Abstracts

Berglund: I will describe a new approach to the rational homotopy theory of mapping spaces which is inspired by Getzler's Lie theory for nilpotent L-infinity algebras. Combined with a recent characterization of spaces that are simultaneously formal and coformal, this yields small chain complexes for calculating the rational homotopy groups of the space aut(X) of homotopy automorphisms of such spaces. In joint work with Madsen, we use this to calculate explicitly the rational homotopy groups of the homotopy automorphisms of highly connected manifolds. If time admits, I will also discuss work in progress towards the calculation of the rational cohomology ring of the classifying space Baut(X).

Gepner: The space of units GL₁(R) of a commutative ring spectrum R is the infinite loop space of a spectrum gl₁(R). Typically, this spectrum is taken to be connective, meaning that it has no nonzero negative homotopy groups. However, there are other interesting deloopings of GL₁(R) which carry important algebraic information about R. One in particular has π_-1 gl₁(R) = π₀Pic(R), the Picard group of R, and π_-2 gl₁(R) = π₀ Br(R), the Brauer group of R. If R is connective, there is a spectral sequence for computing the homotopy groups of Pic(R) and Br(R) which reveals a close relationship between π₀ Br(R) and Br(π₀ R).

Scherer: This is joint work with Boris Chorny. I will explain a very basic construction in homotopy theory, namely that of Whitehead products, and present the Lawvere algebraic theory defining nilpotency of class less than n. I will show that loop spaces which are "nilpotent up to homotopy" enjoy a vanishing property for iterated Whitehead products and move then to a more contemporary notion of homotopy nilpotency due to Biedermann and Dwyer. It is strongly related to Goodwillie calculus and we will see that the understanding of the stages in the Goodwillie tower yields a new vanishing result for iterated Whitehead products.

Müller: These talks give an overview of my thesis. They cover an introduction to classical monadic descent and an important example, Grothendieck descent. Then we examine the question, how this setting can be generalized to the case of categories enriched over model categories and give the general theory of homotopic descent. This includes cosimplicial structures and their totalizations, derived completion and the descent spectral sequence associated to homotopic descent. Finally we give a definition of a generalized Adams spectral sequence as an example of the theory. In particular, we will examine its relationship to the descent spectral sequence.

Turner: Khovanov homology is an abelian-group-valued knot invariant related to the Jones polynomial. Like many other modern knot invariants, the underlying topological or geometrical properties that are being measured remain rather obscure. To better understand the situation it is natural to ask: is it possible to associate to each knot a topological space whose classical invariants (for example cohomology, homotopy groups etc) give Khovanov homology? In this talk I will discuss one approach to this question using homotopy theory.

Seal: Joint work with Dirk Hofmann and Frédéric Mynard. Exponentiable topological spaces showcase a remarkable link between topological and ordered structures, a feature that can be reinterpreted in terms of underlying monadic structures. Indeed, on one hand, exponentiable objects in TOP are those topological spaces whose set of opens forms a continuous lattice. On the other, topological spaces are monoids in the Kleisli category of the filter monad, while continuous lattices form its Eilenberg-Moore algebras. In this talk, I will present this categorical interpretation of topological structures, and illustrate it with well-known and new results.

Finster: Recently a surprising connection has been discovered between homotopy theory and a class of well studied formal languages used in logic and computer science know as dependent type theories. This observation, made independently by Vladimir Voevodsky and the logician Steve Awodey, is the basis for Voevodsky's "Univalent Foundations" program, whose goal is to provide an alternative to set theory in which all homotopy types are regarded as basic entities. In this talk, I will describe some of the ideas that motivate dependent type theories, as well as show how to use them to express homotopy theoretic ideas.

Hess: Let M be a monoidal category endowed with a distinguished class of weak equivalences and with appropriately compatible classifying bundles for monoids and comonoids. We define and study homotopy-invariant notions of normality for maps of monoids and of conormality for maps of comonoids in M. These notions generalize both principal bundles and crossed modules and are preserved by nice enough monoidal functors, such as the normaliized chain complex functor. We provide several explicit classes of examples of homotopy-normal and of homotopy-conormal maps, when M is the category of simplicial sets or the category of chain complexes over a commutative ring.

Berglund (2012): I will talk about joint work in progress with Ib Madsen concerning the problem of calculating the rational cohomology ring of the diffeomorphism group of a (d-1)-connected 2d-dimensional manifold M. Roughly speaking, our approach is to use rational homotopy theory to obtain information about the monoid of homotopy self-equivalences aut(M), and then use surgery theory to measure the difference between homotopy self-equivalences and diffeomorphisms. One can obtain very precise information about the classifying space Baut(M). A surprising result is that its rational homotopy groups only depend on d and the rank of the middle dimensional homology group H_d(M). To approach the cohomology of the diffeomorphism group, we prove a stability theorem analogous to the classical Harer stability theorem for diffeomorphism groups of surfaces. The cohomology in the stable range was recently calculated by Galatius and Randal-Williams.

