The Jyväskylä Summer School

Mathematics Courses

 

 

MA1: Tangent of null sets, differentiability of Lipschitz functions, and other problems in geometric measure theory

Time: 5. - 13.8.2009, 14h lectures + 4 h exercises

Credits: 5 ECTS

Coordinator: Dr. Ville Suomala

Lecturer: Prof. Marianna Csörnyei (University College, London)

Abstract: Abstract (pdf)

The lectures are based on an ongoing joint research with
G. Alberti and D. Preiss.

The starting point of this research is the observation that the solutions of several problems of seemingly different nature can be derived by a simple covering result for null sets in the plane. These problems include the so-called rank-one property of $BV$ functions, and the construction of Lipschitz maps with large non-differentiability sets. This will lead us to the notion of tangent fields to null sets, and provide a new understanding of the structure of sets of small measure.

Many results and proofs presented in the lectures are connected to a possibility of decomposing certain small sets, or perhaps even all Lebesgue null sets, in a way reminiscent of the decompositions of finite sets in combinatorial results. Some of them are direct corollaries of elementary results in combinatorics. For other problems we will need a finer, continuous version of the combinatorial results whose proofs also use techniques that are not available in the discrete world, they are purely analytic.

Our main aim is to show how these results and techniques can be used to solve various problems in analysis. Some of the highlights are:

  1. We prove that one can assign to every null set $N$ in the plane a (Borel measurable) vectorfield $\tau$ such that, given any $C1$ curve $\gamma$ in the plane, $\tau$ agrees with the tangent vector of $\gamma$ at almost every point of the the intersection $\gamma\cap N$. Thus $\tau$ can be regarded as a tangent vectorfield to $N$, and is uniquely determined up to purely unrectifiable sets.
  2. We show a new, geometric proof that every planar set of positive measure can be deformed onto a ball by a Lipschitz deformation (previously known proofs used techniques of complex analysis and Hahn-Banach theorem).
  3. We give a full description of non-differentiability sets of Lipschitz maps on $\R^n$ and show that the planar Rademacher theorem is optimal in the sense that there exist Lipschitz deformations with arbitrarily prescribed null sets of non-differentiability.
  4. We will also discuss many problems that our results left open.

Prerequisites: The lectures are suitable to graduate students and non-experts, with only a basic knowledge of measure theory and real analysis. All necessary concepts and results of analysis and combinatorics will be introduced.

The course is organized in cooperation with the Graduate school in Mathematical Analysis and its applications, the Finnish Academy of Science and Letters (Mathematics Foundation) and the Finnish Centre of Excellence in Analysis and Dynamics Research.

Korppi code: MATS520

Korppi link: 

https://korppi.jyu.fi/kotka/course/student/generalCourseInfo.jsp?course=65821


MA2: Percolation Theory

Dates: 5. - 13.8.2009 (Lectures 14h, exercise sessions 4h )

Credits: 5 ECTS

Level: The course is aimed at graduate students but strong advance undergraduate students with the appropriate background might also find it suitable.

Coordinator: Dr. Ville Suomala

Lecturer: Prof. Jeffrey Steif (Chalmers University of Technology, Göteborg)

Course notes: Handout (pdf)

Abstract: Percolation theory, an area of probability theory, is a beautiful subject where the problems are easily stated but usually require ingenuity for their solution. One often deals with large random systems and tries to understand the global structure.

The percolation model is one of the simplest such possible systems. One sees the concepts of 'phase transition' and 'critical values' in a very simple and transparent situation. More specifically, it is preliminary planned that the following topics will be covered.

In reality, most likely, some subset of these topics will be covered.
(1) The phase transition for the percolation model.
(2) Uniqueness of the infinite cluster.
(3) Continuity of the percolation function in the supercritical regime.
(4) The second moment method and the exact critical value for trees.
(5) An outline of the exact critical value for the 2 dimensional lattice.
(6) Subexponential decay of the cluster size in the supercritical regime.
(7) An overview of conformal invariance and the critical exponents for critical 2-dimensional percolation. 

Prerequisites: One should be familiar with very basic concepts in probability such as independence, etc. One does not need many of the concepts taught in a first year graduate course in probability although it would be helpful to have had some advanced undergraduate courses in probability theory so that one knows a little bit how to think probabilistically. Mathematical maturity is, as usual, good to have. The best way to see what is needed is to look at the notes for the course which should be up on the homepage 3 months before the course start.

The course is organized in cooperation with the Graduate school in Mathematical Analysis and its applications, the Finnish Academy of Science and Letters (Mathematics Foundation) and the Finnish Centre of Excellence in Analysis and Dynamics Research.

Korppi code: MATS521

Korppi link:

https://korppi.jyu.fi/kotka/course/student/generalCourseInfo.jsp?course=65823