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Rustam Sadykov

  1. Invariants of singular sets of smooth maps.

    Втр, 04/06/2010 - 15:01 — SomNambul
    Authors: Rustam Sadykov
    Subjects: Geometric Topology
    Abstract

    A singular point of a smooth map F: M -> N of manifolds is a point in M at
    which the rank of the differential dF is less than the minimum of dimensions of
    M and N. The classical invariant of the set S of singular points of F of a
    given type is defined by taking the fundamental class [\bar{S}]\in H_*(M) of
    the closure of S. We introduce and study new invariants of singular sets for
    which the classical invariants may not be defined, i.e., for which \bar{S} may
    not possess the fundamental class.

  2. The bordism version of the h-principle.

    Втр, 02/09/2010 - 16:01 — SomNambul
    Authors: Rustam Sadykov
    Subjects: Geometric Topology
    Abstract

    In view of the Segal construction each category with operation gives rise to
    a cohomology theory. We show that similarly each open stable differential
    relation R determines cohomology theories k^* of solutions and h^* of stable
    formal solutions of R. We prove that k^* and h^* are equivalent under a mild
    condition.

  3. Singular cobordism categories.

    Пнд, 02/01/2010 - 16:01 — SomNambul
    Authors: Rustam Sadykov
    Subjects: Algebraic Topology
    Abstract

    Recently Galatius, Madsen, Tillmann and Weiss identified the homotopy type of
    the classifying space of the cobordism category of embedded d-dimensional
    manifolds [7] for each positive integer d. Their result lead to a new proof of
    the generalized standard Mumford conjecture. We extend the main theorem of [7]
    to the case of cobordism categories of embedded d-dimensional manifolds with
    prescribed singularities, and explain the relation of singular cobordism
    categories to the bordism version of the Gromov h-principle.

  4. Stable characteristic classes of smooth manifold bundles.

    Втр, 10/27/2009 - 16:01 — SomNambul
    Authors: Rustam Sadykov
    Subjects: Geometric Topology
    Abstract

    Characteristic classes of oriented vector bundles can be identified with
    cohomology classes of the disjoint union of classifying spaces BSO_n of special
    orthogonal groups SO_n with n=0,1,... A characteristic class is stable if it
    extends to a cohomology class of a homotopy colimit BSO of classifying spaces
    BSO_n.

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