## Mathematical Physics and Harmonic Analysis Seminar

**Date:** September 29, 2017

**Time:** 1:50PM - 2:50PM

**Location:** BLOC 628

**Speaker:** Gerardo Mendoza, Temple University

**Title:** *Free real actions, invariant CR structures, hypoellipticity, and Kodaira's vanishing theorem (joint with Several Complex Variables Seminar)*

**Abstract:** Suppose M is a compact CR manifold with a nowhere vanishing real transverse vector field T that preserves the structure and admits an invariant metric which is Hermitian on the CR structure. Then -iT commutes with the Laplacians of the dee-bar complex and defines a selfadjoint operator on the kernel, H^q, of the Laplacian in each degree q with discrete spectrum without finite points of accumulation. Assuming non-degeneracy of the Levi form, we prove that only finitely many eigenvalues of -iT lie on the positive (or negative, depending on q and the signature of the Levi form) real axis. Finiteness of spectrum on one side or the other of 0 is strongly related to Kodaira's vanishing theorem.