Analysis Seminar Fall 2021

    Abstract: The first of a series of expository lectures aimed at graduate students.
The goal of this series of lectures is to survey some of the classical results of Lewy, Nirenberg, Hormander, and Treves and others on local (non)solvability and to give an informal introduction to pseudodifferential operators, parametrices, and propagation of singularities.

    Abstract: This week we will talk about Malgrange's proof of the Newlander-Nirenberg theorem on recognizing/characterizing when a system of PDEs is the Cauchy-Riemann equations in disguise (that is, when we can make a smooth local change of coordinates so that our system of PDEs becomes the Cauchy-Riemann equations).

    Abstract: This week we will talk about Malgrange's proof of the Newlander-Nirenberg theorem characterizing when we have the Cauchy-Riemann equations written in different coordinates. (Last time we talked about the classical one complex variable case and then the real analytic case in higher dimensions. This time we will prove the full result.)

    Abstract: In this talk, I will discuss how to compute the cohomology of a left-invariant CR structure on a compact Lie group. This is a generalization of a result of Pittie. Pittie proved that the Dolbeault cohomology of all left-invariant complex structures on compact Lie groups can be computed by looking at the Dolbeault cohomology induced on a conveniently chosen maximal torus. We use the algebraic classification of left-invariant CR structures of maximal rank on compact Lie groups by Charbonnel and Khalgui to generalize Pittie’s result to left-invariant Levi-flat CR structures of maximal rank on compact Lie groups. This is a joint work with Howard Jacobowitz from Rutgers University.

    Abstract: Classical volume comparison for Ricci curvature is a fundamental result in Riemannian geometry. In general, scalar curvature as the trace of Ricci curvature, is too weak to control the volume. However, with the additional stability assumption on the closed Einstein manifold, one can obtain a volume comparison for scalar curvature. In this talk, we investigate a similar phenomenon for Q-curvature, a fourth-order analogue of scalar curvature. In particular, we prove a volume comparison result of Q-curvature for metrics near strictly stable Einstein metrics by variational techniques and a Morse lemma for infinite dimensional manifolds. This is a joint work with Wei Yuan.

    Abstract: The geometry of conformally-embedded hypersurfaces in general compact manifolds is important for the study of boundary value PDEs. As part of a larger program studying the conformal geometry of such embeddings, I will construct a sequence of conformally invariant tensors that generalize the second fundamental form. Like the second fundamental form, these tensors encode the local extrinsic curvatures of such an embedding. In particular, our main result shows that these tensors characterize the failure of a conformally compact manifold to have an asymptotic Poincare-Einstein structure. The frequency with which these tensors appear in the calculus of conformally-embedded hypersurfaces suggests a deeper picture which is yet to be fully understood.

    Abstract: In this talk, we will introduce a new approach to study the behavior of the (weighted) composition operators acting on Bergman spaces via sparse dominations. New criteria of boundedness and compactness will be characterized. Moreover, we will also derive a new type of weighted estimates associated to the (weighted) composition operators. This talk is based several joint works with Brett Wick, Songxiao Li, Zhenghui Huo and Yecheng Shi.

    Abstract: Asymptotically locally hyperbolic (ALH) manifolds are a class of manifolds whose sectional curvature converges to −1 at infinity. If a given ALH manifold is asymptotic to a static reference manifold, the Wang-Chruściel-Herzlich mass integrals are well-defined for it, which is a geometric invariant that essentially measure the difference from the reference manifold. In this talk, I will present a recent result with L. -H. Huang, which characterizes ALH manifolds that minimize the mass integrals. The proof uses scalar curvature deformation results for ALH manifolds with nonempty compact boundary. Specifically, we show the scalar curvature map is locally surjective among either (1) the space of ALH metrics that coincide exponentially toward the boundary or (2) the space of ALH metrics with arbitrarily prescribed nearby Bartnik boundary data. As a direct consequence, we establish the rigidity of the known positive mass theorems.

    Abstract: I will give a brief introduction to symplectic and spectral geometry of finite dimensional integrable Hamiltonian systems, with an emphasis on recent results about systems of toric and semitoric type.