Loredana Lanzani (Syracuse University) Harmonic analysis techniques in several complex variables Abstract: This talk concerns the application of relatively classical tools from real harmonic analysis (namely, the $T(1)$theorem for spaces of homogenous type) to the novel context of several complex variables. Specifically, I will present recent joint work with E. M. Stein on the extension to higher dimension of Calderón's and CoifmanMcIntoshMeyer's seminal results about the Cauchy integral for a Lipschitz planar curve (interpreted as the boundary of a Lipschitz domain $D\subset{\mathbb C}$). From the point of view of complex analysis, a fundamental feature of the 1dimensional Cauchy kernel $H(w, z) = \tfrac{1}{2\pi i}(wz)^{1}dw$ is that it is holomorphic as a function of $z\in D$. In great contrast with the onedimensional theory, in higher dimension there is no obvious holomorphic analogue of $H(w, z)$. This is because geometric obstructions arise (the Levi problem), which in dimension one are irrelevant. A good candidate kernel for the higher dimensional setting was first identified by Jean Leray in the context of a $C^\infty$smooth, convex domain $D$: while these conditions on $D$ can be relaxed a bit, if the domain is less than $C^2$smooth (never mind Lipschitz!) Leray's construction becomes conceptually problematic. In this talk I will present (i) the construction of the CauchyLeray kernel and (ii) the $L^p(bD)$boundedness of the induced singular integral operator under the weakest currently known assumptions on the domain's regularity  in the case of a planar domain these are akin to Lipschitz boundary, but in our higherdimensional context the assumptions we make are in fact optimal. The proofs rely in a fundamental way on a suitably adapted version of the socalled "$T(1)$theorem technique" from real harmonic analysis. Time permitting, I will describe applications of this work to complex function theory  specifically, to the Szego and Bergman projections (that is, the orthogonal projections of $L^2$ onto, respectively, the Hardy and Bergman spaces of holomorphic functions). 
