Abstract: This week we continue with:

---The index problem for elliptic PDE's, characteristic classes via Chern-Weil theory,

---Miraculous cancellations, Getzler's supersymmetric proof of the Atiyah-Singer index theorem, special cases: Gauss-Bonnet-Chern, Hirzebruch signature theorem, and Riemann-Roch.

Abstract: This talk will consider the problem of characterizing the sum of CR functions from two competing (oppositely-oriented) CR structures sharing the same maximal complex subspace, in two specific scenarios.

In the first scenario the two structures are simply conjugate to each other and the functions in question are pluriharmonic boundary values. (This problem has an extensive history, but some new results will be presented.) In the second scenario the two structures are related by projective duality considerations.

In both cases special attention will be paid to two-dimensional circular domains. This is joint work with Dusty Grundmeier.