\documentclass[12pt]{amsart}
\usepackage{amssymb,amsmath,latexsym}


\usepackage{amsfonts}
\usepackage{amscd}

\usepackage{graphicx,array}
\usepackage[all]{xy}
\numberwithin{equation}{section}

\setlength{\oddsidemargin}{0.1in}
\setlength{\evensidemargin}{0.1in}
\setlength{\textwidth}{6.3in}
\setlength{\baselineskip}{14pt}
\setlength{\parskip}{6pt}

% ---------------------------------------------------
% some useful spacing
% ---------------------------------------------------

\newcommand{\s}{\smallskip}
\newcommand{\m}{\medskip}
\renewcommand{\b}{\bigskip}
%\newcommand{\diag}{\operatorname{diag}}

% ---------------------------------------------------
% German (Fraktur) letters, in alphabetical order for
% convenience of checking list or adding more of them
% ---------------------------------------------------

\def\ga{\mathfrak{a}}
\def\gb{\mathfrak{b}}
\def\gc{\mathfrak{c}}
\def\gd{\mathfrak{d}}
\def\mfe{\mathfrak{e}}
\def\gf{\mathfrak{f}}
\def\mfg{\mathfrak{g}}
\def\gh{\mathfrak{h}}
\def\gi{\mathfrak{i}}
\def\gj{\mathfrak{j}}
\def\gk{\mathfrak{k}}
\def\gl{\mathfrak{l}}
\def\gm{\mathfrak{m}}
\def\gn{\mathfrak{n}}
\def\go{\mathfrak{o}}
\def\gp{\mathfrak{p}}
\def\gq{\mathfrak{q}}
\def\gr{\mathfrak{r}}
\def\gs{\mathfrak{s}}
\def\gS{\mathfrak{S}}
\def\gt{\mathfrak{t}}
\def\gu{\mathfrak{u}}
\def\gv{\mathfrak{v}}
\def\gw{\mathfrak{w}}
\def\gx{\mathfrak{x}}
\def\gX{\mathfrak{X}}
\def\gy{\mathfrak{y}}
\def\gz{\mathfrak{z}}

% ---------------------------------------------------
% And fix \gg for positive definite
% ---------------------------------------------------

\def\ggg{> \hskip -5 pt >}

% ---------------------------------------------------
% some useful shortcuts for Lie theory
% ---------------------------------------------------

\def\Aut{{\rm Aut}}
\def\Int{{\rm Int}}
\def\Ad{{\rm Ad}}
\def\ad{{\rm ad}}
\def\rank{{\rm rank\,}}
\def\trace{\mathrm{Tr}\,}
\def\Ind{{\rm Ind\,}}
\def\mult{{\rm mult\,}}
\def\Hom{{\rm Hom}}
\newcommand{\res}{\mathrm{res}}
\newcommand{\Wk}{W (\fg_k, \fh_{k})}
\newcommand{\Wn}{W (\fg_n, \fh_{n})}
\newcommand{\Wkn}{W_{\fh_{n,\R}} (\fg_k,\fh_{k})}
\renewcommand{\Im}{\mathop{\rm Im} }
% ---------------------------------------------------
% Blackboard Bold letters, in alphabetical order for
% convenience of checking list or adding more of them
% ---------------------------------------------------

\def\A{\mathbb{A}}
\def\B{\mathbb{B}}
\def\C{\mathbb{C}}
\def\D{\mathbb{D}}
\def\E{\mathbb{E}}
\def\F{\mathbb{F}}
\def\G{\mathbb{G}}
\def\H{\mathbb{H}}
\def\I{\mathbb{I}}
\def\J{\mathbb{J}}
\def\K{\mathbb{K}}
\def\L{\mathbb{L}}
\def\M{\mathbb{M}}
\def\N{\mathbb{N}}
\def\O{\mathbb{O}}
\def\P{\mathbb{P}}
\def\Q{\mathbb{Q}}
\def\R{\mathbb{R}}
\def\T{\mathbb{T}}
\def\V{\mathbb{V}}
\def\W{\mathbb{W}}
\def\X{\mathbb{X}}
\def\Y{\mathbb{Y}}
\def\Z{\mathbb{Z}}

\newcommand{\id}{\mathrm{id}}
\newcommand{\gL}{\Lambda }

% ---------------------------------------------------------
% Calligraphic (script) letters, in alphabetical order for
% convenience of checking list or adding more of them
% ---------------------------------------------------------

