Schedule of talks
upcoming Talks Spring 2023
February 9
Andy Raich
Title: Analysis and Geometry for the $\bar\partial$-problem in Several Complex Variables
Abstract: In this *introductory* talk, I will review the concepts from analysis and geometry that I will use in the subsequent seminar. This talk is geared for graduate students and anyone who is interested in how these topics come together in the $\bar\partial$-Neumann problem. Time permitting, I will define/introduce the following topics: holomorphic functions, differential forms, $L^2$-spaces, Hilbert spaces, unbounded operators on Hilbert spaces and their adjoints, the $\bar\partial$-problem, the basic identity and estimate, pseudoconvexity, coercive estimates, and ellipticity. In essence, I want to review the concepts and terminology that I will second in my second talk on maximal estimates in the $\bar\partial$-problem.
February 16
Andy Raich
Title: Maximal Estimates for the $\bar\partial$-Neumann Problem on Non-pseudoconvex domains
Abstract: It is well known that elliptic estimates fail for the $\bar\partial$-Neumann problem. Instead, the best that one can hope for is that derivatives in every direction but one can be estimated by the associated Dirichlet form, and when this happens, we say that the $\bar\partial$-Neumann problem satisfies maximal estimates. In the pseudoconvex case, a necessary and sufficient geometric condition for maximal estimates has been derived by Derridj (for $(0,1)$-forms) and Ben Moussa (for $(0,q)$-forms when $q\geq 1$).
In this talk, I will discuss necessary conditions and sufficient conditions for maximal estimates in the non-pseudoconvex case. Additionally, I will talk about when the necessary conditions and sufficient conditions agree and provide examples. Our results subsume the earlier known results from the pseudoconvex case. This work is joint with Phil Harrington of the University of Arkansas.
March 9
Tuoc Phan, University of Tennessee Knoxville
Title: On regularity theory in Sobolev spaces for linear elliptic and parabolic equations with singular or degenerate coefficients
Abstract: We study regularity in Sobolev spaces for a several classes of linear elliptic and parabolic equations in which the coefficients can be singular or degenerate. Under some optimal regularity conditions on the leading coefficients, generic weighted and mixed-normed weighted Sobolev spaces will be derived in which the existence, uniqueness, and regularity estimates of solutions are proved. Main ideas and techniques in the proofs will be highlighted, and future research directions will be mentioned.
April 20
Patrick Phelps Dissertation defense
May 4
Sean Curry, Oklahoma State University
Title: The local CR embedding problem
Abstract: The definition of a Cauchy-Riemann (CR) manifold abstracts the notion of being a real submanifold in \mathbb{C}^n. A natural question is therefore whether a given CR manifold is locally embeddable as such a real submanifold. We discuss background and recent work in progress on this problem for hypersurface-type CR manifolds.