Analysis and PDE Seminar Homepage, Winter 2013

Location: 383N

Time: Friday, 1-2pm.

• Wednesday, January 9th, 3pm, Room 383N (special date/time):

  Kay Kirkpatrick (University of Illinois, Urbana-Champaign)

  ``Bose-Einstein condensation, the nonlinear Schrödinger equation, and a central limit theorem''

  Abstract: Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called Bose-Einstein condensation (BEC), that behaves like a giant quantum particle. The rigorous connection has recently been made between the physics of the microscopic many-body dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schr&ounl;dinger equation (NLS). I'll discuss recent progress with Gerard Ben Arous and Benjamin Schlein on a central limit theorem for the quantum many-body dynamics, a step towards large deviations for Bose-Einstein condensation.

• Friday, February 8:

  Christian Baer (Potsdam)

  ``Some properties of solutions to weakly hypoelliptic equations''

  Abstract: A linear different operator L is called weakly hypoelliptic if any local solution u of Lu=0 is smooth. We allow for systems, i.e. the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which covers all elliptic, overdetermined elliptic, subelliptic and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence and Riemann's first removable singularity theorem. In the case of constant coefficients we show that Liouville's theorem holds, any bounded solution must be constant and any L^p-solution must vanish.

• Wednesday, February 13, 3-4pm, Room 383N (Note special date/time):

  Raphael Ponge (Seoul National University)

  ``The logarithmic singularities of the Green functions of the conformal powers of the Laplacian''

  Abstract: Green functions play an important role in conformal geometry. In this talk, we shall explain how to compute explicitly the logarithmic singularities of the Green functions of the conformal powers of the Laplacian. These operators the Yamabe and Paneitz operators, as well as the conformal fractional powers of the Laplacian arising from scattering theory for Poincare-Einstein metrics. The results are formulated in terms of Weyl conformal invariants defined via the ambient metric of Fefferman-Graham.

• Friday, February 22, 2-3pm and 4-5pm, Bay Area Microlocal Analysis seminar at Berkeley, 736 Evans Hall

  Israel Michael Sigal (University of Toronto)

  ``Asymptotic completeness of Rayleigh scattering''

  Abstract: Experiments on scattering of photons on atoms (Rayleigh scattering) and on free electrons (Compton scattering) led, in the beginning of 20th century, to our understanding of composition of matter and eventually to creation of quantum mechanics. Though these experiments reproduced central physical phenomena, and though quantum mechanics provided a well defined mathematical framework for describing these processes, their mathematical theory is still missing. (The mathematical framework mentioned is given by the Schroedinger equation of the non-relativistic quantum electro-dynamics.) In recent works, jointly Jean-Francois Bony and Jeremy Faupin, we succeeded in proving asymptotic completeness of Rayleigh scattering. This proof assumes a bound on the average photon number, which is proven in a special case of finite-dimensional quantum systems. In this talk, I describe recent results. I will not assume a prior knowledge of quantum field theory and will provide all necessary definitions in the talk.

  and

  Ben Dodson (UC Berkeley)

  ``The energy critical NLS in an exterior domain''

  Abstract: In this talk we will discuss the energy-critical nonlinear Schroedinger equation outside a convex obstacle in four space dimensions (that is, a cubic NLS), with Dirichlet boundary conditions. We will explain how the results of Visan for the defocusing energy critical NLS in four dimensions can be used to study this problem.

• Monday, February 25, 4-5pm, Room 380F (Note special date/time/room):

  Israel Michael Sigal (University of Toronto)

  ``Magnetic Vortices, Nielsen-Olesen - Nambu strings and theta functions''

  Abstract: The Ginzburg - Landau theory was first developed to explain and predict properties of superconductors, but had a profound influence on physics well beyond its original area. It had the first demonstration of the Higgs mechanism and it became a fundamental part of the standard model in the elementary particle physics. The theory is based on a pair of coupled nonlinear equations for a complex function (called order parameter or Higgs field) and a vector field (magnetic potential or gauge field). They are the simplest representatives of a large family of equations appearing in physics and mathematics. (The latest variant of these equations is the Seiberg - Witten equations.) Geometrically, these are equations for the section of a principal bundle and the connection on this bundle. Besides of importance in physics, they contain beautiful mathematics (some of the mathematics was discovered independently by A. Turing in his explanation of patterns of animal coats). In this talk I will review recent results involving key solutions of these equations - the magnetic vortices and vortex lattices, their existence, stability and dynamics, and how they relate to the modified theta functions appearing in number theory.

• Friday, March 1:

  Simeon Reich (Technion)

  ``Generic properties of continuous differential inclusions and the Tonelli method of approximate solutions''

  Abstract: We study Cauchy problems for differential inclusions in Banach spaces and show that most such problems (in the sense of Baire's categories) have solutions. We consider separately the cases where the point images of the right-hand side are compact and convex, and where they are merely bounded, closed and convex. This is joint work with F. S. de Blasi and A. J. Zaslavski.

• Friday, March 8:

  Dean Baskin (Northwestern)

  ``Strichartz estimates on exterior polygonal domains''

  Abstract: Strichartz estimates are mixed L^p (in time) and L^q (in space) estimates for solutions of Schrodinger and wave equations. These estimates provide a measure of dispersion for the solutions and are useful in the study of semilinear dispersive equations. In this talk I will discuss recent work establishing Strichartz estimates for the Schrodinger equation exterior to non-trapping polygons. This is joint work with Jeremy Marzuola and Jared Wunsch.

• Friday, March 15, 1:15pm:

  Svitlana Mayboroda (U Minnesota)

  ``Boundary value problems for elliptic operators with real non-symmetric coefficients''

  Abstract: One of the simplest and the most important results in elliptic theory is the maximum principle. It provides sharp estimates for the solutions to elliptic PDEs in L^\infty in terms of the corresponding norm of the boundary data. It holds on arbitrary domains for all (real) second order divergence form elliptic operators - div A \nabla$. The well-posedness of boundary problems in L^p, p<\infty, is a far more intricate and challenging question, even in a half-space. In particular, it is known that some smoothness of A in t, the transversal direction to the boundary, is needed. In the present talk we shall discuss the well-posedness in L^p for elliptic PDEs associated to matrices A of real (possibly non-symmetric) coefficients independent on the transversal direction to the boundary. In combination with our earlier perturbation theorems, this result shows that the Dirichlet and Regularity boundary value problems are well-posed in some L^p, 1<p<\infty, whenever (roughly speaking) |A(x,t)-A(x,0)|^2 dxdt/t is a small Carleson measure. This is joint work with S. Hofmann, C. Kenig, and J. Pipher.

• Schedule for Autumn 2012.
• Schedule for Winter/Spring 2012.
• Schedule for Autumn/Winter 2010-2011.
• Schedule for Winter-Spring 2010.
• Schedule for Autumn 2009.
• Schedule for Spring 2009.
• Schedule for Winter 2009.
• Schedule for Spring 2008.
• Schedule for Winter 2008.
• Schedule for Autumn 2007.