The Analysis Seminar

Fall 2011

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.

Organizer: Paul Loya

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September 14: Paul Loya (Binghamton University)
Title: An introduction to the Atiyah-Singer Index Theorem.
Abstract: In this talk we give a very elementary introduction to the Atiyah-Singer index theorem. The talk should be accessible to first year graduate students and even advanced undergraduates.

September 21: Xianjin Xu (Binghamton University)
Title: Differential Harnack inequalities on Riemannian manifolds.
Abstract: This talk is based on the joint paper with Junfang Li(UAB) (Advances in Mathematics 226 (2011) 4456–4491). In the first part of this paper, we get new Li–Yau type gradient estimates for positive solutions of heat equation on Riemannian manifolds with Ricci(M)>= −k. As applications, several parabolic Harnack inequalities are obtained and they lead to new estimates on heat kernels. In the second part, we establish a Perelman type Li–Yau–Hamilton differential Harnack inequality for heat kernels on manifolds with Ricci(M) >= −k. As applications, we obtain new Harnack inequalities and heat kernel estimates on general manifolds. We also obtain various entropy monotonicity formulas for all compact Riemannian manifolds.

October 5: Andrey Gogolev (Binghamton University)
Title: Cohomological equations and the bootstrap argument.
Abstract: A dynamical system is called elliptic if nearby orbits stay together or diverge apart at a slow rate. A dynamical system is called hyperbolic if nearby orbits diverge from each other exponentially fast. We will give an introduction to cohomological equations over elliptic and hyperbolic systems and discuss applications to smooth classification of hyperbolic (Anosov) diffeomorphisms. The talk will be accessible to graduate students, we will provide the necessary background from dynamics.

October 19: Adam Weisblatt (Binghamton University)
Title: A probabilistic look at the Feynman integral.
Abstract: Path integration was discovered to better understand the comparison between the global structure of quantum and the local structure of classical dynamics. There exist a lot of folklore surrounding the Feynman integral. For instance, it provides a tool to understand solutions of PDE's, the development of quantum finance, and Witten's quantum topology. We will show a derivation of Feynman Kac, and how the path integral makes an appearance. We will then have an outline, at least, of the powerful math needed to understand the existence of Feynman's formula.

October 26: Ye Li (Binghamton University)
Title: Convergence theorems in Riemannian geometry.
Abstract: In this talk we will discuss smoothing Riemannian metrics in dimension 4 and its application in the convergence of compact Riemannian manifolds with bounded Ricci curvature.

November 2: Camil Muscalu (Cornell University)
Title: Triangular Fourier series and physical reality.
Abstract: Every product of n classical Fourier series splits naturally as a sum of n factorial "triangular Fourier series". The goal of the talk is to describe their theory and their natural connections with problems from "physical reality".

November 9: Magdalena Czubak (Binghamton University)
Title: Conservation Laws, Virial Identities and Morawetz estimates for the Schrodinger equations.
Abstract: In this talk we introduce the wellposedness and scattering theory of the Nonlinear Schrodinger equation (NLS). In particular, we discuss local conservation laws, Morawetz estimates and their applications to the scattering theory. In the end, we present new interaction Morawetz estimates for the magnetic NLS. This part of the talk will be based on joint work with J. Colliander (Toronto) and J. Lee (Minnesota). The talk will be accessible to graduate students.