Analysis Research Group

Analysis  Seminar (Spring 07) Wednesday 4:30 - 5:20 pm Cardwell 120 (except when otherwise noted)

Organizer: Charles Moore Previous semesters: Spring 06 - Fall 06

Wednesday, February 7 - Chuck Moore Random Riemann Sums Abstract: We discuss a probabilistic approach to integration.This talk will be reasonably elementary and accessible to most graduate students. Wednesday, February 14 - Weidong Chen. An efficient algorithm for band-limited extrapolation by regularization. Wednesday, February 21 - Marianne Korten Stefan problems, the weak jump condition, and Hele-Shaw problems. Wednesday, February 28 - Ryan Berndt Recent developments on the two weight problem for the Fourier Transform Abstract: I will discuss some very recent developments on finding sufficient conditions for the Fourier transform to map weighted Lp into a different weighted Lp. Central to my argument is a pointwise estimate for the Fourier transform in terms of the decreasing rearrangement of the Hardy-Littlewood Maximal function. Thursday, March 8 (at 10:30 a.m. in Cardwell 023) - Patti Bauman, Purdue University    On a variational model for high-temperature superconductors Abstract: We describe an energy functional that models high-temperature (layered) superconductors. The model has both two and three-dimensional features: it contains a three-dimensional vector field (called the magnetic potential) defined on R3, and N+1 order parameters (complex-valued functions) defined on N+1 two-dimensional bounded parallel domains, respectively. The components of these functions satisfy a nonlinear coupled elliptic system of partial differential equations. Supported in part by an ADVANCE Departmental Initiative Grant Wednesday, March 14 - Svetlana Roudenko, Arizona State University Concentration properties for the blow up solutions for the cubic nonlinear Schrodinger equation in 2 and 3D. Abstract: First, I will discuss Bourgain's mass (L2) concentration phenomenon for the cubic nonlinear Schrodinger (NLS) equation in 2D which heavily uses the Fourier restriction theorems. I will focus on the refinement of this technique which links the divergence of Strichartz norms with the size of the concentration window. As a consequence, a logarithmic lower bound on the blow up rate of the relevant Strichartz norm will be obtained. Next, I will consider the focusing cubic NLS in 3D and discuss the L3 concentration phenomenon and consequences related to the blow up dynamics. Wednesday, March 27 - Andrea Bertozzi, UCLA   (at 10:30 a.m. in Cardwell 144) Finite time blowup of solutions of an aggregation equation in Rn Abstract: We consider an aggregation equation in all space dimensions consisting of transport of a nonnegative density u by a vector field that is the gradient of an averaging kernel applied to u. We are interested in biologically relevant kernels which can have Lipschitz points at the origin. In one space dimension this problem is known to produce finite time singularities in which the density goes to infinity. We present a proof of this fact in all space dimensions larger than one. The proof uses potential theory estimates and has a connection to theory of solutions of the Euler equations for incompressible flow. Supported in part by an ADVANCE Departmental Initiative Grant Friday, March 30 - Maria Calle, Courant Institute   (at 10:30 a.m. in Cardwell  143) Non-proper limits of minimal surfaces in 3-manifolds Abstract: Roughly speaking, it is expected that the only two types of singular laminations that can occur as limits of closed embedded minimal surfaces in a 3-manifold with positive scalar curvature are accumulations of catenoids and non-proper helicoid-like limits. T. Colding and C. DeLellis constructed an example of the first type. I will present a construction of the second type: we show that there exists a metric with positive scalar curvature on S2 X  S1 and a sequence of embedded minimal tori that converges to a minimal lamination that, in a neighborhood of a strictly stable 2-sphere, is smooth except at two helicoid-like singularities on the 2-sphere. This is a joint work with Darren Lee.   Supported in part by an ADVANCE Departmental Initiative Grant Wednesday, April 4  - Monica Visan, Institute for Advanced Study The mass-critical nonlinear Schrodinger equation Abstract:  We prove global wellposedness and scattering for the mass-critical NLS at critical regularity in high dimensions and for radial initial data.  This is joint work with Terence Tao and Xiaoyi Zhang. Wednesday, April 11 - Ray Treinen, University of Toledo Floating Drops Abstract: Consider three fluids in equilibrium, ordered by their densities: $\rho_0<\rho_1<\rho_2$.  There is a surface tension associated with each interface between fluids. Matching solutions of ODE's will be discussed as a method of determining the shape of the interfaces, and both analytical and numerical results will be presented.  I will briefly comment on several conjectures that seem plausable given the numerical results. A minimizer of energy is found for a bounded container using the theory of functions of bounded variation.  Then a limiting argument is used to show the existence of a minimizer over an unbounded domain.                          Wednesday, April 18 - Virginia Naibo Mixed norm estimates for the k-plane transform. Abstract: I will start by introducing the k-plane transform, of which the X-ray transform and the Radon transform are particular cases. Such transformations have many applications, in particular in tomography. I will prove sharp mixed norm inequalities for the k-plane transform when acting on radial functions and for potential-like operators supported in k-planes. Wednesday, April 25 - Maria Reguera, University of Missouri The characteristic function of the paraboloid is not a bounded bilinear multiplier Abstract: We consider a bilinear Fourier multiplier operator on the 4-dimensional euclidean space whose symbol is the characteristic function of a certain paraboloid. We use a Kakeya type counterexample to show that such a bilinear operator is unbounded from  Lp X  Lq  to   Lr  in the 2-dimensional euclidean space outside the local   L2  case. Supported in part by an ADVANCE Departmental Initiative Grant Wednesday, May 2 - Diego Maldonado Paraproducts as bilinear Calderon-Zygmund operators Abstract: The core of the talk will be a remark about how paraproducts can be realized as bilinear C-Z operators. Then, we will use the multilinear C-Z theory developed by Grafakos and Torres to prove new bounds, as well as to recover knownones, for these operators in several function spaces. This is joint work with A. Benyi, A. Nahmod, and R. Torres.