Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

In this talk, I will discuss the n-dimensional Dirac-Dunkl operator associated with the reflection group Z_2^{n}. I will exhibit the symmetries of this operator, and describe the invariance algebra they generate. The symmetry algebra will be identified as a rank-n generalization of the Bannai-Ito algebra. Moreover, I will explain how a basis for the kernel of this operator can be constructed using a generalization of the Cauchy-Kovalevskaia extension in Clifford analysis, and how these basis functions form a basis for irreducible representations of Bannai-Ito algebra. Finally, I will conjecture on the role played by the multivariate Bannai-Ito polynomials in this framework.

We consider (complex) Gaussian analytic functions on a horizontal strip, whose distribution is invariant with respect to horizontal shifts (i.e., "stationary"). Let N(T) be the number of zeroes in [0,T] x [a,b]. First, we present an extension of a result by Wiener, concerning the existence and characterization of the limit N(T)/T as T approaches infinity. Secondly, we characterize the growth of the variance of N(T). We will pose to discuss analogues of these results in a few other settings, such as zeroes of real-analytic Gaussian functions and winding of planar Gaussian functions, pointing out interesting similarities and differences. For the last part, we consider the "persistence probability" (i.e., the probability that a function has no zeroes at all in some region). Here we present results in the real setting, as even this case is yet to be understood. Based in part on joint works with Jeremiah Buckley and Ohad Feldheim.

This talk concerns recent joint work with E. M. Stein on the extension to higher dimension of Calder\'on's andCoifman-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:\vskip-1.0em$$H(w, z) = \frac{1}{2\pi i}\frac{dw}{w-z}$$\smallskip\vskip-0.7em\noindent is that it is holomorphic (that is, analytic) as a function of $z\in D$. In great contrast with the one-dimensional theory, in higher dimension there is no obvious holomorphic analogueof $H(w, z)$. This is because of geometric obstructions (the Levi problem) that in dimension 1 are irrelevant. A good candidate kernel for the higher dimensional setting was first identified by Jean Lerayin 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 (much less Lipschitz!) Leray's construction becomes conceptually problematic.In this talk I will present {\em(a)}, the construction of theCauchy-Leray kernel and {\em(b)}, 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 Szeg\H o and Bergman projections (that is, the orthogonal projections of $L^2$ onto, respectively, the Hardy and Bergman spaces of holomorphic functions).

For $\mathbb B^n$ the unit ball of $\mathbb C^n$, we consider Bergman-Orlicz spaces of holomorphic functions in $L_\alpha^\Phi(\mathbb B^n)$, which are generalizations of classical Bergman spaces. Weobtain their atomic decomposition and then prove weak factorization theorems involving the Bloch space and Bergman-Orlicz space and also weak factorization involving two Bergman-Orlicz spaces. This talk is based on joint work with D. Bekolle and A. Bonami.