%%%%%%%%%%%%%%% Lie groups %%%%%%%%%%%%%%%%
\newcommand{\SO}{\mathrm{SO}}
\newcommand{\Sp}{\mathrm{Sp}}
\newcommand{\Spin}{\mathrm{Spin}}
\newcommand{\SU}{\mathrm{SU}}
\newcommand{\U}{\mathrm{U}}
\newcommand{\diag}{\mathrm{diag}}
\newcommand{\Tr}{\mathrm{Tr}\, }
\newcommand{\so}{\mathfrak{so}}
\newcommand{\lsp}{\mathfrak{sp}}
\newcommand{\lsl}{\mathfrak{sl}}
\def\cA{\mathcal{A}}
\def\cB{\mathcal{B}}
\def\cC{\mathcal{C}}
\def\cD{\mathcal{D}}
\def\cE{\mathcal{E}}
\def\cF{\mathcal{F}}
\def\cG{\mathcal{G}}
\def\cH{\mathcal{H}}
\def\cI{\mathcal{I}}
\def\cJ{\mathcal{J}}
\def\cK{\mathcal{K}}
\def\cL{\mathcal{L}}
\def\cM{\mathcal{M}}
\def\cN{\mathcal{N}}
\def\cO{\mathcal{O}}
\def\cP{\mathcal{P}}
\def\cQ{\mathcal{Q}}
\def\cR{\mathcal{R}}
\def\cS{\mathcal{S}}
\def\cT{\mathcal{T}}
\def\cU{\mathcal{U}}
\def\cV{\mathcal{V}}
\def\cW{\mathcal{W}}
\def\cX{\mathcal{X}}
\def\cY{\mathcal{Y}}
\def\cZ{\mathcal{Z}}



%%%%%%%%%% From the paper with Henrik %%%%%%%
%%%%%%%%%% with "overlines" extended by Joe %%%%%%%
\newcommand{\Br}{B_r}
\newcommand{\Brc}{\overline{B_r(0)}}
\newcommand{\Dr}{D_r(o)}
\newcommand{\Drc}{\overline{D_r(o)}}
\newcommand{\Bsc}{\overline{D_s(o)}}
\newcommand{\Exp}{\mathrm{Exp}}
\newcommand{\PW}{\mathrm{PW}}
\newcommand{\LM}{\Lambda^+ (M)}
\renewcommand{\Re}{\mathrm{Re}}
\renewcommand{\thesection}{\arabic{section}}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\renewcommand{\theparagraph}{\thesection.\arabic{equation}}
\newtheorem{theorem}[equation]{Theorem}
\renewcommand{\thetheorem}{\thesection.\arabic{equation}}
\newtheorem{lemma}[equation]{Lemma}
\renewcommand{\thelemma}{\thesection.\arabic{equation}}
\newtheorem{corollary}[equation]{Corollary}
\renewcommand{\thecorollary}{\thesection.\arabic{equation}}
\newtheorem{proposition}[equation]{Proposition}
\renewcommand{\theproposition}{\thesection.\arabic{equation}}
\newtheorem{note}[equation]{}
\renewcommand{\thenote}{\thesection.\arabic{equation}}
\newtheorem{definition}[equation]{Definition}
\renewcommand{\thedefinition}{\thesection.\arabic{equation}}
\newtheorem{example}[equation]{Example}
\renewcommand{\theexample}{\thesection.\arabic{equation}}
\newtheorem{remark}[equation]{Remark}
\renewcommand{\theremark}{\thesection.\arabic{equation}}
\newtheorem{convention}[equation]{Convention}
\renewcommand{\theconvention}{\thesection.\arabic{equation}}
\newtheorem{query}[equation]{Query}
\renewcommand{\thequery}{\thesection.\arabic{equation}}
\newtheorem{noname}[equation]{}
\renewcommand{\thenoname}{\thesection.\arabic{equation}}

