School of Mathematics
Loughborough University

LE11 3TU

Alexey Bolsinov - Research Interests

  • Symplectic geometry and Poisson manifolds: symplectic and topological invariants of Lagrangian foliations, compatible Poisson structures.
  • Topology of dynamical systems and integrability: topological entropy, obstructions to integrability, integrable tops, bi-Hamiltonian systems.
  • Singularities: singularities of the momentum mapping, their invariants and algorithmic classification.
  • Riemannian geometry and Lie groups: projective equivalence, geodesic flows on Lie groups and homogeneous spaces, symmetries and reduction, Lie pencils.

  • Main results and research directions

  • In [BT1,BT2] we describe a new phenomenon (the so-called "integrable chaos") in dynamical systems. Namely, we have constructed a series of examples of integrable geodesic flows with positive topological entropy. These systems, in particular, are counterexamples to the Paternain conjecture on non-integrability of geodesic flows on compact manifolds with fundamental groups of exponential growth. In [BDV] we have studied the spectral problem for these examples. In particular, we have described the spectrum ot the Laplace operator, its eigenfunctions and the quantum monodromy phenomenon.

    [BT1] Bolsinov A. V., Taimanov I. A. Integrable geodesic flows with positive topological entropy, // Invent. Math., 140(2000), pp. 639-650.

    [BT2] Bolsinov A. V., Taimanov I. A. Integrable geodesic flows on suspensions of automorphisms of tori // Proc. of the Steklov Institute of Mathematics, 231(2000), pp.42-58.

    [BDV] Bolsinov A., Dullin H., Veselov A. Spectra of SOL-manifolds: arithmetic and quantum monodromy // Comm. Math. Phys., 2006, V.264, pp.583-611.

  • The following question is classical in modern Riemannian geometry: what are smooth compact manifolds that admit integrable geodesic flows? In [BJ1, BJ2] we generalize a series of results obtained by Mishchenko, Thimm, Paternain, Spatzier and Bazaikin by proving that all homogeneous spaces G/H as well as bi-quotients K\G/H of a compact Lie group G admit integrable geodesic flows.

    [BJ1] Bolsinov A., Jovanovic B. Noncommutative integrability, moment map and geodesic flows// Annals of Glob. Anal. and Geom., 2003, Vol. 23, pp. 305-322.

    [BJ2] Bolsinov A., Jovanovic B. Complete involutive algebras of functions on cotangent bundles of homogeneous spaces// Math. Zeit., 2004, Vol. 246, pp. 213-236.

  • Two Riemannian metrics g and g' on a manifold M are said to be projectively equivalent if they have the same geodesics (considered as unparameterized curves). In [BM1] we have found a new condition for two metrics to be projectively equivalent and show that the class of projectively equivalent geodesic flows coincide with the so-called Benenti systems. In particular, such geodesic flows are always bi-hamiltonian.

    [BM1] Bolsinov A.V., Matveev V.S. Geometrical interpretation of Benenti systems // Jour. Geometry and Physics, 2003, Vol. 44, pp. 489-506.

  • In [B1] we study non-degenerate singularities of Lagrangian foliations on four-dimensional symplectic manifolds. The main result of this paper is the topological classification of saddle-saddle singularities of complexity two (i.e., with 2 critical points on the singular leaf). We have shown that there are exactly 39 different topological types of such singularities. This classification turns out to be quite useful for the topological analysis of integrable Hamiltonian systems with two degrees of freedom. In particular, in [BRF] we use the results and ideas of [B1] to obtain the complete topological description of the Kovalevskaya top in rigid body dynamics.

    [B1] Bolsinov A.V. Methods of calculation of Fomenko-Zieschang topological invariant// In book : Advances in Sov. Math., 1991, Vol. 6, AMS, Providence, pp. 147-183.

    [BRF] Bolsinov A.V., Richter P.H., Fomenko A.T. Loop molecule method and the topology of the Kovalevskaya top// Sbornik: Mathematics, 191(2000) No. 2, pp. 151-188.

