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Series: Analysis Seminar

Consider Hermitian matrices A, B, C on an n-dimensional Hilbert space such that C=A+B. Let a={a_1,a_2,...,a_n}, b={b_1, b_2,...,b_n}, and c={c_1, c_2,...,c_n} be sequences of eigenvalues of A, B, and C counting multiplicity, arranged in decreasing order. Such a triple of real numbers (a,b,c) that satisfies the so-called Horn inequalities, describes the eigenvalues of the sum of n by n Hermitian matrices. The Horn inequalities is a set of inequalities conjectured by A. Horn in 1960 and later proved by the work of Klyachko and Knutson-Tao. In these two talks, I will start by discussing some of the history of Horn's conjecture and then move on to its more recent developments. We will show that these inequalities are also valid for selfadjoint elements in a finite factor, for types of torsion modules over division rings, and for singular values for products of matrices, and how additional information can be obtained whenever a Horn inequality saturates. The major difficulty in our argument is the proof that certain generalized Schubert cells have nonempty intersection. In the finite dimensional case, it follows from the classical intersection theory. However, there is no readily available intersection theory for von Neumann algebras. Our argument requires a good understanding of the combinatorial structure of honeycombs, and produces an actual element in the intersection algorithmically, and it seems to be new even in finite dimensions. If time permits, we will also discuss some of the intricate combinatorics involved here. In addition, some recent work and open questions will also be presented.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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).

Series: Analysis Seminar

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.

Series: Analysis Seminar

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?

Series: Analysis Seminar

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).

Series: Analysis Seminar

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.