Project no. 144014

Geometry and Topology of Manifolds and Integrable Dynamical Systems

Ministry of Science, Technologies and Development of Republic of Serbia

Project leader: |
Vladimir Dragovic |

Coodinating institution: |
Mathematical Institute SANU |

Period: |
1 January 2006 – 31 December 2010 |

Researchers:

Name |
Title |
Institution |

Vladimir Dragovic | full research professor | Mathematical Institute SANU |

Rade Zivaljevic | full research professor | Mathematical Institute SANU |

Bozidar Jovanovic | associate research professor | Mathematical Institute SANU |

Borislav Gajic | assistant research professor | Mathematical Institute SANU |

Milena Radnovic | assistant research professor | Mathematical Institute SANU |

Jelena Stanojevic | junior assistant | Faculty of Economy, Belgrade |

Vladimir Grujic | assistant professor | Faculty of Mathematics, Belgrade |

Maja Rabrenovic | assistant | Faculty of Mathematics, Belgrade |

Aleksandar Lipkovski | full professor | Faculty of Mathematics, Belgrade |

Vladimir Jankovic | associate professor | Faculty of Mathematics, Belgrade |

Goran Djordjevic | assistant professor | Faculty of Science and Mathematics, Nis |

Ljubisa Nesic | assistant professor | Faculty of Science and Mathematics, Nis |

Dragoljub Dimitrijevic | junior research assistant | Faculty of Science and Mathematics, Nis |

Jasmina Jeknic | junior research assistant | Faculty of Science and Mathematics, Nis |

Boban Marinkovic | assistant professor | Faculty of Mining and Geology, Belgrade |

Katarina Kukic | assistant | Faculty of Transport and Traffic Engineering, Belgrade |

Milica Stojanovic | assistant professor | Faculty of Management, Belgrade |

Jelena Jockovic | junior asssistant | Faculty of Pharmacy, Belgrade |

**Abstract: **The objective of the project is coordinated high level research involving application of complex multidisciplinary techniques from algebraic and differential geometry and topology to some of the currently leading research themes in the theory of integrable dynamical systems and geometric combinatorics. Expected results are grouped into several research themes including the following. Axiomatization of the Hess-Appel'rot systems. Isoholomorphic integrable systems, Lax representations and geometry of Prym varieties. Integrable magnetic flows on homogeneous spaces. Foundations of the theory of combinatorial groupoids (pioneered by M. Joswig (TU Berlin)) and the development of "parallel 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 complexes 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

**Subject description and importance of research: **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.

**Research Goal: **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