Abstracts for some talks (Fall Semester 2011)
Tuesday, September 13, 4:00 p.m., VV B139.
Roman Shvydkoy (UIC)
Convex integration for a class of active scalar equations.
The method of convex integration takes it origin in the classical
C^1-immersion theory of Nash and Kuiper. Largely extended in subsequent
years, it has been applied to various variational problems and more recently
to equations of incompressible fluids to construct examples of "wild
solutions" that behave counter-intuitively. In this talk we will explore to
which extent this method can be adopted to a general class of
non-dissipative active scalar equations including 2D or 3D porous media
equations, equations arising in magnetostrophic turbulence in the Earth’s
fluid core, etc, in each case demonstrating non-uniqueness of solutions in
the class of bounded scalars. We will discuss some limitations of the method
as well by showing, for example, its inapplicability to the classical
surface quasi-geostrophic equations.
Tuesday, September 20, 4:00 p.m., VV B139.
Luis Silvestre (University of Chicago)
Title: On drift-diffusion equations with fractional diffusion.
Abstract: We prove a new Holder estimate for parabolic equations with a
bounded drift and fractional diffusion. There is no assumption on the
divergence on the drift. This estimate allows us to study the regularity of
solutions to some nonlinear problems like the Hamilton-Jacobi equation
or scalar conservation laws with critical fractional diffusion
Tuesday, September 27, 4:00 p.m., VV B139.
Brian Street (UW Madison)
Title: Multi-parameter singular integrals
Abstract: The Calder\'on-Zygmund theory of singular integrals associates to a family of "balls" B(x,delta) a notion of a singular integral
operator. This talk concerns the question: what is a good notion of a singular integral operator associated to balls with multiple radius
parameters, B(x,delta_1,...,delta_nu)? We present an answer in the special case that the balls are Carnot-Carath\'eodory balls. These singular
integrals have applications to some partial differential operators which are defined by vector fields.
Tuesday, October 4, 4:00 p.m., VV B139.
Jiri Lebl (UW Madison)
Title: Bounding the rank of Hermitian forms and rigidity for CR
mappings of hyperquadrics
Abstract:
This is joint work with Dusty Grundmeier and Liz Vivas. We prove that
the rank of a Hermitian form on the space of holomorphic polynomials
can be bounded by a constant depending only on the maximum rank of the
form restricted to affine manifolds. As an application we prove a
generalization of the Baouendi-Huang and Baouendi-Ebenfelt-Huang
rigidity theorems for CR mappings between hyperquadrics. If we have a
real-analytic CR mapping of a hyperquadric not equivalent to a sphere
to another hyperquadric $Q(A,B)$, then either the image of the mapping
is contained in a complex affine subspace or $A$ is bounded by a
constant depending only on $B$. Finally, we will prove a stability
result about existence of non-trivial CR mappings of hyperquadrics.
That is, as long as both $A$ and $B$ are sufficiently large and
comparable, then there exist CR mappings whose image is not contained
in a hyperplane.
Tuesday, October 11, 4:00 p.m., VV B139.
Richard Oberlin (LSU)
Title: On some estimates in harmonic analysis motivated by the return times problem.
Abstract: The return times problem, from ergodic theory, asks
whether certain sequences given by orbits of a function are "good weights"
for ergodic averages and Hilbert transforms. This is related to the
question of boundedness for associated maximal operators in harmonic analysis.
Tuesday, October 18, 4:00 p.m., VV B139.
Wei Min Wang (Orsay)
Spectral gap and PDE
Abstract: We develop a new geometric L2 method for nonlinear PDE
with applications to the energy supercritical NLS, NLW
and other equations.
Tuesday, October 25, 4:00 p.m., VV B139.
Alexander Fish (UW Madison)
Geometric properties of Intersection Body Operator.
Abstract: The notion of an intersection body of a star body was introduced
by E. Lutwak: K is called the intersection body of L if the radial function
of K in every direction is equal to the (d-1)-dimensional volume of the
central hyperplane section of L perpendicular to this direction.
The notion turned out to be quite interesting and useful in Convex Geometry
and Geometric tomography. It is easy to see that the intersection body of a
ball is again a ball. E.Lutwak asked if there is any other star-shaped body
that satisfies this property. We will present a solution to a local version
of this problem: if a convex (or star) body K is closed to a unit ball and
the intersection body of K is equal to K, then K is a unit ball.