  • In [BF1], [B2] we have constructed an invariant (more precisely, a family of invariants) that allows one to classify integrable Hamiltonian systems with two degrees of freedom up to orbital equivalence. By applying this technics, in [BF2] we have found a new non-trivial orbital isomorphism between two classical systems: Euler top in rigid body dynamics and the geodesic flow on the ellipsoid (Jacobi problem). Then, using smooth orbital invariants, in [BD] we have shown that this isomorphism is only continious: in the smooth sense the two systems are not equivalent.

    [BF1] Bolsinov A.V., Fomenko A.T. Orbital equivalence of integrable systems with two degrees of freedom. The classification theorem. Part I, II// Russian Acad. Sci. Sb. Math. 81(1995), No. 2, pp. 421-465 and 82(1995), No. 1, pp. 21-63.

    [B2] Bolsinov A.V. A smooth trajectory classification of integrable Hamiltonian systems with two degrees of freedom // Sbornik: Mathematics 186(1995) No. 1, pp. 1-27.

    [BF2] Bolsinov A.V., Fomenko A.T. Orbital classification of geodesic flows on two-dimensional ellipsoids. The Jacobi problem is orbitally equivalent to the integrable Euler case in rigid body dynamics // Functional Analysis and its Appl., 29(1995) No. 3, pp. 149-160.

    [BD] Bolsinov A.V., Dullin H. On the Euler case in rigid body dynamics and the Jacobi problem // Regular and Chaotic Dynamics, 2(1997) No. 2, pp. 13-25 (in Russian).

  • It is well-known that a pair of compatible Poisson brackets on a manifold M always generates a commutative subalgebra A in the Poisson algebra of all smooth functions on M. A natural problem appearing very often in applications is to find out whether A is complete (i.e., guarantees the complete Liouville integrability). In [B3, B4, B5] I have obtained a completeness criterion which allows one to answer efficiently this question. A particular case of this theorem gives necessary and sufficient conditions for completeness of some well-known commutative subalgebras in the Poisson-Lie algebra P(g) of a finite-dimensional Lie algebra g (see [B3, B4, B5]) . These results are closely related to the Mischenko-Fomenko conjecture (1978) saying that P(g) always admits a complete commutative subalgebra. In [B6] we give a constructive proof of this conjecture following the method suggested recently by S. Sadetov.

    [B3] Bolsinov A.V. A completeness criterion for a family of function in involution constructed by the argument shift method// Sov. Math. Dokl., 38(1989) No. 1, pp. 161-165.

    [B4] Bolsinov A.V. Compatible Poisson brackets on Lie algebras and completeness of families of functions in involution// Math. USSR Izvestiya, 38(1992) No. 1, pp. 69-89.

    [B5] Bolsinov A.V. Commutative families of functions related to consistent Poisson brackets// Acta Appl. Math., 24(1991), pp. 253-274.

    [B6] Bolsinov A. Complete commutative families of polynomials in Poisson-Lie algebras: A proof of the Mischenko-Fomenko conjecture // In book: Tensor and Vector Analysis, Vol. 26, Moscow State University, 2005, pp. 87-109. (in Russian).

  • In [BM2] we study symplectic invariants of singular Lagrangian foliations on simplectic 4-manifolds and then apply them to the classical rigidity problem in Riemannian geometry. Namely, we construct new nontrivial examples of Riemannian metrics with conjugate geodesic flows on 2-surfaces of arbitrary genus. In [BV] we study semilocal symplectic invariants of non-degenerate singularities of Lagrangian foliations of arbitrary dimension. In particular, we obtain a symplectic classification of non-degenerate singularities of complexity one.

    [BM2] Bolsinov A., Matveev V. Symplectic invariants of Liouville foliations and conjugacy of geodesic flows (in preparation).

    [BV] Bolsinov A., Vu Ngoc San, Symplectic equivalence for integrable systems with common action integrals (in prepration).

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