Mathieu Anel (CRM)
Title: Fundamental infinity-groupoid of stacks, and applications
Abstract: (joint work with J. Heller (UWO)) Using results of
Dugger and Isaksen, we define an extension of the functor of
singular chains from the category Top of locally contractible
topological spaces to the model category of topological stacks
(simplicial presheaves on Top with the projective local model
structure) and establish its link with the cohomology with constant
coefficients. As an application, given a topological group G, we
compare three notions of classifying object for G: the
classifying space, the classifying stack and the classifying
homotopy type. To compare them we embed them in the category of
topological stack and prove that although there are non equivalent
as stacks, they have the same fundamental infinity-groupoid and thus
the same cohomology with constant coefficients. Another application
is a nice interpretation of the results of Henriques on integrating
L-infinity algebras.
Ulrich Bunke (Universität Regensburg)
Title: Two-periodic twisted cohomology for topological stacks (I + II)
Abstract: Two periodic twisted de Rham cohmology has been
introduced as the target of the Chern character from twisted,
K-theory or to measure H-fluxes in string theory models with
B-field. I will review a set-up for sheaf theory on topological
stacks and then explain how one can construct a topological analog
of twisted de Rham cohomology. Among other interesting features are
a T-duality isomorphism and calculations for coefficients different
from Q or R.
David Carchedi (Universiteit Utrecht)
Title: Foliations and Mapping Stacks of Groupoids
Abstract: This talk is about research in progress concerning
the construction of a smooth moduli space for foliations of a
manifold. It will be shown how the language of Lie Groupoids and
smooth stacks gives a nice description of foliations. It will also
be explained how the existence of an internal Hom in the 2-category
of smooth Deligne-Mumford stacks could be used to construct a smooth
Deligne-Mumford stack classifying all foliations of a fixed
manifold. An internal Hom is then introduced in the 2-categories of
topological orbispace stacks and topological Deligne-Mumford stacks
by constructing a topological groupoid presentation of each mapping
stack through the use of Hilsum-Skandalis bibundles. It is then
shown that each of these 2-categories of topological stacks is
Cartesian closed. Finally, it is indicated how this machinery can
then be used to construct a topological Deligne-Mumford stack
classifying all foliations of a smooth manifold.
Erratum from David Carchedi (2009-07-27): The 2-category of topological stacks
is not cartesian closed in the naive sense. For the correct statement
see ArXiv:0907.3925.
David Gepner (University of Sheffield)
Title: Orbivariant homotopy theory (Talk I)
Abstract: The basic building blocks of equivariant homotopy
theory are the principal homogeneous G-spaces G/H, for H a closed
subgroup of a fixed topological group G. For many purposes,
however, one must allow G to vary, in which case it may be more
convenient to consider the following generalization: let Orb denote
the category of orbit stacks (of which the G-orbits G/H are special
cases), and consider the category of presheaves of spaces on Orb,
viewed as a topological category. This category has a natural
homotopy theory which simultaneously encapsulates all of the
equivariant categories as well as the homotopy theory
of orbifolds. This is joint work with André Henriques.
David Gepner (University of Sheffield)
Title: Elliptic cohomology of orbifolds (Talk II)
Abstract: Recent work of J. Lurie on topological modular forms
shows that elliptic cohomology is naturally defined on orbifolds, or
even the more general subcategory of stacks whose stabilizers are
compact Lie groups. In this talk we will review some aspects of the
theory of elliptic cohomology and its orbivariant extensions. In
particular we will focus on the Witten genus and its topological
manifestation as the string orientation.
Jeffrey Giansiracusa (University of Oxford)
Title: Pontrjagin-Thom maps and the homology of the
moduli stack of stable curves (I+II)
Abstract: A Pontrjagin-Thom map is a homotopy-theoretic
wrong-way map: an embedding f: M→ N leads to a homotopy
class from N to the one-point compactification of the normal
bundle of f. If f is merely an immersion then one obtains a
stable homotopy class. This construction can be extended to the
category of local quotient stacks. The boundary of the
Deligne-Mumford compactification is a union of substacks which are
images of immersions. The Pontrjagin-Thom maps associated to these
immersions produce large new families of classes in the mod p
homology of these moduli stacks.
Gustavo Granja (IST, Lisboa)
Title: The stack of complex structures
Abstract: I will discuss work in progress aiming at obtaining
conditions for the existence of a smooth holomorphic stack
classifying complex structures on a closed smooth manifold. I will
also explain how it follows from work with Abreu and Kitchloo that
in certain simple examples the homotopy type associated to certain
open substacks of this stack is the classifying space of the
symplectomorphism group determined by a suitable symplectic form on
the manifold.
