-Ekedahl, Beck-Zaslavski etc.). Order complexes of posets of geometric origin (Bjorner, Vassilev, Zivaljevic) etc. Complexes of geometric origin including the moment-angle complexes (after Buchstaber-Ray-Panov), arc complexes (after Panov-Sullivan), graph complexes (after Vassilev, Lovasz-Babson-Kozlov, ribbon graph complexes etc.). Combinatorics and geometry of Hurwitz spaces (after Dubrovin), Hurwitz problem and associated combinatorial problems (after Arnold, Ekedahl, Lando, Shapiro, Vainshtein etc.). Graph combinatorics and 2d quantum gravity (after Dubrovin, Manin, Di Francesco). Topological combinatorics, including applications of diagrams of spaces over posets, geometry and topology of vector bundles, e.g. applications of Koschorke's "exact singularity sequence for normal bordism groups" (Zivaljevic, arXiv:math.CO/0412483)etc. Development of combinatorial groupoids (after Joswig), cf. Zivaljevic math.CO/0506075 etc. Necessary and sufficient conditions of extremum (collaboration with MGU, V. Jankovic, V.M. Tihomirov) etc.
Axiomatization of the systems of Hess-Appel'rot type, their geometry and dymanics, study of isoholomorphic integrable systems, Lax representations and geometry of Prym varieties. Magnetic flows on homogeneous spaces, Lax representation and compatible Poisson structures. Foundation of the theory of combinatorial groupoids Foundations of the theory of combinatorial grupoids (pioneered by M. Joswig (TU Berlin)) and the development of "paralel transport" of Hom-complexes (graph complexes) over these groupoids with applications to the generalized Lovasz conjecture (see math.CO/0506075 for initial results in this direction). Further application of these methods should provide new insight about cubical compexes non-embeddable into cubical latices (a problem of S. P. Novikov which arose in connection with the 3-dimensional Insig model). Applications of (normal) bordisms and the combinatorics of Gray codes to the problem of equipartitions of measures in R^d by d hyperplanes (Grundbaum 1960, Hadwiger 1966), see arXiv:math.CO/0412483 for preliminary results for measures with additional symmetries. Geometry of Stiefel varieties, integrable geodesic flows and potential systems. Dynamics of integrable billiards and geometry of pencils of quadrics. Hamiltonization of the multidimensional Chaplygin rolling ball problem.