%%%%%%%%%%%% From GO
\newcommand{\fa}{\mathfrak{a}}
\newcommand{\fb}{\mathfrak{b}}
\newcommand{\fc}{\mathfrak{c}}
\newcommand{\fg}{\mathfrak{g}}
\newcommand{\fh}{\mathfrak{h}}
\newcommand{\fk}{\mathfrak{k}}
\newcommand{\fl}{\mathfrak{l}}
\newcommand{\fm}{\mathfrak{m}}
\newcommand{\fn}{\mathfrak{n}}
\newcommand{\fo}{\mathfrak{o}}
%\newcommand{\fp}{\mathfrak{p}}
\newcommand{\fq}{\mathfrak{q}}
\newcommand{\fs}{\mathfrak{s}}
\newcommand{\fu}{\mathfrak{u}}
\newcommand{\fz}{\mathfrak{z}}
\newcommand{\Supp}{\mathrm{Supp}}

%\newcommand{\cF}{\mathcal{F}}
\newcommand{\rS}{\mathrm{S}}
\newcommand{\rI}{\mathrm{I}}

%Miscellaneous
\newcommand{\te}{\text}% same as \mathrm command.
\newcommand{\tei}{\textit}
\newcommand{\im}{\implies}
\newcommand{\ti}{\times}
\newcommand{\exi}{\ \exists \ }
\newcommand{\ther}{\therefore}
\newcommand{\be}{\because}
\newcommand{\ep}{\epsilon}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\ol}{\overline}
\newcommand{\ul}{\underline}
\newcommand{\q}{\quad}% spacing
\newcommand{\qq}{\qquad}% morepspacing
\newcommand{\tit}{\textit}
\newcommand{\nd}{\noindent}
\newcommand{\bs}{\backslash}
\newcommand{\pa}{\partial}
\newcommand{\para}{\parallel}
\newcommand{\sm}{\setminus}



\def\sideremark#1{\ifvmode\leavevmode\fi\vadjust{\vbox to0pt{\vss% the remark
 \hbox to 0pt{\hskip\hsize\hskip1em%                          will appear only
\vbox{\hsize2cm\tiny\raggedright\pretolerance10000%          on the side
 \noindent #1\hfill}\hss}\vbox to8pt{\vfil}\vss}}}%
                                                   %          in 2cm

                                                   %          wide box
                                                   %

\newcommand{\edz}[1]{\sideremark{#1}}
\parindent0cm
\begin{document}
\begin{center}
\Large\textbf{
Abstracts}
\end{center}

\begin{center}
\textbf{Akram Aldroubi:
Union of subspace clustering, sparse approximations and dimensionality reduction}
\end{center}

The subspace clustering problem proposes to find a union of subspaces that are closest to a given set of data.  This is a generalization of the classical least squares, and It has many applications in computer science and engineering, such as face recognition and motion tracking in videos. However, the subspace clustering problem  is highly non-linear and the data often lives in a very high dimensional space while the subspaces model consists of low dimensional vector spaces.
In this talk we will set up the problem, relates some of the theoretical results about existence of solutions, algorithms for solving it, and how to reduce the problem to a more tractable low dimensional one.
\bigskip

\begin{center}
\textbf{Gaik Ambartsoumian:
Exterior Problem of Acoustic Reflectivity Imaging}
\end{center}

In various applications of acoustic reflection tomography the support of the image function lies outside of the data acquisition set, which is usually a closed curve in 2D or a closed surface in 3D. The problem of image reconstruction in this setup is highly unstable, yet necessary and important (e.g. in intravascular ultrasound). Although one can not expect stable reconstruction of the whole image, microlocal considerations show that certain image singularities can be reconstructed correctly. There are some uniqueness results and exact inversion formulas for a few restricted cases of this exterior problem, however no robust inversion algorithm has been developed so far to recover the "visible" singularities. The talk will discuss the known results and recent advances in this direction.



\begin{center}
\textbf{
Gregory Backus: A Categorization of Mexican Free-Tailed Bat (Tadarida brasiliensis) Chirps
}
\end{center}

Male Mexican Free-tailed Bats (Tadarida brasiliensis) attract mates and defend territory using multi-phrase songs that follow a structured set of rules.  A subjective view of their spectrograms shows similarity and dissimilarity between the chirps (a syllable within the song) of different males.  We developed an algorithm to characterize the shapes of these chirps.  In order to compress data, the discrete Fourier transform and a four level Daubechies 2 wavelet decomposition allowed us to divide the chirps into segments based on frequency information after we cut the chirps into time-based segments.  After calculating these segments' energies, we compressed the large data vectors into a single data point using multidimensional scaling.  When comparing our first dimension of this multidimensional scaling, wavelet analysis gave us categorizations that most closely resembled our subjective categorizations.
\bigskip

