Seminar Calendar
for events the day of Tuesday, April 11, 2017.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, April 11, 2017

Topology Seminar
11:00 am   in 345 Altgeld Hall,  Tuesday, April 11, 2017
 Del Edit Copy
Submitted by cmalkiew.
 Kate Ponto (U Kentucky)To Be Announced

Probability Seminar
2:00 pm   in 347 Altgeld Hall,  Tuesday, April 11, 2017
 Del Edit Copy
Submitted by wangjing.
 Maxim Raginsky (UIUC)To Be Announced

Algebraic Geometry Seminar
3:00 pm   in 243 Altgeld Hall,  Tuesday, April 11, 2017
 Del Edit Copy
Submitted by rtramel.
 Deepam Patel (Purdue University)To Be Announced

Mathematics Colloquium
4:00 pm   in 245 Altgeld Hall,  Tuesday, April 11, 2017
 Del Edit Copy
Submitted by kapovich.
 Loredana Lanzani (Syracuse University)Harmonic analysis techniques in several complex variablesAbstract: 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 Coifman-McIntosh-Meyer'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 1-dimensional Cauchy kernel $H(w, z) = \tfrac{1}{2\pi i}(w-z)^{-1}dw$ is that it is holomorphic as a function of $z\in D$. In great contrast with the one-dimensional 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 Cauchy-Leray 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 higher-dimensional context the assumptions we make are in fact optimal. The proofs rely in a fundamental way on a suitably adapted version of the so-called "$T(1)$-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).