The Analysis Seminar

Fall 2012

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.

September 12: Vamsi Pingali (Stony Brook)
Title: A generalised Monge-Ampère equation
Abstract: Monge-Ampère equations play an important role in both real and complex geometry. We shall describe a generalised version of these motivated by Chern-Weil theory and by a conjecture of X.X. Chen (which in turn came from Kähler geometry). We also consider a real "model" of this PDE in a special case so as to study the difficulties involved in making the estimates. In doing so, a connection to an old theorem of Krylov shall be pointed out.

September 19: Xiangjin Xu (Binghamton)
Title: Introduction to Schoen and Yau's book, Lectures on Differential Geometry, Part 5.
Abstract: This continues Ye Li's talks in Spring on discussing Schoen - Yau's book, Lectures on Differential Geometry. In this talk, I will focus on Yau's gradient estimates of heat kernel, Harnack inequality and estimates for heat kernel.

October 3: Xiangjin Xu (Binghamton)
Title: Introduction to Schoen and Yau's book, Lectures on Differential Geometry, Part 6.
Abstract: This continues to discussing Schoen - Yau's book, Lectures on Differential Geometry. In this talk, I will apply Li-Yau type gradient estimates of heat equations to study Harnack inequality and estimates for heat kernel.

October 17: Paul Loya (Binghamton)
Title: Witten's holonomy theorem for manifolds with corners.
Abstract: In the 1980's Daniel Quillen introduced determinant line bundles and about the same time Edward Witten derived a remarkable formula for the holonomy of the determinant line bundle of a Dirac operator using something called the \eta invariant" of Atiyah, Patodi, and Singer. In the physics literature, the holonomy of the determinant line bundle is called the "global anomaly". Witten's derivation was later made rigorous by Bismut and Freed and also by Cheeger. In this talk I will give an introduction to eta invariants and Witten's holonomy theorem, and then I will discuss recent work concerning generalizations of this theorem to situations quite different from the original results.

October 24: Marius Beceanu (IAS)
Title: Solutions to the energy-critical wave equation near of the soliton.
Abstract: We characterize certain solutions of the energy-critical wave equation in a neighborhood of the soliton.

November 14: Chongchun Zeng (Georgia Tech)
Title: Small BGK waves and Landau damping
Abstract: In this talk, we discuss the Landau damping -- the asymptotic stability of the linearly stable homogeneous states of the Vlasov-Possion system. It has been proved that solutions to the system linearized at stable homogeneous states decay algebraically in time. In such a Hamiltonian system, this decay is not caused by any dissipation. The nonlinear asymptotic stability is open until recently when Mouhot and Villani proved it of solutions in the Gevery class. The problem in Sobolev spaces remains open. We show that the nonlinear damping does not happen in Sobolev space with too low regularity by constructing BKG waves -- traveling waves -- arbitrarily close to stable homogeneous states. In the contrary, in Sobolev spaces with higher regularity, we show that there are no invariant structures -- including BGK waves -- near any stable homogeneous states and thus the same obstacle for the damping as in the rough Sobolev spaces does not appear. Similar results have also been proved for the Euler equation near Couette flow. These are joint works with Zhiwu Lin.

November 28: Jason Metcalfe (UNC-Chapel Hill)
Title: Local well-posedness for quasilinear Schrodinger equations with rough data
Abstract: We will discuss recent joint works with J. Marzuola and D. Tataru that focus on local existence for generic quasilinear Schrodinger equations with data in low regularity spaces. The Mizohata integrability conditions plays an essential role in choosing appropriate spaces, and in order to incorporate the necessary decay, we choose Sobolev spaces that are adapted to include a summability over a partition of space into cubes. The primary estimate is a local smoothing estimate, which is adapted to these spaces.

December 5: Raluca Felea (RIT)
Title: Microlocal analysis of SAR with moving objects
Abstract: We consider four particular cases of Synthetic Aperture Radar imaging with moving objects. In each case we analyze the forward operator F which maps the image to the data and the normal operator F*F which is used to recover the image. In general, by applying the backprojection operator F* to the data, artifacts appear in the reconstructed image. We describe these artifacts and show how to microlocally reduce their strength to obtain a better image.