Ib Madsen

Diffeomorphism groups from a homotopical viewpoint

The first lecture will concentrate on surgery with special emphasis on Sullivan's contributions: the surgery exact sequence, calculation of normal invariants, and the Adams conjecture. The next lectures will outline the solution of the generalized Mumford conjecture about the stable homology of the mapping class group and a recent generalization thereof, namely the stable homological structure of diffeomorphism groups of (d-1)-connected 2d-dimensional manifolds for large d. The latter uses the techniques from the surface case, surgery and Waldhausen's algebraic K-theory of spaces.

Kevin Costello

Supersymmetric, holomorphic and topological field theories in dimensions 2 and 4

Applications of quantum field theory to mathematics often involve topological twists of supersymmetric field theories. However, for a mathematician, the physics presentation of supersymmetric field theories and their twists can be difficult to understand.

In this series of talks I will explain how many of the twisted supersymmetric field theories of relevance in mathematics have very natural interpretations in terms of derived geometry. I will, in particular, explain how to construct the A and B models of mirror symmetry, and the P^{1} of twisted N=4 gauge theories which appear in Kapustin and Witten's work on the Langlands program.

John Morgan

Rational homotopy theory

Dennis' work on rational homotopy theory had several motivations. One was to give a fairly simply algebraic category that is equivalent to the rational homotopy category. It was known that there were models for rational homotopy theory that were elegant algebraic categories, for example Quillen's category of differential graded Lie algebras. Dennis, for reasons having to do with geometric applications wanted a model closer to differential forms. This turned out to be the category of (connected) differential graded algebras over **Q**, up to quasi-isomorphism. A central ingredient of his construction was the notion of a minimal, free object in every equivalence class, an object he called the minimal model. It turns out that the minimal model faithfully reflects the (rational) postnikov tower of the space, so for example it is easy to read off the homotopy groups and even the *k*-invariants from the minimal model.

Another part of the program was to connect this construction of a category equivalent to the homotopy category to the differential forms on a manifold, so that one could use the differential forms to extract not just cohomological information but also homotopy theoretic information. For this one needed to construct differential forms over **Q** for any simplicial complex and show their minimal model was the one associated to the Postnikov tower of the space. These are the so-called piecewise polynomial forms, which can be defined over **Q**. The last step was to connect smooth differential forms on a smooth manifold to the piecewise polynomial forms on some smooth triangulation. Once this bridge had been established one could apply results about differential forms to establish homotopy theoretic results. Maybe the most striking of these is the applications to compact Kahler manifolds, e.g. non-singular projective varieties and more generally to open non-singular varieties.

In these three lectures, we will give some of the homotopy theoretic background, describing the difference between rational and integral homotopy theory, explaining the significance of the Steenrod squaring operation and the closely related fact that the Whitney cochain cup product is not commutative. We will also talk about *A _{∞}* structures and some of the recent developments in integral homotopy theory. We will introduce the minimal model and show how it is related to the Postnikov tower of a space. We will introduce the

**Q**piecewise polynomial forms on a simplicial complex and show their relationship with the rational homotopy type of the space. Then we will turn to geometric applications especially the results on Kahler manifolds, arising from the Hodge decomposition on cohomology and the so-called d-d bar lemma.

Alexander Shnirelman

Fluid dynamics

__Lecture 1:__General notions

Group

*D*of volume preserving diffeomorphisms;

*D*as a Riemannian manifold; the Least Action Principle; geodesics on

*D*, Lagrange and Euler equations; conservation laws and vorticity equations; local existence theorem for the Euler equations; singularity problem; known results.

__Lecture 2:__Long-time behavior of 2-d flows

Global solvability in the 2-d case; partial analyticity of solutions; stability of steady flows; Arnold stable and minimal flows; irreversibility of 2-d fluid dynamics; Liapunov functions and wandering domains; Generalized Minimal Flows as final states.

__Lecture 3.__Weak solutions of the Euler Equations

Formal definition; paradoxical weak solutions; energy dissipation due to the irregularity of weak solutions; construction of energy dissipating weak solutions; existence problem of physically meaningful weak solutions.

