Southern California Analysis
and PDE Conference
Registration and local information
We will need an approximate head count, so unless you prefer to stand during the talks, please either arrive very early or register here. Seating will be reserved for all who register.
Saturday, December 3, 2011, Natural Sciences 2, 1201 (Not the math building!)
9:30am-10:25am Assaf Naor (Courant) - Grothendieck's inequality and the propeller conjecture.
10:30am-11:25am Abel Klein (UCI) - Bounds on the density of states for Schrodinger operators.
11:30am- 12:10pm Helge Krueger (Caltech) - Skew-shift CMV matrices.
12.10pm -1:45pm Lunch Break
1:45pm-2.40pm Monica Visan (UCLA) - The energy-supercritical nonlinear wave equation
2:45pm-3.45pm Yuval Peres (Microsoft Research) - Tug-of-war and infinity Laplace equation with vanishing Neumann boundary condition
3.45pm-4.15pm Coffee Break
4.15pm-4.50pm Christoph Marx (UCI) - Analytic quasi-periodic Schroedinger operators and rational frequency approximants
4:55pm-5.50pm Leonid Parnovski (University College London) - Integrated density of states of Schroedinger operators with periodic or almost-periodic potentials
The information about local hotels with UCI discounts can be obtained here
For maps, and other travel information click on the "Campus Map" link to the left or above. The parking lot closest to NS2 (building 402 on the map) is 12A .
Driving directions and parking:
From the north: Take I-405 to 73 south, and exit at Bison (there is no toll, despite the confusing signs).
From the south: Take I-5 to 73 north, and exit 73 at Bison (this part of 73 is a toll road).
Take Bison east into campus. Continue straight ahead to lot 16. You can buy a parking permit from the machine at the entrance to lot 16. Park in lot 16, 12A, or 12B.
To park on campus you will need to purchase a parking permit. Parking may cost as much as $10 for the full day. You can buy a permit from one of the dispensers shown on the map. Most of the kiosks take credit cards. Quarterly or annual parking permits from other UC campuses are honored at UCI.
For students and postdocs who are on the market this year or next:
You are invited to bring your preprints and participate in our professionalization workshop. You can get advice from the experts on writing of cover letters, research and teaching statements, preparing resumes and take part in a mock interview. Contact szhitomi@uci.edu for details.
Funding
The SCAPDE is supported by a grant from the National Science Foundation
Funding is available for graduate students and postdocs who do not have NSF or other grant or school support. People interested in applying for travel funds should check the appropriate box in the electronic registration form and send an e-mail request to Jacob Sterbenz <jsterben@math.ucsd.edu>.
The deadline for sending in requests for travel support is November 24.
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Local organizers:
A. Farhat, S. Jitomirskaya, R. Mavi, W. Yessen
Global organizer: J. Sterbenz
Past meetings:
Spring 2011, UCSD
Fall 2010, UCLA
Winter 2010, UC San Diego
Fall 2009, UC Irvine
Winter 2009, UCLA
Winter 2008, UC San Diego
Fall 2002, UCLA
Spring 2000,UC San Diego
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Abstracts
A. Klein Bounds on the density of states for Schrodinger operators
I will describe bounds on the density of states of Schrodinger operators. These are deterministic results that do not require the existence of the integrated density
of states. The results are stated in terms of a "density of states outer-measure", which always exists. We prove log-Holder continuity for the density of states
in one, two, and three dimensions for continuous Schrodinger operators ., and in any dimension for discrete Schrodinger operators. In particular, we recover the
Craig-Simon result of log-Holder continuity of the the integrated density of states for discrete and one-dimensional continuous ergodic Schrodinger operators, and
prove log-Holder continuity of the the integrated density of states for two and three dimensional continuous ergodic Schrodinger operators. The proof for the two
and three dimensional cases relies on estimates of the local behavior of approximate solutions of the stationary Schr\" odinger equation, and on the
quantitative unique continuation principle for the same equation. (Joint work with J. Bourgain)
H. Krueger Skew-shift CMV matrices.
CMV matrices are a natural unitary analog to
Schroedinger operators. I will be concerned with the
family of CMV matrices with Verblunsky coefficients,
i.e. the analog of the potential, given by
alpha_n = lambda exp(2 pi i omega n^k)
for 0 < |lambda| < 1, omega irrational, and
k \geq 2 an integer. I will show in this case that
the spectrum is the entire unit circle and the Lyapunov
exponent is positive. Furthermore, I will discuss the
eigenvalue statistics in the case k = 2 and present
numerical evidence what the right answer should
be for k \geq 3.