Based on a joint work with Nazarov, Ryabogin and Zvavitch.
Tuesday, November 1, 4:00 p.m., VV B139.
Dmitri Bilyk (Minnesota)
Harmonic analysis methods in discrepancy theory and related problems.
Abstract:
Discrepancy theory is primarily concerned with approximations of
uniform distributions by discrete ones and estimating the errors that
necessarily arise in such approximations. Harmonic analysis methods
(such as Fourier transform, Fourier series, Riesz products, wavelets,
and, more recently, Littlewood-Paley theory) have traditionally played
a pivotal role in the subject. We shall discuss some classical results
and conjectures, current progress, and connections to other fields,
such as probability and approximation theory.
Tuesday, November 8, 4:00 p.m., VV B139.
Aynur Bulut (University of Texas)
The defocusing cubic nonlinear wave equation in the energy-supercritical
regime
Abstract:
In this talk, we will present some recent results in the study of the
nonlinear wave equation with cubic defocusing nonlinearity,
describing the completion of a program to
establish global well-posedness and scattering in the energy-supercritical
regime under an assumed a priori uniform-in-time control of the critical norm.
In particular, we discuss our recent results in which the
concentration-compactness approach of Kenig and
Merle is combined with sophisticated tools from harmonic analysis to yield
insight in this class of problems.
Tuesday, November 15, 4:00 p.m., VV B139.
Dean Baskin (Northwestern University)
Radiation fields and the critical semlinear wave equation on R^3
Abstract: Radiation fields are rescaled restrictions to null infinity of
solutions to the wave equation. They provide a concrete realization of the
translation representations of Lax and Phillips and can be thought of as a
generalization of the Radon transform. We consider the radiation fields for
the critical defocusing semilinear wave equation \Box u + f(u) = 0 in R^3,
where f(u) is odd, smooth, arises as the derivative of a potential energy,
and "behaves like u^5". We show that if the radiation field of a finite
energy solution u is smooth, radial, supported in |s|\leq R, and has mean
zero, then the initial data of u are smooth, radial, and supported in |z|
\leq R. Time permitting, I will discuss the problem of recovering f from
the nonlinear scattering matrix associated to the radiation fields. This is
joint work with Antonio Sa Barreto.
Monday, November 21, 4:00 p.m., VV B139
Betsy Stovall (UCLA)
Scattering for the cubic Klein--Gordon equation in two dimensions
We will discuss recent work concerning the cubic Klein--Gordon equation
u_{tt} - \Delta u + u \pm u^3 = 0
in two space dimensions with real valued initial data in the energy space,
u(0) \in H^1, u_t(0) \in L^2. We show that in the defocusing case, solutions
are global and have finite L^4 norm (in space and time). In the focusing case,
we characterize the dichotomy between such behavior and finite time
blowup for initial data having energy less than that of the ground state.
In this talk, we will pay particular attention to connections with certain
questions arising in harmonic analysis.
This is joint work with Rowan Killip and Monica Visan.
Monday, November 28, 4:00 p.m., VV B239
Colloquium
Burglind Joericke (Institut Fourier, Grenoble)
Title: Analytic knots, satellites and the 4-ball genus
After introducing classical geometric knot invariants and satellites
I will concentrate on knots or links in the unit sphere in
$\mathbb C^2$ which bound a complex curve (respectively, a smooth complex
curve) in the unit ball. Such a knot or link will be called analytic
(respectively, smoothly analytic). For analytic satellite links of
smoothly analytic knots there is a sharp lower bound for the 4-ball
genus. It is given in terms of the 4-ball genus of the companion and
the winding number. No such estimate is true in the general case.
There is a natural relation to the theory of holomorphic mappings
from open Riemann surfaces into the space of monic polynomials
without multiple zeros. I will briefly touch related problems.
Tuesday, November 29, 4:00 p.m., VV B139.
Thomas Hangelbroek (Vanderbilt University)
Zeros estimates and kernel approximation
Abstract at:
http://www.math.wisc.edu/~seeger/abstracts/hangelbroek-11.pdf
Tuesday, December 13, 4:00 p.m., VV B139.
Hart Smith (University of Washington)