Bruce Kleiner

Hyperbolic groups and analysis on metric spaces

These lectures will survey some developments connecting analysis on metric spaces with the asympototic geometry of Gromov hyperbolic spaces. The roots of this topic go back to Mostow rigidity on the group theory side, and the classical theory of quasiconformal homeomorphisms on the analytical side. Seminal work by Heinonen-Koskela and Cheeger in the late 90's created the possibility of extending the classical framework for quasiconformal/hyperbolic geometry to the much broader setting of metric measure spaces, with potential applications to group theory and rigidity. After reviewing the relevant background, the lectures will cover the subsequent progress along these lines.

Richard Canary

Sullivan's dictionary

__Talk 1__: Kleinian groups and the Sullivan Dictionary I

The Sullivan dictionary provides a conceptual framework for understanding the connections between the dynamics of rational functions and Kleinian groups. We will survey the basic theory of Kleinian groups with an emphasis on the quasiconformal deformation theory where the analogies between the two theories are closest.

We then discuss the three major conjectures which drove many of the major developments in Kleinian groups since Thurston revolutionized the field in the 1970s. Marden's Tameness Conjecture predicts that every hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, i.e. homeomorphic to the interior of a compact 3-manifold. Thurston's Ending Lamination Conjecture proposed a classification of all hyperbolic 3-manifolds with finitely generated fundamental group. The Bers-Sullivan-Thurston Density Conjecture asserted that every finitely generated Kleinian group is a limit of geometrically finite Kleinian groups. Each of these conjectures has been resolved in the last decade.

__Talk 2__: Kleinian groups and the Sullivan dictionary II

We will continue our discussion of the three major conjectures. We will then discuss applications of Marden's Tameness Conjecture. We will focus on dynamical applications, the most prominent of which is the resolution of Ahlfors' Measure Conjecture. We will also discuss applications to the dynamics of geodesic flows of hyperbolic 3-manifolds and to limit sets of Kleinian groups. Marden's Tameness conjectures also has topological and group-theoretic applications which we will discuss if time permits.

__Talk 3__: Kleinian groups and the Sullivan dictionary III

We discuss the space AH(M) of (marked) hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M. One may view this as a natural generalization of Teichmueller space in the 3-dimensional setting. The resolution of Thurston's Ending Lamination Conjecture gives a classification of the manifolds in AH(M), but the topology of AH(M) remains elusive since the invariants in this classification do not vary continuously over AH(M). We will survey recent work which shows that the topology of AH(M) is actually quite pathological.

AH(M) naturally sits inside the character variety X(M) of conjugacy classes of representation of the fundamental group of M into PSL(2,C). The outer automorphism group of the fundamental group of M acts naturally on both AH(M) and X(M). If time permits, we will discuss recent work on the dynamics of this action.

Artur Avila and Misha Lyubich

Renormalization

__1st lecture__(by M. Lyubich)

In this introductory lecture a general overview of the idea of renormalization and its various incarnations in low-dimensional dynamics will be given.

__2nd and 3rd Lectures__(by Artur Avila)

One of the main themes in low-dimensional dynamics is the investigation of the interplay between order (periodic or KAM behavior) and chaos (nonuniform hyperbolicity). In the best understood cases, the analysis involves the description of the dynamics of a renormalization operator acting on parameter space and presenting an attractor.

We will discuss two incarnations of this general idea. The first concerns unimodal dynamics, and we will focus on the proof of the existence of a global renormalization attractor (its applications, such as the "regular or stochastic dichotomy", being discussed in Lyubich's talk). The second concerns perhaps the simplest class of dynamical systems compatible with both KAM behavior and nonuniform hyperbolicity: one-frequency cocycles. We will explain the emerging global picture for the parameter space and its application to certain Schrödinger operators (the "spectral dichotomy"), and then describe the role played by a (non-longer global) renormalization attractor in the measure-theoretical analysis of the phase-transition.