Limit Theorems for Horocycle Flows, Giovanni Forni (University of Maryland)
We discuss joint results with A. Bufetov on limit
distributions of suitably normalized ergodic integrals for classical
horocycle flows on compact hyperbolic
surfaces. We prove that limit distributions exist under some
conditions, that they
always have compact support and that they can be described in terms of
a class of
finitely additive measures on the phase space. These results describe
in some detail
the fine behavior of ergodic integrals of smooth functions for one of the main
examples of renormalizable parabolic flow.
Entire functions and spectral problems, Alexei Poltoratski (Texas A&M University)
The Krein - de Branges theory of Hilbert spaces of entire functions was created in 1940-60's to treat spectral problems for second order differential operators. It translates such problems into the language of complex and harmonic analysis, entire functions and singular integrals.
Since its creation the theory outgrew its original purpose. Multiple relations
with other areas of analysis have been discovered, including connections with analytic number theory and
the Riemann Hypothesis.
In my talk I will discuss the basics of the theory and some of its newly found applications.
Analytic one-frequency cocycles, Svetlana Jitomirskaya (UC Irvine)
We describe a new approach to the proof of joint continuity of
Lyapunov exponents of holomorphic quasiperiodic cocycles in frequency and
cocycle, at irrational frequency, first proved for SL(2,C) cocycles in
Bourgain-Jitom., 2002. The approach is powerful enough to handle singular
and multidimensional cocycles, thus establishing the above continuity in
full generality. This has important consequences including that on a dense
open set of analytic quasiperiodic one-frequency matrix cocycles in
arbitrary dimension Oseledets filtrationis either dominated or trivial.
The underlying mechanism is therefore different from that of the
Bochi-Viana theorem for continuous cocycles, which links non-domination
with *discontinuity* of the Lyapunov exponent. Using approximations by
rational frequencies, domination can be characterized by a gap in the
Lyapunov spectrum and certain regular behavior of the Lyapunov exponent
when extending to complex phases, which in turn always occurs for small
nonzero complexifications under the gap condition. It is a report on a
joint work with A. Avila and C. Sadel.
Modulo-automorphic Hardy spaces for Kotani-Last problem, Alexander Volberg (Michigan State University)
We make an exposition of a theory of Hardy spaces in infinitely connected domains,
and we show what properties of such spaces are exactly responsible for the existence of
ergodic family of Jacobi matrices with absolutely continuous spectrum, but not almost periodic.
Small scale creation in 2D Euler equation for ideal flow, Alexander Kiselev (University of Wisconsin)
The global existence of smooth solutions for 2D Euler equation in smooth
bounded domain has been known since 1933. The equation is in some sense
critical, as the needed estimates barely close and the upper bound on the
possible growth of the gradient of vorticity is a double exponential in
time. There has been much research on whether this bound is sharp, but
until recently the gap between the best infinite time growth example and double
exponential remained huge. I will provide a review of the subject; then I will discuss a recent construction which gives a solution for 2D Euler in a disk where the growth in vorticity gradient is indeed a double exponential in time. This proves sharpness of the upper bound and illustrates an important role of boundaries in creation of small scales.
Strichartz inequality for orthonormal functions, Rupert Frank (Caltech)
We prove a Strichartz inequality for a system of orthonormal functions,
with an optimal behavior of the constant in the limit of a large number of
functions. The estimate generalizes the usual Strichartz inequality, in
the same fashion as the Lieb-Thirring inequality generalizes the Sobolev
inequality. As an application, we consider the Schr\"odinger equation in a
time-dependent potential and we show the existence of the wave operator in
Schatten spaces.
The talk is based on joint work with M. Lewin, E. Lieb and R. Seiringer.
Magnetic Vortices, Nielsen-Olesen - Nambu strings and theta functions, Israel Michael Sigal (University of Toronto)
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 connection on a principal bundle and the section of the associated vector bundle. 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.
Spectral methods for Quantum walks, Alberto Gruenbaum (UC Berkeley)
Quantum walks can be studied with different methods. In the case
of non-constant coins, or even in the case with no coins, the spectral
method that goes back to Schur, Szego, Riesz, etc. gives one way
to study relevant physical quantities.
Existence and stability of traveling pulse solutions for the FitzHugh-Nagumo equation, Hans Koch (University of Texas at Austin)
The FitzHugh-Nagumo model describes the propagation of electrical signals
in biological tissues.
It is a nonlinear reaction-diffusion equation, coupled to an ODE.
Of particular interest are traveling pulse solutions.
For large diffusion constant,
the flow factors approximately into a slow and a fast part.
In this case, existence and stability of the pulse
was proved in the 1970s and 80s.
The results presented here cover "realistic" parameter values.
We prove existence and stability for pulse solutions
both on the circle and the real line.
The non-compact case, stability is proved
by estimating the associated Evans function.
Due to the non-perturbative nature of the problem,
we had to enlist another collaborator: a computer.
Since this approach has gained significant popularity
in the past few years, we will also describe some of the currently
used techniques in computer-assisted proofs.
(Joint work with Gianni Arioli.)
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