Abstracts for some talks (Spring Semester 2012)
Tuesday, January 31, 4:00 p.m., VV B139.
Alexandru Ionescu (Princeton University)
On the global stability of the constant equilibrium solution of the
Euler-Poisson system
Abstract: The Euler-Poisson system is a model used to
describe the dynamics of a two-fluid plasma. I will discuss
some recent work on the question of
global existence of solutions of this system, in the case of small
perturbations of a constant background.
Friday, February 3, 2012, 4:00 p.m., VV B239.
Colloquium
Akos Magyar (University of British Columbia)
On prime solutons to linear and quadratic equations
Abstract: The classical results of Vinogradov and Hua establishes prime
solutions of linear and diagonal quadratic equations in sufficiently many
variables. In the linear case there has been a remarkable progress
over the past few years by introducing ideas from additive
combinatorics. We will discuss some of the key ideas, as well
as their use to obtain multidimensional extensions of the theorem
of Green and Tao on arithmetic progressions in the primes. We
will also discuss some new results on prime solutions to
non-diagonal quadratic equations of sufficiently large rank.
Most of this is joint work with B. Cook.
Joint PDE/Analysis seminar
Monday, February 6, B115, 3:30 p.m.
Yao Yao (UCLA)
Degenerate diffusion with nonlocal aggregation: behavior of solutions
Abstract:
Thee Patlak-Keller-Segel (PKS) equation models the collective motion of cells
which are attracted by a self-emitted chemical substance. While the global
well-posedness and finite-time blow up criteria are well known, the asymptotic
behavior of solutions is not completely clear. In this talk I will present
some results on the asymptotic behavior of solutions when there is global existence. The key tools used in the paper are maximum-principle type arguments
as well as estimates on mass concentration of solutions. This is a joint
work with Inwon Kim.
Tuesday, February 21, 4:00 p.m., VV B139.
Malabika Pramanik (University of British Columbia)
Maximal operators and differentiation on sparse sets
Friday, February 24, 2012
Colloquium
Malabika Pramanik (University of British Columbia)
Analysis on sparse sets
http://www.math.wisc.edu/~seeger/abstracts/malabika2012.pdf
Tuesday, February 28, 4:00 p.m., VV B139.
Dmitri Ryabogin (Kent State University)
On problems of Bonnesen and Klee.
Abstract: This is joint work with Fedor Nazarov and Artem Zvavitch.
We will discuss some results related to the questions of Bonnesen about unique determination
of convex (non-symmetric) bodies given the volumes of their
maximal sections and projections.
We will show that if $d\ge 4$ is even, then one can find two
essentially different convex bodies such that the volumes of their maximal
sections, central sections, and projections coincide for all
directions.
Tuesday, March 6, 2012, 4:00 p.m., VV B139.
Han Peters (KdVI, University of Amsterdam)
Cusps and Complex Dynamics
Sufficiently small perturbations of singular holomorphic mappings from
one to two complex dimensions have the property that their images
intersect the original image. After discussing how to prove this, we
show how this result can be used to obtain new results in two
dimensional complex dynamics.
This is joint work with Misha Lyubich.
Friday, March 16, 2012, 4:00 p.m., VV B239.
Colloquium
Burak Erdogan (University of Illinois - Urbana-Champaign)
Smoothing for the KdV equation and Zakharov system on the torus
Tuesday, March 20, 4:00 p.m., VV B139.
Stephen Wainger (UW)
Discrete singular operators on discrete subgroups of nilpotent Lie groups
Monday, March 26, 4:00 p.m., VV B115, PDE seminar.
Vlad Vicol (University of Chicago)
Shape dependent maximum principles and applications
Abstract: We present a non-linear lower bound for the fractional
Laplacian, when evaluated at extrema of a function. Applications to
the global well-posedness of active scalar equations arising in
fluid dynamics are discussed. This is joint work with P. Constantin.
Friday, March 30, 2012, 4:00 p.m., VV B239.