\begin{center}
\textbf{B. A. Bailey:  Multivariate Polynomial Interpolation and Sampling in Paley-Wiener Spaces}\\
\bigskip
\end{center}
In this presentation, an equivalence between existence of particular exponential Riesz bases for multivariate bandlimited functions and existence of certain polynomial interpolants for these bandlimited functions is given.  For certain classes of unequally spaced data nodes and corresponding $\ell_2$ data, the existence of these polynomial interpolants allows for a simple recovery formula for multivariate bandlimited functions which demonstrates $L_2$ and uniform convergence on $\mathbb{R}^d$.  A simpler computational version of this recovery formula is also given, at the cost of replacing $L_2$ and uniform convergence on $\mathbb{R}^d$ with $L_2$ and uniform convergence on increasingly large subsets of $\mathbb{R}^d$. As a special case, the polynomial interpolants of given $\ell_2$ data converge in the same fashion to the multivariate bandlimited interpolant of that same data.  Concrete examples of pertinant Riesz bases and unequally spaced data nodes are also given.
\bigskip

\begin{center}
\textbf{Jens Christensen: Sampling of band limited functions for Gelfand pairs}
\end{center}

We present a generalization of the classical sampling theorem of band limited functions for the Fourier transform on $L^2(G/K)$ when $(G,K)$ is a Gelfand pair.
\bigskip

\begin{center}
\textbf{Brad Currey (joint work with Azita Mayeli)}
\end{center}

Let $G$ be a locally compact, second countable group. A unitary representation $\pi$ of $G$ acting in a Hilbert space $\mathcal H$ is said to be admissible if there is $\psi \in \mathcal H$ such that the mapping
$$W_\psi : f \mapsto \langle f, \pi(\cdot) \psi\rangle$$
is an isometry of $\mathcal H$ into $L^2(G)$; in this case, $W_\psi$ is a generalization of the continuous wavelet transform. We are interested in both necessary and sufficient conditions that a representation be admissible, and for admissible representations, explicit constructions for (generalized) continuous wavelets $\psi$. We present results for various Lie groups, including very recent joint work with Vignon Oussa.
\bigskip


\begin{center}
\textbf{
Rim Gouia: Inversion of the circular Radon transform on an annulus.
}
\end{center}


The circular Radon transform $Rf$ puts into correspondence to a
given function $f$ its integrals along circular trajectories. The
need for such a transform arises in several contemporary problems of
medical imaging, synthetic aperture radar and non destructive testing.\\


The major problems related to the circular Radon transform are the
existence and uniqueness of its inversion, inversion formulas and
the range description of the transform. In the case, when the
circular Radon transform $Rf$ is known for circles of all possible
radii, there are well developed theories now addressing most of the
questions mentioned above. However, many of these questions are
still open when $Rf$ is available for only half of all possible
radii, or when the support of $f$ is outside the circle.\\


The aim of my presentation is to discuss some new results about the
existence and uniqueness of the representation of a function by its
circular Radon transform with radially partial data. A new inversion
formula is presented in the case of the circular acquisition
geometry for both interior and exterior problems when the Radon
transform is known on an annulus.
\bigskip

\begin{center}
\textbf{Eric Grinberg: :  The Admissibility Problem for Radon Transforms on Projective Spaces}
\end{center}


The admissibility problem, first posed by I.M. Gelfand in the context of the Plancherel Measure in the context of Lie groups, asks for classification of minimal invertible limited data problems for a given full-data Radon transform. We will discuss the admissibility problem for affine and projective spaces with an emphasis on the projective line transform in a finite projective space of dimension three. This is joint work with David Feldman.
\bigskip

\begin{center}
\textbf{
Bin Han: Symmetric Complex Orthonormal Wavelets and Matrix Extension with Symmetry
}
\end{center}

\noindent
Abstract: In this talk we shall discuss symmetric orthonormal complex wavelets. It is well-known that except the discontinuous Haar wavelet, compactly supported real-valued orthonormal wavelets cannot have symmetry. In this talk, we first study symmetric orthonormal dyadic complex wavelets such that the orthonormal refinable functions have high linear-phase moments and the wavelets have high vanishing moments. Such wavelets lead to real-valued symmetric tight framelets with desirable moment properties, and are related to coiflets which are of interest in numerical algorithms. Then we shall address symmetric orthonormal complex wavelets with a general dilation factor. This problem is related to the interesting matrix extension problem with symmetry, which plays a fundamental role in many areas. We shall present two families of compactly supported symmetric orthonormal complex wavelets with arbitrarily high vanishing moments or arbitrarily high linear-phase moments.
\bigskip

\begin{center}
\textbf{Alex Iosevich: Regular value theorem in a fractal setting.}
\end{center}

