## Seminars and Colloquia by Series

Wednesday, September 14, 2016 - 14:05 , Location: Skiles 005 , Michael Lacey , Georgia Tech , Organizer: Shahaf Nitzan
The Ricci-Stein theory of singular integrals concerns operators of the form \int e^{i P(y)} f (x-y) \frac {dy}y.The L^p boundedness was established in the early 1980's, and the weak-type L^1 estimate by Chanillo-Christ in 1987.  We establish the weak type estimate for the maximal truncations. This method of proof might well shed much more information about the fine behavior of these transforms.  Joint work with Ben Krause.
Wednesday, September 7, 2016 - 14:05 , Location: Skiles 005 , Robert Kesler , Georgia Tech , Organizer: Shahaf Nitzan
Multilinear singular integral operators associated to simplexes arise naturally in the dynamics of AKNS systems. One area of research has been to understand how the choice of simplex affects the estimates for the corresponding operator. In particular, C. Muscalu, T. Tao, C. Thiele have observed that degenerate simplexes yield operators satisfying no L^p estimates, while non-degenerate simplex operators, e.g. the trilinear Biest, satisfy a wide range of L^p estimates provable using time-frequency arguments. In this talk, we shall define so-called semi-degenerate simplex multipliers, which as the terminology suggests, lie somewhere between the degenerate and non-degenerate settings and then introduce new L^p estimates for such objects. These results are known to be sharp with respect to target Lebesgue exponents, unlike the best known Biest estimates, and rely on carefully localized interpolation arguments
Wednesday, August 31, 2016 - 14:05 , Location: Skiles 005 , Amalia Culiuc , Georgia Tech , , Organizer: Shahaf Nitzan
In this talk we discuss two weight estimates for well-localized operators acting on vector-valued function spaces with matrix weights. We will show that the Sawyer-type testing conditions are necessary and sufficient for the boundedness of this class of operators, which includes Haar shifts and their various generalizations. More explicitly, we will show that it is suficient to check the estimates of the operator and its adjoint only on characteristic functions of cubes. This result generalizes the work of Nazarov-Treil-Volberg in the scalar setting and is joint work with K. Bickel, S. Treil, and B. Wick.
Wednesday, April 27, 2016 - 14:05 , Location: Skiles 005 , Vincent Genest , MIT , Organizer: Plamen Iliev
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.
Wednesday, April 20, 2016 - 14:05 , Location: Skiles 005 , Dustin Mixon , Ohio state University , Organizer: Shahaf Nitzan
Recently, Awasthi et al proved that a semidefinite relaxation of the k-means clustering problem is tight under a particular data model called the stochastic ball model. This result exhibits two shortcomings: (1) naive solvers of the semidefinite program are computationally slow, and (2) the stochastic ball model prevents outliers that occur, for example, in the Gaussian mixture model. This talk will cover recent work that tackles each of these shortcomings. First, I will discuss a new type of algorithm (introduced by Bandeira) that combines fast non-convex solvers with the optimality certificates provided by convex relaxations. Second, I will discuss how to analyze the semidefinite relaxation under the Gaussian mixture model. In this case, outliers in the data obstruct tightness in the relaxation, and so fundamentally different techniques are required. Several open problems will be posed throughout.This is joint work with Takayuki Iguchi and Jesse Peterson (AFIT), as well as Soledad Villar and Rachel Ward (UT Austin).
Wednesday, April 13, 2016 - 14:05 , Location: Skiles 005 , Naomi Feldheim , Stanford University , Organizer: Shahaf Nitzan
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.
Wednesday, April 6, 2016 - 14:05 , Location: Skiles 005 , Krystal Taylor , Ohio State University , Organizer: Michael Lacey
We use Fourier analysis to establish $L^p$ bounds for Stein's  spherical    maximal theorem in the setting of compactly supported Borel measures $\mu, \nu$   satisfying natural local size assumptions $\mu(B(x,r)) \leq Cr^{s_{\mu}}, \nu(B(x,r)) \leq Cr^{s_{\nu}}$.  As an application, we address the following geometric problem: Suppose that $E\subset \mathbb{R}^d$ is a union of translations of the unit circle, $\{z \in \mathbb{R}^d: |z|=1\}$, by points in a set $U\subset \mathbb{R}^d$.  What are the minimal assumptions on the set $U$ which guarantee that the $d-$dimensional Lebesgue measure of $E$ is positive?
Wednesday, March 30, 2016 - 14:00 , Location: Skiles 005 , Loredona Lanzani , Syracuse University , Organizer: Michael Lacey
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).
Wednesday, March 16, 2016 - 14:05 , Location: Skiles 005 , Alex Powell , Vanderbilt University , Organizer: Shahaf Nitzan
Consistent reconstruction is a method for estimating a signal from a collection of noisy linear measurements that are corrupted by uniform noise.  This problem arises, for example, in analog-to-digital conversion under the uniform noise model for memoryless scalar quantization.  We shall give an overview of consistent reconstruction and prove optimal mean squared error bounds for the quality of approximation.  We shall also discuss an iterative alternative (due to Rangan and Goyal) to consistent reconstruction which is also able to achieve optimal mean squared error; this is closely related to the classical Kaczmarz algorithm and provides a simple example of the power of randomization in numerical algorithms.
Wednesday, March 9, 2016 - 14:00 , Location: Skiles 005 , Edgar Tchoundja , University of Yaounde , Organizer: Michael Lacey
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.