Colloquium
Wilhelm Schlag (University of Chicago)
Invariant manifolds and dispersive Hamiltonian equations
Abstract: We will review recent work on the role that center-stable
manifolds play in the study of dispersive unstable evolution equations.
More precisely, by means
of the radial cubic nonlinear Klein-Gordon equation we shall exhibit a
mechanism in
which the ground state soliton generates a center-stable manifold which
separates a region
of data leading to finite time blowup from another where solutions
scatter to a free wave
in forward time. This is joint work with Kenji Nakanishi from Kyoto
University, Japan.
Monday, April 9, 3:30 p.m., VV B115
Joint PDE/Analysis Seminar
Charles Smart (MIT)
Title: PDE methods for the Abelian sandpile
Abstract: The Abelian sandpile growth model is a deterministic
diffusion process for chips placed on the $d$-dimensional integer
lattice. One of the most striking features of the sandpile is that it
appears to produce terminal configurations converging to a peculiar
lattice. One of the most striking features of the sandpile is that it
appears to produce terminal configurations converging to a peculiar
fractal limit when begun from increasingly large stacks of chips at
the origin. This behavior defied explanation for many years until
viscosity solution theory offered a new perspective. This is joint
work with Lionel Levine and Wesley Pegden.
Monday, April 16, 3:30 p.m., VV B115.
PDE Seminar
Jiahong Wu (Oklahoma State)
The 2D Boussinesq equations with partial dissipation
Abstract:
The Boussinesq equations concerned here model geophysical flows such as
atmospheric fronts and ocean circulations. Mathematically the 2D Boussinesq
equations serve as a lower-dimensional model of the 3D hydrodynamics equations. In fact, the 2D Boussinesq equations retain some key features of the
3D Euler and the Navier-Stokes equations such as the vortex stretching
mechanism. The global regularity problem on the 2D Boussinesq equations
with partial dissipation has attracted considerable attention in the last
few years. In this talk we will summarize recent results on various cases
of partial dissipation, present the work of Cao and Wu on the 2D Boussinesq
equations with vertical dissipation and vertical thermal diffusion, and
explain the work of Chae and Wu on the critical Boussinesq equations
with a logarithmically singular velocity.
Tuesday, April 17, 4:00 p.m., VV B139.
Ilya Kossovskiy (University of Western Ontario)
Analytic Continuation of Holomorphic Mappings
From Non-Minimal Hypersurfaces
Abstract:
The classical result of H. Poincare states that a local
biholomorphic mapping from an open piece of the 3-sphere in
$\mathbb{C}^2$ onto another open piece extends analytically to a
global holomorphic automorphism of the sphere. This theorem was
generalized by H. Alexander to the case of a sphere in $\mathbb{C}^n$ ($n\geq 2$),
then later by S. Pinchuk for the case of
strictly pseudoconvex hypersurface in the preimage and a sphere in
the image, and finally by R. Shafikov and D. Hill for the case of an
essentially finite hypersurface in the preimage and a quadric in
the image. In this joint work with R. Shafikov we consider the -
essentially new - case when a hypersurface $M$ in the
preimage contains a complex hypersurface. We demonstrate that the above
extension results fail in this case, and prove the following
analytic continuation phenomenon: a local biholomorphic mapping from
$M$ onto a non-degenerate hyperquadric in $\mathbb{CP}^n$ extends
to a punctured neighborhood of the complex hypersurface,
as a multiple-valued local biholomorphic mapping.
We also establish an interesting interaction between non-minimal
spherical real hypersurfaces and linear differential equations
with an isolated singular point.
Tuesday, May 8, 4:00 p.m., VV B139.
Sergey Denisov
Analysis of the instability in two-dimensional fluids
Abstract:
For the two-dimensional Euler equation describing incompressible nonviscous
fluid, we obtain lower bounds for the growth of Sobolev norms. We also
explain the merging mechanism for the dynamics of centrally symmetric vortex
patches.