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Theme for TIFR Centre For Applicable Mathematics, Bangalore

Abstract:  In sharp contrast to $\mathbb{C}$, there is a deep interplay between geometry and the function theory on domains in  $\mathbb{C}^n$, $n>1$. We will begin with a quick overview of some of these ideas; focusing, in particular, on pseudoconvexity, finite-type, and circular symmetries. We will then proceed to show that Lipschitz holomorphic functions exhibit a gain in Lipschitz regularity in the complex tangential directions. Finally, we will discuss a smoothing property of the Bergman projection. Along the way, we will highlight connections to the $\overline{\partial}$-Neumann problem and the extension of biholomorphic mappings.