The regular value theorem in differential geometry says that if the map $F$ from an $n$-dimensional smooth manifold $X$ to an $m$-dimensional smooth manifold $Y$ is a submersion on the set $F^{-1}(y)=\{x \in X: F(x)=y\}$, then $F^{-1}(y)$ is either empty or is a smooth sub-manifold of $X$ of dimension $n-m$. We shall that if $Y$ is replaced by ${\Bbb R}$ and $X$ is replaced by $E \times E'$, where $E, E'$ are compact Ahlfors-David regular subsets of ${\Bbb R}^d$ of Hausdorff dimension $s$, $0<s<d$. We shall prove that if $F$ is sufficiently regular and the Hausdorff dimension of $E$ and $E'$ is sufficiently large with respect to the regularity of $F$, then the upper Minkowski dimension of
$\{(x,y) \in E \times E': F(x,y)=t \}$ is at most $s+s'-1$. Connections with the theory of regularity of the Fourier integral operators and the Falconer distance conjecture will also be discussed.
\bigskip


\begin{center}
\textbf{Deguang Han: Frames for Group Representations}
\end{center}

The Feichtinger frame conjecture states that every bounded (from
below) frame is a finite  union of basic Riesz sequences. This
conjecture turns out to be equivalent to the Kadison- Singer pure
state extension problem and several other well-known unsettled
problems. In this talk I will focus on a few aspects of this
conjecture related to group representation frames and exponential
frames for fractal measures. Additionally I will also discuss its
connection with a new duality principle for group representations
and the $II_{1}$ factor classification problem. (This talk is
based on collaborations with Dorin Dutkay, Gabriel Picioroaga
David Larson,  Qiyu Sun and Eric Weber).
\bigskip

\begin{center}
\textbf{Hongyu He: Uniform Bounds on Smooth Matrix Coefficients on $L^2$-spaces}
\end{center}
\bigskip


In this talk, I shall discuss a Cowling-Haagerup-Howe type uniform bound for smooth matrix coefficients of semisimple group acting on $L^2$-spaces.
I will then discuss how the uniform bounds can infer the spectrum of homogeneous spaces.
\bigskip

\begin{center}
\textbf{John Jasper: Frames with prescribed norms and frame operator}
\end{center}

Abstract: We present an extension of Kadison’s theorem describing diagonals of self-adjoint operators
with 3 point spectrum. As a consequence, we characterize sequence of norms of a frame whose frame operator
has 2 point spectrum.

\begin{center}
\textbf{David A. Jimenez: On the Match of Point Configurations
}
\end{center}

On many computer vision applications, it is often necessary to match a given image to one in an indexed set. One way this may done is by the identification and comparison of landmarks on the image. We call this discrete set of points in $\mathbb{R}^d$ a point configuration. We analyze different algorithms to decide whether or not two point configurations are identical up to a rigid motion. We call this identification, a match. We will also compare and analyze the possibility of using these algorithms for the problem of fingerprint identification.
\bigskip

\begin{center}
\textbf{Emily King}
\end{center}

This talk will present new multi-dimensional transformations which generalize shearlet transforms both anisotropically and isotropically to $L^2(\mathbb{R}^d)$.  This is achieved by means of explicit constructions of families of reproducing Lie subgroups of the symplectic groups.  The properties of these new families will be discussed.  In particular, an analog of the Calder\'{o}n admissibility condition will be provided as well as a discussion of the related co-orbit space theory.
\bigskip


\begin{center}
\textbf{
Dominic Kramer: Frame Based Steganography}
\end{center}


\noindent
Steganography is the art of hiding secret data
in various data sources such as images. This talk
will discuss a technique, which incorporates the
use of frames, for hiding an element of a Hilbert
space in the coefficients of another element
of the space.
\bigskip

\begin{center}
\textbf{Nishu Lal: Product structure of the spectral zeta function of the Sturm-Liouville operator on fractals}
\end{center}


