topology@ub

Research Group in Algebraic Topology

Summer seminar 2010


    Introduction to Conformal Field Theories

    By Leandro Lombardi (Universidad de Buenos Aires)

    In this series of talks we will try to build a bridge between theoretical physics and mathematics.

    Talk 1 (June 4, 16.00, aula T1)
    A glimpse to physicists' viewpoint of Field Theory as a motivation of the axioms of quantum field theories introduced by Atiyah and Segal.

    Talk 2 (June 7, 16.00, aula T1)
    Mathematical aspects of two-dimensional conformal field theory with an emphasis on Segal's approach to the subject.

    Talk 3 (June 11, 16.00, aula T1)
    Vertex operator algebras and their relation with two-dimensional conformal field theory.

    An effort will be made to keep talks independent from each other.


    Motivic Homotopy By Paul Arne Østvaer (University of Oslo)

    Talk 1 (June 8, 12.00, aula T1)
    K-THEORETIC BACKGROUND
    We survey classical and recent results in K-theory, broadly interpreted. Keywords are: the Grothendieck construction; vector bundles; classifying spaces; quadratic forms; Bökstedt squares.

    Talk 2 (June 9, 12.00, aula B7)
    MOTIVIC HOMOTOPY I
    Motivic or A1-homotopy theory is designed to study invariants of algebraic varieties such as motivic cohomology, K-theory and Witt theory. We focus on the Milnor conjecture, relating Galois cohomology and Milnor K-theory, and the so-called slice filtration introduced by means of motivic stable homotopy theory. Other keywords include: motives; a strict model for K-theory; the Milnor conjecture on quadratic forms.

    Talk 3 (June 10, 12.00, aula B7)
    MOTIVIC HOMOTOPY II
    In this talk we discuss recent computations of motivic stable stems, algebraic cobordism and algebraic K-theory over various base fields of characteristic zero. Parts of this are joint work with Kyle Ormsby.


    Rational Homotopy of Mapping Spaces By Urtzi Buijs (Universitat de Barcelona)

    Talk 1 (June 14, 16.00, aula T1)
    RATIONAL HOMOTOPY THEORY
    1.1. Some history
    1.2. Algebraic preliminaries
    1.3. Homotopical algebra
    1.4. Bridge between Topology and Algebra
    1.5. Short dictionary of topological invariants

    Talk 2 (June 15, 12.00, aula T1)
    ALGEBRAIC MODELS OF MAPPING SPACES
    2.1. The Haefliger-Brown-Szczarba model
    2.2. The homotopy Lie algebra of mapping spaces
    2.3. Lie models
    2.4. L-infinity models

    Talk 3 (June 17, 12.00, aula T1)
    APPLICATIONS, EXAMPLES AND PROBLEMS
    3.1. L-S-category of mapping spaces
    3.2. H-space structures on mapping spaces
    3.3. Upper bounds for the Whitehead length of mapping spaces

    Yagita Invariants of Symplectic Groups By Cornelia Busch (ETH Zürich)

    Talk 1 (June 16, 16.00, aula T1)
    THE YAGITA INVARIANT OF SOME SYMPLECTIC GROUPS
    If a group G of finite virtual cohomological dimension has p-periodic Farrell cohomology, then the Yagita invariant of this group equals its p-period. We compute Yagita invariants of symplectic groups that do not have p-periodic Farrell cohomology.

    Talk 2 (June 18, 12.00, aula T1)
    INVARIANTS IN THE COHOMOLOGY OF GROUPS
    As a complement to the previous talk, we will give some examples for the computation of the p-period in Farrell cohomology and of the Yagita invariant.

    Motivic Homotopy By Markus Spitzweck (University of Oslo)

    Tuesday, July 6, 16.00, aula T2
    MOTIVIC TWISTED K-THEORY
    We will discuss a motivic version of the twisted K-theory spectrum. In topology, a twist is given by an integral third cohomology class, or equivalently by a map to K(Z, 3) = BBS1. In the motivic context, a twist will be given by a map from a motivic space to BBGm. Analogously to Khorami's result, we will give a Künneth formula for motivic twisted K-theory. We will construct a Chern character to motivic twisted cohomology and finally a spectral sequence converging from motivic cohomology to motivic twisted K-theory. As an example, we will discuss twists of the motivic (3, 1)-sphere. This is joint work with Paul Arne Østvaer.




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