Thur., May 31 2:00 pm J. Stewart Library **Note date&time change	Boris Chorny (Australian National Univ., AU) Brown representability for space-valued functors In this talk we will discuss two theorems which resemble the classical cohomological and homological Brown representability theorems. The main difference is that our results classify small contravariant functors from spaces to spaces up to weak equivalence of functors. In more detail, we will show that every small contravariant functor from spaces to spaces converting coproducts to products, up to homotopy, and taking homotopy pushouts to homotopy pullbacks is naturally weekly equivalent to a representable functor. This theorem may be considered as a contravariant analog of Goodwillie's classification of linear functors, see his [Calculus II, III] papers. The interpretation of the current result in terms of Homotopy Calculus is still a challenge. Homological representability theorem states: every contravariant continuous functor from the category of finite simplicial sets to simplicial sets is equivalent to a restriction of a representable functor. This theorem is essentially equivalent to Goodwillie's classification of linear functors.
Wed., May 2 1:30 pm J. Stewart Library	Georg Biedermann L-stable functors We generalize and greatly simplify the approach of Lydakis and Dundas-R\"ondigs-{\O}stv{\ae}r to construct an L-stable model structure for small functors from a closed symmetric monoidal model category V to a V-model category M, where L is a small cofibrant object of V. For the special case V=M=S_* pointed simplicial sets and L=S^1 this is the classical case of linear functors and has been described as the first stage of the Goodwillie tower of a homotopy functor. We show, that our various model structures are compatible with a closed symmetric monoidal product on small functors. We compare them with other L-stabilizations described by Hovey, Jardine and others. This gives a particularly easy construction of the classical and the motivic stable homotopy category with the correct smash product. We establish the monoid axiom under certain conditions.
Wed., Apr. 25 1:30 pm J. Stewart Library	TBA
Wed., Apr. 18 1:30 pm J. Stewart Library	Mircea Voineagu On the Lawson Homology of projective hypersurfaces
Wed., Apr. 4 1:30 pm J. Stewart Library	Markus Severitt Motivic homotopy types of projective curves The talk deals with the classification of smooth projective curves, abelian varieties and Severi-Brauer varieties over a field k up to isomorphism in the motivic homotopy category H(k). For this we need the motivic homotopy invariance of the genus of a curve and the motivic homotopical behaviour of A^1-rigid schemes. Furthermore Nikita A. Karpenko classified Severi-Brauer varieties up to motivic equivalence as Chow motives. To use this, we have to make a connection between motivic homotopy theory and Chow motives.
Due to the intensity of the program in March, no talks are scheduled for this month
Wed., Feb. 28 1:30 pm J. Stewart Library	Jeremiah Heller (Univ. of Western Ontario, CA) Semi-topological cohomologies for varieties Motivated by Friedlander-Walker's construction of semi-topological K-theory and Friedlander-Lawson's morphic cohomology, we construct and study an (oriented) cohomology theory defined on smooth complex varieties, called "semi-topological cobordism". The construction combines the semi-topological construction of Friedlander-Walker and motivic stable homotopy theory. One of its features is that with finite coefficients semi-topological cobordism agrees with Voevodsky's algebraic cobordism, on the other hand for certain nice classes of varieties (e.g. smooth cellular varieties) the semi-topological cobordism of X agrees integrally with the topological cobordism of X.
Tues., Feb. 27 10:00 am J. Stewart Library	Aristide Tsemo (Ryerson University, CA) Geometric representation of cohomological classes Let C be a site, L a sheaf defined on C, elements of H^0(C,L), H^1(C,L), and H^2(C,L) classify or are geometric representations of respectively, sections, torsors, and gerbes defined on C bounded by L. This geometric objects are used in differential geometry where torsors or differential bundles, allow to study topological and differential properties of manifolds, in algebraic geometry, gerbes or stacks are fundamental objects in the study of moduli spaces. In theoretical physics the action which describe the evolution of a string is a function of the holonomy of a gerbes. Representations of higher classes are motivated by many examples as to find a geometric description of the action which describe the evolution of a brane in physics. The purpose of this talk is to describe how we can use a sequence of fibered categories to define geometric representations of cohomological classes.
Wed, Feb. 21 1:30 pm J. Stewart Library	Oliver Roendigs (Univ. of Bielefeld, DE) A model category version of Goodwillie's calculus of functors This is joint work with Georg Biedermann and Boris Chorny. We construct a tower of model structures on the category of functors from spaces to spaces (or spectra). The n-th stage models n-excisive homotopy functors. If the functors take values in spectra, we are able to take the fibre model structure at the n-th stage, which we show to be Quillen equivalent to the category of spectra with an action of the symmetric group on n letters.
Tues., Feb. 20 10:00 am J. Stewart Library	Thomas Fiore (Univ. of Chicago, US) Double Categories and Pseudo Algebras In this talk I will recall Ehresmann's notion of double category, examine several examples, and sketch a 2-equivalence between double categories with folding and certain pseudo algebras. This equivalence can be viewed as a generalization of some results of R. Brown and collaborators. These double structures arise naturally when one categorifies the notion of category to the notion of pseudo algebra over the 2-theory of categories as in the context of conformal field theory.
Wed, Feb. 14 1:30 pm J. Stewart Library	Bruce Bartlett (Univ. Sheffield, UK) Finite groupoids, gerbes and topological quantum field theory A 'topological quantum field theory' (TQFT) is a functor from the n-dimensional cobordism category (objects are (n-1)-dimensional manifolds, morphisms are cobordisms) to vector spaces. These were introduced by Atiyah and Segal in the 90's, when they were trying to make sense of what the physicists were doing. If one thinks hard about them, one finds that all sorts of higher category ideas naturally pop out. These ideas have deep connections to algebraic topology: for instance, the Freed-Teleman-Hopkins theorem about equivariant K-theory is a theorem about TQFT's. I will introduce the simplest non-trivial model, based on a finite group, and use it to give an introduction to (what is now called) 'extended' TQFT.
Wednesday, Feb. 7 1:30 pm room 230	Mathieu Anel (Univ. Western Ontario) Classifying space vs classifying stack ! Given a topological group G, the purpose of the talk will to compare the classifying space of G with the classifying stack of G-torsors.
Wednesday, Jan. 31 1:30 pm room 230	Georg Biedermann (Univ. Western Ontario) On the homotopy theory of n-types Extending the Quillen equivalence between simplicial sets and groupoids enriched in simplicial sets by Dwyer and Kan I achieve a classification of n-types of simplicial presheaves in terms of (n-1)-types of presheaves of groupoids enriched in simplicial sets. This can be viewed as a different description of the homotopy theory of higher stacks. As a special case we obtain a good substitute for the homotopy theory of (weak) higher groupoids.