In this talk, we will discuss the spectral zeta function of a self-similar Sturm-Liouville operator on the half real line and C. Sabot's work on connecting the spectrum of this operator with the iteration of a rational map of several complex variables.
The Sturm-Liouville operator on $[0,\infty)$ is viewed as a limit of the sequence of operators $\frac{d}{dm_{<n>}}\frac{d}{dx}$ with Dirichlet boundary condition on $I_{<n>}=[0,\alpha^{-n}]$ which are the infinitesimal generators of the Dirichlet form $(a_{<n>}, m_{<n>})$.  In particular, it is defined in terms of a self-similar measure $m$ and Dirichlet form $a$, relative to a suitable iterated function system (IFS) on $I = [0,1]$.  In the case of the Sierpinski gasket, as was shown by A. Teplyaev, extending the known relation by M. Lapidus for fractal strings, the spectral zeta function of the Laplacian has a product structure with respect to the iteration of a rational map on the complex plane which arises from the decimation method.  In the case of the above self-similar Sturm-Liouville problem, we obtain an analogous product formula, but now expressed in terms of the (suitably defined) zeta function associated with the dynamics of the corresponding `renormalization map', viewed as a rational function on the 2-dimensional projective space.  This is joint work with Michel Lapidus.
\bigskip

\begin{center}
\textbf{David Larson: Operator-Valued Measures, Dilations, and The Theory of Frames}
\end{center}

Abstract:  We show that there are some natural associations between the theory of frames (including continuous frames and framings), the theory of operator-valued measures on sigma-algebras of sets, and the theory of normal linear mappings between von Neumann algebras. In this connection frame theory itself is identified with the special case in which the domain algebra for the mapping is commutative. Some of the more important results and  proofs for mappings in this case extend naturally to the case where the domain algebra is non-commutative.
This happens frequently enough, and in profound enough ways, to justify defining a noncommutative frame to be an arbitrary ultraweakly continuous linear mapping between von Neumann algebras. It has been known for a long time that a sufficient condition for a unital bounded linear map between C*-algebras to have a Hilbert space dilation to a bounded homomorphism is that the mapping is that the map be completely bounded.
Our theory shows that under suitable hypotheses even if it is not completely bounded it still has a Banach space dilation to a homomorphism, and the Banach space can be rather nice.  We view this as a generalization of the known result that arbitrary framings have Banach dilations.  This is joint work with Deguang Han, Bei Liu and Rui Liu.
\bigskip


\begin{center}
\textbf{Shidong Li:
Fusion frame in action: High resolution image fusion}
\end{center}

The notion of Fusion Frames was introduced to study ways in which functions or signals from a set of subspaces can be combined coherently regardless how complicated subspaces are related.  We have argued that fusion frames have a deep root in data fusion applications for distributed systems such as sensor networks. It is time that we put fusion frame in action and show exactly how it can be applied in sensor array data fusion. Formulations of high resolution image fusion using fusion frames will be presented. These techniques use the impulse response function of cameras as the building block of the mathematical frames and fusion frames in the fusion process. By taking realistic camera physics into consideration, the proposed approach provides a natural and realistic modeling of the high-resolution image fusion problem. Deterministic and iterative fusion algorithms will be discussed. The fusion frame approach for high-resolution image fusion is also seen to be robust to realistic fusion problems from inhomogeneous image measurements (taken at different space or time or by different cameras). The effectiveness of this approach is demonstrated through both simulated and realistic examples. This is a joint work with Zhenjie Yao and Weidong Yi.
\bigskip

\begin{center}
\textbf{
Eileen Martin: Continuously Moving Parseval Frames on Smooth Manifolds.
}
\end{center}

\noindent
Moving bases on manifolds are important in the study of differential geometry and are applied in mathematical physics, but moving bases do not exist on all manifolds, for instance, the sphere. An alternative to a moving basis is a Parseval frame of unit-length vectors. We examine the existence of such frames on the M\"{o}bius strip, the Klein bottle, and $n$-dimensional spheres. We prove the existence of a continuously moving, unit-length Parseval frame on $S^n$ when $n$ is an odd integer. More generally, we investigate the relationship between the existence of a nowhere zero vector field and that of a continuously moving Parseval frame of unit length. One potentially useful tool in studying this relationship is the frame force associated with the frame potential. To better understand this possible method, we are led to a study of the dynamical properties of the frame force.
\bigskip

\begin{center}
\textbf{Azita Mayeli: Band-limited wavelets and Besov space norms in Hilbert spaces}
\end{center}

