There will be five minicourses, aimed at PhD students in various branches of analysis and given by Piotr Hajlasz, Pekka Koskela, Grzegorz Swiatek, Xavier Tolsa and Michel Zinsmeister. Their titles and abstracts are available below. A preliminary schedule (update of Sept. 6, 2009) is available as a PDF file.
Piotr Hajlasz: Sobolev mappings between manifolds and metric spaces (4 hours)
In the course I will discuss several topics related to the theory of Sobolev mappings. In particular I will discuss: (1) the problem of approximation of Sobolev mappings between manifolds by smooth mappings; (2) Sobolev spaces on metric-measure spaces and Sobolev mappings between metric spaces; (3) approximation of Sobolev mappings between metric spaces by Lipschitz mappings; (4) Sobolev surjections onto metric spaces with the particular emphasis on highly regular surjective mappings between Carnot groups; (5) differentiability of metric space valued Sobolev mappings; (6) Sard theorm for Sobolev mappings between manifolds.
Pekka Koskela: Planar mappings of finite distortion (3 hours)
We give a brief introduction to a generalization of the theory of planar quasiconformal mappings. These are mappings where, instead of a uniform bound on the infinitesimal distortion of round shapes, one only controls the distortion in average. Typically, this means that the pointwise distortion is exponentially integrable.
Grzegorz Swiatek: Applications of Quasiconformal Maps in Complex Dynamics (3 hours)
One aspect of the beauty of complex function theory is the fine balance between rigidity and flexibility, for example the identity principle and the Riemann mapping theorem. In dynamics, one usually arrives with conditions on the asymptotic behavior of the iterates, local or even global, and the question arises whether they are rigid or not. Some notion of deformation in complex dynamics is needed to address these issues. This deformation space should preserve dynamical features of the system and many spaces commonly used in function theory are incompatible with this requirement.
Quasiconformal mappings and the measurable Riemann mapping theorem provide such tools. There are several ideas used here, including deforming a system by means of its quasiconformal conjugacy to another one, analysis of the invariance of the Beltrami coefficient for a quasiconformal conjugacy, or surgery which consists of constructing an entirely new system from pieces. The goal of this lecture series is to present the power of these ideas in on a series of examples.
Xavier Tolsa: The Cauchy transform, Menger curvature, and analytic capacity (4 hours)
In this course we will review the relationship between the Cauchy transform, Menger curvature, and analytic capacity. More precisely, we will show that Menger curvature is an essential tool to study the boundedness of the Cauchy integral operator in the space of square integrable functions, and we will describe some of the connections with analytic capacity.
Michel Zinsmeister: Planar growth processes (3 hours)
1) Lowner differential equation(s): we will show how every growth process in the plane can be encoded by a one-parameter family of positive probability measures m_t on the circle.
2) We will discuss when this family consists of Dirac masses, ie when m_t=\delta_\xi(t) (the function \xi is then called the driving function). A sufficient condition is when the process consists of growth of a simple curve. This is not necessary but we will see conditions on the driving function that imply it. We will illustrate this discussion by the phases of SLE (Rohde and Schramm) and by Lind's deterministic analogue.
3) On the opposite we will discuss growth processes where the measures m_t are AC wrt Lebesgue measure. Hele-Shaw flows are such and we will discuss a recent and remarkable link between random normal matrices and Hele-Shaw flows discovered by Wiegmann-Elbau-Felder-Makarov-Hedenmalm. If time allows it we will discuss random analogue of Hele-Shaw flow, ie DLA.