C. Marx Analytic quasi-periodic Schroedinger operators and rational
frequency approximants
Consider a quasi-periodic Schroedinger operator with analytic potential and irrational frequency. We show that up to sets of zero Lebesgue measure, the absolutely continuous spectrum can be obtained asymptotically from the intersections over the phase of the spectra of periodic operators associated with the continued fraction expansion of the frequency. Similarly, from the asymptotics of the unions over the phase of such spectra, one recovers the spectrum of the quasiperiodic operator.
A. Naor Grothendieck's inequality and the propeller conjecture.
We will start with a description of the classical Grothendieck
inequality, and the corresponding Grothendieck constant. We will then
indicate a few of the numerous applications of the Grothendieck
inequality, and review the work on the problem of estimating the
Grothendieck constant. In particular, we will describe the 1977
Krivine conjecture on the exact value of the Grothendieck constant,
and a related conjecture of Koenig, on maximizers of a certain oscillatory
kernel, that was made as a step towards proving the Krivine conjecture.
The main new result that we will discuss is the solution of Krivine's
conjecture (joint with M. Braverman, K. Makarychev and Y. Makarychev).
While Krivine asked for a specific counter example showing that his bound
is sharp, we will obtain an improved Grothendieck inequality for all
matrices, showing that the Grothendieck constant is strictly smaller than
Krivine's bound. The new analytic contribution here is establishing the
usefulness of higher dimensional rounding schemes, and we will discuss
their impact on approximation algorithms. We will also discuss recent
works (one joint with Khot and the other joint with Heilman and Jagannath)
on a useful extension of the Grothendieck inequality that relates to an
isoperimetric problem called the propeller conjecture; this conjecture has
been recently proved in R^3.
L. Parnovski Integrated density of states of Schroedinger operators with periodic or almost-periodic potentials
I will discuss the asymptotic behaviour of the integrated density
of states of Schroedinger operators with periodic or almost-periodic
potentials when the value of the spectral parameter is large. I will outline
the proof of the existence of the complete asymptotic expansion of the IDS
when the potential is either smooth periodic, or generic quasi-periodic
(finite liner combination of exponentials), or belongs to a large class of
almost-periodic functions. This is a joint result with R.Shterenberg.
Y. Peres Tug-of-war and infinity Laplace equation with vanishing Neumann boundary condition
Abstract: We study a version of the stochastic "tug-of-war" game, played on graphs and smooth domains, with an empty set of terminal states. We prove that, when the
running payoff function is shifted by an appropriate constant, the values of the game after n steps converge. Using this we prove the existence of solutions to the
infinity Laplace equation with vanishing Neumann boundary condition. In earlier work with Schramm, Sheffield and Wilson (http://arxiv.org/abs/math/0605002, JAMS
2009), we related a tug of war game to the infinity Laplacian equation with Dirichlet boundary conditions- I will survey that work as well as the version for the
p-Laplacian in http://arxiv.org/abs/math/0607761
(Talk based on joint work with Tonæi Antunoviæ, Scott Sheffield, Stephanie Somersille http://arxiv.org/abs/1109.4918 to appear in Comm. PDE)
T. Tao Localization and compactness properties of the Navier-Stokes
global regularity problem (talk cancelled)
Abstract: Using some new localized energy and enstrophy inequalities
for the three-dimensional Navier-Stokes problem, we establish some
equivalences between the periodic and non-periodic versions of these
problems (at least if one allows a smooth forcing term). We also show
that (due to a Galilean invariance) the homogeneous and inhomogeneous
periodic Navier-Stokes regularity problems are equivalent, and (due to
compactness properties of the flow) various qualitative and
quantitative formulations of the problem are also equivalent. While
these equivalences do not actually establish any definitive answer to
the Navier-Stokes regularity problem in any of its formulations, they
suggest that the precise form of the problem is not of primary
importance.
M. Visan The energy-supercritical nonlinear wave equation
We discuss global well-posedness and scattering for the
energy-supercritical NLW in three space dimensions. This is joint
work with Rowan Killip.