In this talk we first introduce inhomogeneous Besov norms for abstract Hilbert spaces. Using band-limited wavelets, we then establish a correlation between the frequency content of a function in a Hilbert space, in terms of its spectral Fourier transform, and its smoothness as described its by Besov norm.
\bigskip


\begin{center}
\textbf{
John Myers: An Algorithm for Unitary Equivalence of Matrices and a Path-Connectedness Application
}
\end{center}

\noindent
Two iterative algorithms are developed to transform a given matrix to a unitarily equivalent matrix with constant main diagonal: one if the matrix has elements in $\mathbb{R}$ and one for elements in $\mathbb{C}$. It will be shown that the algorithms will converge in finitely many iterations if the matrices are of size $2^n \times 2^n$. One possible application of the algorithms is in proving that certain sets of projections are path-connected. Hence, continuity is of interest and several results will be given regarding continuity.


\bigskip


\begin{center}
\textbf{
Patrick J. Orchard: Orthogonal and Maximal Sets for Bernoulli Measures
}
\end{center}

\noindent
Joint work with: Bryan Archer, University of Oklahoma; Rees Dooley, University of Oklahoma;
Reid Kelley, University of Oklahoma; Alyssa Leone, University of Oklahoma;
Patrick Orchard, University of Oklahoma

\title{Orthogonal and Maximal Sets for Bernoulli Measures}
\author{Bryan Archer, University of Oklahoma\\Andrew Holmes, University of Oklahoma\\Patrick Orchard, University of Oklahoma}
\date{}
\maketitle
We consider orthogonal and maximal sets on $L^2(X_\lambda,\mu_\lambda)$ where $\mu_\lambda$ is the Hutchinson measure associated with the Bernoulli Iterated Function System (IFS) for $\lambda\in(0,1)$ and $X_\lambda$ is the support of the measure. By a previous theorem from Jorgensen and Pedersen, for $\lambda=\frac{1}{2n}$ we have an orthonormal basis of exponential functions $e^{2\pi i\gamma x}$ where $\gamma$ is in the preconstructed set $\Gamma_\frac{1}{2n}$. We investigate sets $c\Gamma_\frac{1}{2n}$ where c is an odd integer dependent on 2n. We prove that the set $3\Gamma_\frac{1}{4}\cup\bigcup_{k=1}^{\infty}-4^k(1+3 \cdot 4\Gamma_\frac{1}{4})$ is an orthogonal and maximal set for the space $L^2(X_\frac{1}{4},\mu_\frac{1}{4})$.
\bigskip


\begin{center}
\textbf{
Elena Ournycheva: On Y. Nievergelt's inversion formula for the Radon transform
}
\end{center}

In 1986 Y. Nievergelt suggested a simple formula which allows to reconstruct a continuous compactly supported function on the $2$-plane from
its Radon transform. This formula falls into the scope of the classical
convolution-backprojection method. We show that elementary tools of fractional calculus can be used to obtain more general inversion formulas for
the k-plane Radon transform of continuous and $L^p$ functions on $R^n$ for all
$1 \leq k < n$.

This is a joint work with Boris Rubin.
\bigskip

\begin{center}
\textbf{
Nguyen Cong Phuc}
\end{center}

We discuss a global weighted estimate for a class nonlinear singular operators arising from  divergence form elliptic operators with BMO coefficients on Reifenberg flat domains. Such an estimate implies new global regularity results in Morrey, Lorentz, and H\"older spaces for solutions of certain quasilinear elliptic equations. Moreover, it can also be used to obtain a capacitary estimate to treat a measure datum quasilinear Riccati type equations with nonstandard growth in the gradient. This talk is based on joint work with Tadele Mengesha.
\bigskip

\begin{center}
\textbf{
John Rock: Partition zeta functions of self-similar measures.
}
\end{center}


For an Iterated Function System (IFS) on the unit interval weighted by a probability vector we define a multifractal spectrum for the self-similar Borel measure uniquely determined by the weighted IFS as the abscissae of convergence of its partition zeta functions.  Partition zeta functions are Dirichlet series determined by the self-similar Borel measure and a naturally defined sequence of partitions. These partition zeta functions are indexed by coarse Holder regularity and we show that the corresponding abscissae of convergence equal the Hausdorff dimensions of corresponding Besicovitch subsets of the support of the measure. In the case of the binomial measure, the classical Hausdorff multifractal spectrum is recovered.
\bigskip


\begin{center}
\textbf{Jos\'e Luis Romero}
\end{center}

Coorbit spaces are functional spaces defined by imposing size and integrability conditions to an integral transform T. In the case of the wavelet transform, the corresponding spaces include the family Lebesgue and Sobolev spaces and, more generally, Besov and Tribel-Lizorkin spaces. For the short-time Fourier transform, the associated coorbit spaces are modulation spaces.

