The Analysis Seminar

Spring 2013

The seminar meets Wednesdays in room LN 2205 at 3:30 p.m. There are refreshments and snacks in the Anderson Reading Room at 3:15.

The seminar is partly funded as one of Dean's Speaker Series in Harpur College (College of Arts and Sciences) at Binghamton University.

February 27th: Dong Li (IAS & University of British Columbia)
Title: Global H^1 solutions to a class of logarithmically regularized 2D Euler equations
Abstract: I will explain some recent work on constructing global H^1 solutions to a class of logarithmically regularized 2D Euler equations in vorticity form.

March 13th: Dan Geba (Rochester)
Title: Ill-posedness results for generalized Boussinesq equations
Abstract: In this talk I will focus on recent work addressing the Cauchy problem for a family of generalized Boussinesq equations, which appear as models in areas such as the study of shape-memory alloys or nonlinear dielectrics. I will show that the associated flow map is not smooth for a range of Sobolev indices, thus providing a threshold for the regularity needed to perform a Picard iteration for these problems. This is joint work with A. Alexandrou Himonas and David Karapetyan.

March 20th: Longzhi Lin (Rutgers)
Title: Uniformity of harmonic map heat flow at infinite time
Abstract: We will discuss an energy convexity along the harmonic map heat flow with small initial energy and fixed boundary data on the unit 2-disk. In particular, this gives an affirmative answer to a question raised by W. Minicozzi asking whether such harmonic map heat flow converges uniformly in time strongly in the W^{1,2}-topology, as time goes to infinity, to the unique limiting harmonic map.

April 17th : Xiangwen Zhang (Columbia)
Title: ABP-Type Estimate on Riemannian Manifolds and its Applications
Abstract: The classical Alexandrov-Bakelman-Pucci (ABP) estimate is essential in the regularity theory for second order fully non-linear elliptic equations. In a joint work with Y.Wang, we extend this estimate to general Riemannian manifolds with Ricci curvature bounded from below. Then the Harnack inequalities for non-linear PDE follow from this ABP type measure estimate via the method of Krylov-Safanov. In particular, this gives a non-divergent proof for the Harnack inequality of Laplacian equations established by Yau. In the second part of the talk, we will discuss how to use this new ABP to approach some special cases of the classical geometric inequalities: the isoperimetric and Minkowski inequality.

April 24th--Dean's Speaker Series : Robert Jerrard (Toronto)
Title: Geometric measure theory approaches to Hamiltonian evolution problems
Abstract: The field of geometric measure theory has developed a variety of tools that have proved very useful for the study of certain geometric problems of elliptic and parabolic type, with the canonical examples being the minimal surface problem and its parabolic counterpart, motion by mean curvature. We investigate, by considering a couple of simple examples, whether these tools, or at least this perspective, can be at all useful in studying geometric evolution problems of Hamiltonian type, including Schroedinger and hyperbolic analogs of the minimal surface problem.

April 25th--Dean's Speaker Series (joint w/ Colloquium) SPECIAL DAY THURSDAY and TIME 4:30pm : Robert Jerrard (Toronto)
Title: Weak solutions of an equation describing vortex filaments
Abstract: An old conjecture, dating back to the early 20th century, holds that vortex filaments in ideal fluids in certain limits can be described by a geometric evolution equation called the binormal curvature flow. Smooth solutions of the binormal curvature flow are mostly well-understood. However, it is interesting and possibly useful to study rough solutions as well, for a couple of (probably orthogonal) reasons: because non-smooth vortex filaments may occur in physical fluids, and because some rough solutions of the binormal curvature flow are conjectured to exhibit very rich, peculiar (and probably nonphysical) behavior with surprising number-theoretic properties. This talk will describe the history of some of these conjectures, which remain completely open, as well as a recent proposal for a notion of very weak solutions of the binormal curvature flow which reveals some previously unexpected stability properties.

May 1st--Dean's Speaker Series (joint w/ Colloquium) SPECIAL TIME 4:40pm : Allan Greenleaf (Rochester)
Title: Is there a general theory of Fourier integral operators?
Abstract: Fourier integral operators (FIOs) are fundamental tools in the analysis of linear partial differential equations. FIOs can, as in Egorov's theorem, be used to conjugate partial differential operators to normal forms, which are then easier to analyze, and they are natural objects for studying the spectral geometry of Riemannian manifolds . These early applications of FIOs have since broadened, first to linear inverse problems, such as the Radon transform and its variants, and more recently to linearizations of nonlinear inverse problems, such as in exploration seismology. However, these applications often lead to situations where the assumptions of the standard FIO calculus are violated. This talk will describe the geometry, formulated in terms of singularity theory, behind these difficulties and assess the prospects for having a satisfactory general theory.

May 2nd--Dean's Speaker Series: SPECIAL DAY THURSDAY and TIME 2:50pm : Allan Greenleaf (Rochester)
Title: Microlocal analysis of some problems in seismic and radar imaging
Abstract: This talk will cover in more detail the linearized inverse problems of exploration seismology and synthetic aperture radar that lead to compositions of FIOs outside of the standard calculus. As in all imaging problems, one wants to create an image of the` scene' of interest that contains the primary features of the scene without introducing extraneous artifacts. To locate, describe and suppress artifacts, one needs good understanding of both the `forward' imaging operator and the corresponding `normal' operator. I will describe how, for some frequently encountered data acquisition geometries, it is possible, by proving new results concerning the compositions of FIOs, to give a precise analysis of the artifacts.