**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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