Homotopy seminar

Organizer: Eric Friedlander (Northwestern)

Wednesdays 3:30-4:30 pm, April 4 - 25, 2007 held at the Fields Institute

Wednesday, Apr. 25 3:30 pm J. Stewart Library	TBA
Wednesday, Apr. 18 3:30 pm J. Stewart Library	TBA
Wednesday, Apr. 11 3:30 pm J. Stewart Library	Paul Arne Ostvaer Rigidity in motivic homotopy theory
Wednesday, Apr. 4 3:30 pm J. Stewart Library	Jens Hornbostel Rigid theorems (a survey)

Graduate Homotopy Workshop
This is a "workshop" by graduate students for graduate students. The speakers are all members of the Homotopy Program, but everybody is invited to the talks.
May 3, 2007: 10:00 A.M. - 5:00 P.M.
Fields Library

Schedule:

10:00-11:00	Oriol Raventos Stabilization of model categories We discuss briefly two constructions of the stabilization of a model category. After that we give some applications in classical homotopy, motivic homotopy and brave new algebra.
11:00-11:15	Coffee
11:15-12:15	Alexander Berglund A glimpse of rational homotopy theory I will talk about the rational homotopy theory of Quillen and Sullivan. There is a Quillen adjunction between simplicial sets and commutative dg (=differential graded) algebras over the rationals that induces an equivalence between full subcategories of the respective homotopy categories whose objects are, on one side, simply connected rational spaces and, on the other, minimal dg-algebras. Roughly speaking, this means that the homotopy theory of rational spaces is reduced to differential homological algebra.
12:15-2:00	Lunch
2:00-3:00	Markus Severitt Motives The aim of the talk is to introduce classical Chow motives as well as Voevodsky's triangulated category of (effective) motives and how these are related to each other. Furthermore the representability of motivic cohomology will be discussed.
3:00-3:30	Coffee
3:30-4:30	Haakon S. Bergsaker The structure of the complex cobordism ring I will try to explain the proof of Quillen's famous theorem stating that the complex cobordism ring is isomorphic to the Lazard ring, over which the universal formal group law is defined.

SCIENTIFIC PROGRAMS AND ACTIVITIES

Seminar Series

January - June 2007 held at the Fields Institute

Homotopy seminar

Schedule:

January - June 2007
held at the Fields Institute