Richard Hepworth (University of Sheffield)
Title: Orbifold Morse Inequalities (Talk I)
Orbifold Morse-Smale-Witten Theory (Talk II)
Abstract: The Morse Inequalities relate the Betti numbers of a
manifold to the critical points of a Morse Function on the manifold.
More powerfully, the Witten Complex computes the homology of a
manifold using the critical points and gradient lines of a
Morse-Smale Function on the manifold. In these two talks I will
explain how these results extend to differentiable Deligne-Mumford
stacks, where we obtain information on the homology of the
classifying space, the inertia stack, and on the inertia string
product of Behrend et. al. I will also explain how this is motivated
by Chen-Ruan cohomology and the `crepant resolution conjecture'.
Sharon Hollander (IST, Lisboa)
Title: Applications of Homotopy Theory of Stacks (I+ II)
Abstract: I will describe the homotopy theory of stacks and
explain how algebraic stacks can be natural seen in this context. A
consequence of this perspective will be certain criteria for the
algebraicity of a stack.
Noah Kieserman (University of Madison-Wisconsin)
Title: Obstruction of coisotropic submanifolds and
Haefliger's integration-over-leaves map
Abstract: The L-infinity-algebra governing deformations of
coisotropic submanifolds of symplectic manifolds is known quite
explicitly, due to Oh and Park. Coisotropics are canonically
foliated, and the L-infinity-operators involve several structures
which may be defined for a general foliated manifold. In proving
obstructedness results for a specific family of examples, we make
heavy use of Haefliger's integration-over-leaves map, raising the
possibility that these structures may be most naturally defined on
some fine model for the leaf space.
Anders Kock (Aarhus Universitet)
Title: Colimit aspects of certain stacks
Abstract: I shall describe how certain stacks associated to
differentiable groupoids may be viewed as a definite kind of
2-dimensional coequalizer in a suitable 2-category.
Behrang Noohi (Florida State University)
Title: String Topology for Stacks (I + II)
Abstract: String topology (Chas-Sullivan, Cohen, Jones,...)
studies the homology of the loop space of a manifold by exploiting
the so-called string operations. With the goal of producing an
equivariant version of the theory, we formulate string topology for
topological stacks and prove the existence of string operations
under certain natural hypotheses. As a consequence, we obtain
equivariant string topology for compact Lie group actions on
manifolds. This is a joint work with K. Behrend, G. Ginot, and P.
Xu. There are two technical tools that are used in this work:
homotopy theory of topological stacks and Fulton-MacPherson
bivaraint theories. I will give a quick introduction to these topics
as well.
Dmitry Roytenberg (IHES)
Title: Differential graded manifolds and higher Lie
theory
Abstract: The notion of differential graded manifold extends
that of a manifold as far as differential calculus is concerned, and
also that of a Lie algebra or algebroid. For example, Hitchin's
"generalized geometry" is the geometry associated to a certain class
of dg-manifolds. Severa recently showed that dg manifolds arise
naturally as first-order approximations to Kan simplicial manifolds.
The natural question is whether every dg-manifold arises in this
way. To this end, for every dg-manifold I define a simplicial
presheaf on the site of smooth manifolds and conjecture that it is
objectwise-fibrant and satisfies descent for hypercovers. One then
asks whether there exists an integer n such that the nth truncation
of this presheaf is equivalent to one presented by a
finite-dimensional simplicial manifold. I will give examples of when
this is known to be the case and discuss possible strategies for
attacking the general problem.
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Zoran Škoda (IRB, Zagreb)
Title: Categorifications of equivariant sheaves and quotients
Abstract: Principal G-bundles have their analogues in higher
categorical situations as well as in noncommutative geometry. Less
studied, but of central importance is to understand the analogues of
equivariant sheaves on the 2-group torsors and on the noncommutative
torsors; as well as the appropriate version of descent along
torsors. There are several technical devices introduced in our work
in progress. First of all, is constructon of an appropriate
2-category of G-equivariant objects in a 2-fibered 2-category, where
G is a monoidal category acting on an object in the base 2-category.
Second, the study of actions and equivariant objects is done in
relative setup, and certain distributive laws play role in such
geometrical situations. The descent along torsors itself is very
parallel to the classical case. I will sketch few examples and
applications.