Schedule 2010

Our meetings take place at noon, unless otherwise noted.

Talks

• An Introduction to Nonlinear Dispersive PDE

  Tim, JCMB 6206, Wednesday 19 May 2010

  Abstract Notes

  Perhaps surprisingly the main tool used to study the existence theory of nonlinear Dispersive PDE is the simple Contraction Mapping Theorem. In this talk I will explain how this works and consider some applications to the nonlinear Schrodinger and Wave equations. I will also cover the basic linear theory of these equations and mention some connections with Harmonic analysis.

• Bochner-Riesz Multipliers

  George, JCMB 4312, Wednesday 5 May 2010

  Abstract Notes

  In my recent PG Colloquium talk, I gave an overview of the Bochner-Riesz conjecture and stated some of the known results. My aim here will be to supply some of the missing details.

  The bulk of this will be showing how a restriction estimate can be used to get some of the best known results (due to Fefferman in the 70s).

  I also hope to mention some more recent developments, and how this conjecture connects to other open problems.

• BMO, The John-Nirenberg Inequality and an application to weights

  Josef, JCMB 4312, Wednesday 28 April 2010

  Abstract Notes

  In this talk we will define the space of bounded mean oscillation. After some examples we will proof the remarkable John-Nirenberg inequality, which leads to some nice corollaries.

  Moreover we will connect BMO and the A_p-weights of Muckenhaupt, from where the question of the week (CHOCOLATE!!!) will arise.

  If there is some time left, we investigate the relation between BMO and Carleson measures as well, otherwise this will be postponed to the next talk.

• An introduction to Singular integrals

  Marina, JCMB 4312, Thursday 22 April 2010

  Abstract Notes

  In this talk we will prove a basic theorem on singular integrals, which states that the convolution of a function with a kernel is a bounded linear tranform on L^p(Rⁿ), for every 1<p<oo, as long as the kernel satisfies several nice properties.

  We will also see some extensions of the theorem, regarding limits of such convolutions, and (hopefully) investigate the particular case when the kernel K is of the form A(x)/|x|ⁿ, where A is homogeneous of degree 0.

• An introduction to restriction theory

  Javier, JCMB 4312, Wednesday 31 March 2010

  Abstract Notes Erratum and question

  We will present the restriction theory. If time permits the proof of the Stein-Tomas theorem will be given.

  But if anyone has a particular interest about a concrete topic in restriction theory, just let me know, and I'll talk about it.

• Littlewood-Paley Theory

  George, JCMB 5326, Monday 22 March 2010

  Abstract Notes

  The Littlewood-Paley Theorem says that the norm of a function in L^p is equivalent to the norm of a related "square function". The essential idea is to break the function into pieces in a particular way, then deal with each of the pieces. In the talk, I will make all this precise, and will probably even prove the theorem!

  We will then apply this approach to study Fourier multipliers, and in particular, to prove the Hormander multiplier theorem. Time permitting, we will also consider connections with function spaces.

• Distributions, Function spaces, and Applications

  Tim, JCMB 5326, Monday 15 March 2010

  Abstract Notes
  I will cover the basic theory of distributions and introduce the Besov-Lipschitz and Triebel-Lizorkin spaces. This scale of spaces includes all the major function spaces in analysis, L^p spaces, Sobolev spaces, Hardy spaces, etc. I will then cover one or two topics from the following,

  • Connections to interpolation theory.
  • Applications to proving product estimates in Sobolev spaces.
  • Function spaces on domains and extension operators.
  • Atomic decomposition of Hardy spaces and applications to boundedness of singular integrals.

  The exact topics I decide to cover will depend firstly on how much preparation I get done before Monday and secondly on what you would all like to hear about :)

• The A_p weights of Muckenhaupt

  Josef, JCMB 6311, Monday 8 March 2010

  Abstract

  It is a well-known fact that the maximal function is bounded on Lp(dx) for all 1 < p < infinity. In the talk we will address the question on which weighted spaces we still have boundedness of the maximal function, i.e. for which weights w the maximal function is bounded on Lp(wdx).

  After a short motivation we will introduce the Ap-weights of Muckenhaupt (and the class A_infinity), for which we will derive some basic properties and the remarkable reverse Holder inequality. To finish the talk we will give an application to the solvability of the Dirichlet problem.

If you would like to be added to this list (and receive e-mail updates about forthcoming talks) please contact George.