Analysis and PDE Seminar Homepage, Winter 2013
Location: 383N
Time: Friday, 12pm.

Wednesday, January 9th, 3pm, Room 383N (special date/time):
Kay Kirkpatrick (University of Illinois, UrbanaChampaign)
``BoseEinstein 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 BoseEinstein condensation (BEC), that
behaves like a giant quantum particle. The rigorous connection has
recently been made between the physics of the microscopic manybody
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 manybody dynamics, a step towards large deviations for BoseEinstein 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^psolution must vanish.

Wednesday, February 13, 34pm, 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 PoincareEinstein metrics. The results are formulated in terms of Weyl conformal invariants defined via the ambient metric of FeffermanGraham.

Friday, February 22, 23pm and 45pm, 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 nonrelativistic quantum electrodynamics.) In recent works, jointly JeanFrancois 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 finitedimensional 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 energycritical 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, 45pm, Room 380F (Note special date/time/room):
Israel Michael Sigal (University of Toronto)
``Magnetic Vortices, NielsenOlesen  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 righthand 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 nontrapping 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
nonsymmetric 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 wellposedness of boundary problems in L^p,
p<\infty, is a far more intricate and challenging question, even in
a halfspace. 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 wellposedness in L^p for
elliptic PDEs associated to matrices A of real (possibly
nonsymmetric) 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 wellposed 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.