Chataur: Intersection homology and cohomology were introduced by Goresky and MacPherson in order to extend the L-genus to singular spaces. These theories provide a generalization of Poincaré duality and are useful in the study of algebraic varieties. In this talk, we will explain how to develop algebraic models for intersection cohomology.

Dessai: We report on joint work with Michael Wiemeler on the classification of low dimensional complete intersections with S¹-symmetry.

Dwyer: This talk, which should be entirely accessible to doctoral students and advanced masters students, will consist of an introduction to localization in model categories, including a number of interesting and important examples.

Scherer (2012): This is joint work with Emmanuel Dror Farjoun. We usually think that the pullback of a fibration along any map is another fibration which cannot become more complicated than the fibration we started with. This is often true and I will recall a few classical examples. However, when we look at "flatness properties" this philosophical principle is not the rule. By an L-flat fibration sequence we mean one which remains a fibration sequence after applying the functor L. The plan is to explain which functors L behave well with pullbacks in the category of spaces and continue with the analogous question for group extensions which started this project.

Finster (2012): The Opetopes are a family of polytopes which are used in several definitions of higher category, and in this sense, constitute a kind of alternative to the more familiar notion of simplices. They arise very naturally from considering pasting diagrams of "many-in one-out" operations, and can be regarded geometrically as the higher dimensional analog of trees. A very succinct definition due to Joyal, Kock and Batanin constructs the Opetopes using the language of polynomial functors on Set, giving them a very close connection to the notion of inductive datatype found in many modern programming languages. In this talk, I will sketch their definition and explain a convenient graphical notation for depicting and manipulating them.

Rovelli: In this talk, we will introduce the definition of triangulated category, with a particular care to similarities and differences between the triangulated context and the abelian one. The approach is that followed by Holm and Jorgensen's article: given an abelian category, we consider both the complexes category and the homotopic category, and we will easily see that the first one is abelian, while for the second one we need to change the environment. In the second part of the talk, we will give an idea of how to build the derived category out of an abelian one, we will explain why it is more useful (for the purpose of studying homology) than the homotopic category, and finally we will discuss whether it is triangulated.

Sabatini: (Joint work with Prof. L. Godinho, IST Portugal) In 2009 Tolman formulated the ``symplectic generalization of Petrie's conjecture": given a compact symplectic manifold M with a Hamiltonian S¹-action and minimal number of fixed points, is it possible to characterize all the possible cohomology rings and Chern classes that can arise? We turn this question into a computational problem, in the following way. We derive a simple algebraic identity involving the
ûrst Chern class. This enables us to construct an algorithm to obtain linear relations among the isotropy weights at the
ûxed points. Since determining the weights at the fixed points determines the (equivariant) cohomology ring and Chern classes, this allows us to give a (positive) answer to the question. In particular, we give a complete list of cohomology rings and Chern classes when dim(M) is less than or equal to 8.

Hausel: We will be looking at the graph of Betti numbers, depicted as a discrete function of cohomological degree, of various non-compact complete hyperkahler manifolds: hyperkahler toric, quiver and character varieties. We find that if they are of large dimension, then the graphs will converge to various distributions, including Gauss, Gumbel and Airy. This is a joint experiment with Fernando Rodriguez Villegas.

Seal (2012): One of the simplest non-cartesian monoidal structure on a category is given by the tensor product of abelian groups. This tensor has the desirable property of representing bilinear maps as linear ones. However, the corresponding general notion of a ``bimorphism'' (that is, of a ``morphism in each variable'') is a priori awkward to express in categorical terms. A further analysis reveals that the ``free abelian group'' monad is monoidal, and that this additional structure facilitates a categorical definition of bimorphisms - and of the corresponding tensor. Hence, under appropriate hypotheses a monoidal monad on a monoidal category induces a monoidal structure on the Eilenberg-Moore category - a fact that invites the study of actions therein.

Bergner: With many definitions being given for (∞, n)-categories, one criterion to check is whether they can be thought of as categories enriched in (∞ , n-1)-categories. In joint work with Charles Rezk, we are establishing a chain of Quillen equivalences from the model structure for Θ_n-spaces and the model structure for categories enriched in Θ_n-1-spaces. This comparison also gives insight into how to compare to other known models.

Dernière mise à jour: 13.08.2012