Mathematical Institute of the University of Bonn

RG Analysis and Partial Differential Equations

S5B1 - Graduate Seminar on Advanced Topics in PDE

Summer term 2016

Instructors:

Dr. Roland Donninger, Prof. Dr. Herbert Koch, Dr. Mariusz Mirek, Prof. Dr. Christoph Thiele.

Lectures

Friday, 14:15 - 16:00, Seminar room 0.011

Description

We will discuss in this seminar various topics in Analysis and PDE. The seminar may lead to research topics in the area that can lead to a Master or PhD thesis. Students interested in participating should contact one of the organizers.

Talks

• April 22, 2016: Mateus Costa de Sousa (IMPA Brazil).
  Title: Regularity of maximal operators.
  Abstract: We will discuss regularity of maximal operators acting in functions of bounded variation or in Sobolev spaces. We will show some new results about convolution type maximal operator associated to a family of elliptic PDE and also about a nontangential maximal operator. This is based on joint work with E. Carneiro and R. Finder.

• May 2, 2016: Pawel Biernat (University of Bonn).
  Title: Type II blow-up mechanism for the supercritical harmonic map flow.
  Abstract: Heat flow for harmonic maps is known to produce finite-time singularities from smooth initial data. These singular solutions arise for a large class of initial data and present a major obstacle in solving the heat flow equation for arbitrarily large times. I will show how to (formally) construct these singular solutions using matched asymptotics and how to prove their existence.

• May 6, 2016: Mariusz Mirek (University of Bonn).
  Title: Variational estimates for operators of Radon type.
  Abstract: We will discuss L^p estimates (p>1) of r-variations (r>2) for averaging operators and truncated singular integrals of Radon type modelled on polynomial mappings. We will show a new method which handles the short variations with the aid of vector-valued estimates. This is a joint work with Elias M. Stein and Bartosz Trojan.

• May 9, 2016: Pavel Zorin-Kranich (University of Bonn).
  Title: Sparse domination of variational operators.
  Abstract: We prove sharp weighted estimates for variational truncations of singular integral operators that are pointwise larger than the maximal truncations. To this end we apply Lacey's pointwise sparse domination technique to a variational refinement of the nontangential maximal function, unifying a number of its previous applications in the process. Joint work with de Franca Silva.

• June 3, 2016: Shaoming Guo (Indiana University at Bloomington).
  Title: Decoupling inequalities related to Vinogradov's mean value theorems in dimension two and three.
  Abstract: I will present some sharp l^p L^p decoupling inequalities associated to certain translation and dilation invariant surfaces. These inequalities imply the sharp bounds for the integer solutions of certain system of Diophantine equations.

• June 10, 2016: Alexander Volberg (Michigan State University).
  Title: Absolutely continuos harmonic measure is rectifiable.
  Abstract: We will present the solution of a question of Chris Bishop raised by him in 1990: if harmonic measure of a domain in \R^d is absolutely continuous with respect to Hausdorff measure of codimension 1, is this true then that harmonic measure is (d-1) rectifiable? The answer is yes, it is a work of a big group of mathematicians: JONAS AZZAM, STEVE HOFMANN, JOSE MARIA MARTELL, SVITLANA MAYBORODA, MIHALIS MOURGOGLOU, XAVIER TOLSA, AND ALEXANDER VOLBERG. I will show how the solution is based on recent achievements of non-homogeneous harmonic analysis.

• June 24, 2016: Victor Lie (Purdue University).
  Title: Long term regularity of the one-fluid Euler-Maxwell system in 3D with vorticity.
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• July 4, 2016: Gennady Uraltsev (University of Bonn). This talk starts at 16:15 in room 2.040
  Title: Sparse domination of the Variation Carleson operator.
  Abstract: In this talk we will recall main ingredients of the outer measure proof of the Variational Carleson Operator and we will show how the embedding maps allow us to do a Lacey style sparse domination of the associated bi-linear form. Sparse domination allows one to recover weighted bounds on the operator. This is an ongoing joint work with Di Pilinio and Do.

• July 8, 2016: Julien Sabin (Universite Paris-Sud).
  Title: Maximizers for the Stein-Tomas inequality.
  Abstract: We provide a necessary and sufficient condition for the pre-compactness up to symmetries of maximizing sequences to the Stein-Tomas inequality on the sphere in any dimension, generalizing the results of Christ-Shao in two dimensions. This condition is related to a conjecture on the optimizers for the Strichartz inequality.

• July 15, 2016: Shantanu Dave, (Wolfgang Pauli Institute Vienna).
  Title: Hypo-ellipticity of certain geometric differential operators.
  Abstract: Ellipticity of operators such as Dirac.Laplace operators plays an important role in geometry and topology. These operators are obtained from geometry Riemannian/spin structures and hence their analysis has consequences on geometric problems. In this talk we shall construct a natural class of differential operators on manifolds with other geometric structures like a contact structure, Engel structures or Cartan structures in 5 dimensions. We shall note that some of these operators have been obtained from representation theoretic methods. Furthermore we shall see that these operators have an underlying hypo-ellipticity thanks to the analysis of nilpotent Lie groups. As a consequence we obtain regularity results as well as results similar to Hodge decomposition. The natural construction suggests that these operators have connections to topology and geometry and we will point out some relevant connections. This talk is based on joint work with Stefan Haller.

• July 22, 2016:

  The first speaker: Tomasz Szarek, (Polish Academy of Science Warsaw).

  Title: Maximal operator of the classical multi-dimensional Laguerre semigroup.
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  The second speaker: Bartosz Langowski, (Technical University of Wroclaw).
  Title: Sobolev spaces related to classical and symmetrized Jacobi expansions.
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