Abstract: K-homology is the dual theory to K-theory. This talk will give the basic definition (following Atiyah, Brown-Douglas-Fillmore, and Kasparov) of K-homology as abstract elliptic operators. A different approach ( due to Baum-Douglas) will also be indicated. This second definition of K-homology is closely connected to the D-branes of string theory. K-homology will then be used to state the BC (Baum-Connes) conjecture.

Abstract: After a brief review on the Schwarz Reflection Principle in one variable, I will discuss the general situation in higher dimensions using the language of the so-called Segre varieties associated with real analytic hypersurfaces in $\mathbb C^n$. I will then explain how to use it for proving boundary regularity results for holomorphic mappings.