Phase-space multipliers are operators defined by applying a mask to the transform T. I will present a result on the characterization of membership to a certain coorbit space in terms of families of phase-space multipliers. In the case of time-frequency analysis, this generalizes recent results to an irregular context.
\bigskip


\begin{center}
\textbf{Lakshmi Roychowdhury: Optimal points for any Cantor distribution}
\end{center}

For over fifty years electrical engineers and mathematicians have been interested in the problem of efficiently ``quantizing'' a probability distribution in the sense of estimating a given probability by a discrete probability supported on a finite set.  This problem arises in signal processing, data compression, cluster analysis, and pattern recognition, and it also has been studied in the context of economics, statistics, and numerical integration.
Given a Borel
probability measure $P$ on $\mathbb R^d$,  the $n$th \textit{quantization
error} is defined by
\[V_n=\inf_{\alpha} \int \min_{a\in\alpha} \|x-a\|^2 dP(x),\]
where the infimum is taken over all subsets $\alpha$ of $\mathbb R^d$ with card $\alpha\leq n$ for $n\geq 1$. We note
that if $\int \| x\|^2 dP(x)<\infty$ then there is some set $\alpha$ for
which the infimum is achieved. The set $\alpha$ for which the infimum is achieved is called the optimal set of $n$-means or $n$-optimal set. It can be shown that for a continuous probability measure $P$ an optimal set of $n$-means always has exactly $n$-elements.
Let $S_1, S_2 : \mathbb R \to \mathbb R$ be defined by
\[S_1(x)=\frac 1 3 x \text{ and } S_2 (x)=\frac 1 3 x +\frac 2 3. \]
Then for a probability vector $(p_1, p_2)$ with $p_1, p_2>0$, there exists a unique Borel probability measure $P$ on $\mathbb R$ such that
$P=p_1 P\circ S_1^{-1}+p_2 P\circ S_2^{-1}$, where $P\circ S_i^{-1}$ denotes the image measure of $P$ with respect to $S_i$ for $i=1, \,2$. For $\sigma=(\sigma_1, \sigma_2, \cdots, \sigma_k) \in \{ 1, 2\}^k$, set $S_{\sigma}=S_{\sigma_1}\circ \cdots \circ S_{\sigma_k}$ and $J_\sigma=S_{\sigma}([0, 1])$. Then the set $C=\prod_{k\in \mathbb N} \bigcup_{\sigma \in \{1, 2\}^k} J_\sigma$ is the Cantor set and equals the support of the probability measure $P$. In this talk, I will show what are the $n$-optimal sets, and the corresponding $n$th quantization error for $n\geq 1$.
\bigskip

\begin{center}
\textbf{Mrinal Kanti Roychowdhury: Quantization dimension for infinite self-similar mappings}
\end{center}
 The term `quantization' in the title originates in the theory of signal processing and denotes a process of discretising signals. As a mathematical theory quantization concerns the best approximation of probabilities by discrete probabilities with a given number of points in their support. Given a Borel probability measure $\mu$
on $\mathbb R^d$, a number $r \in (0, +\infty)$ and a natural number $n
\in \mathbb N$, the $n$th \textit{quantization error} of order $r$ for $\mu$
is defined by
\[V_{n, r}(\mu)=\inf \{\int d(x, \alpha)^r d\mu(x): \alpha \subset \mathbb R^d, \, \text{card}(\alpha) \leq n\},\]
where $d(x, \alpha)$ denotes the distance from the point $x$ to the set
$\alpha$ with respect to a given norm $\|\cdot\|$ on $\mathbb R^d$. Note
that if $\int \| x\|^r d\mu(x)<\infty$ then there is some set $\alpha$
for which the infimum is achieved. The \textit{quantization dimension
of order $r$} for $\mu$ is defined to be
\[D_r=D_r(\mu) =\lim_{n \to \infty} \frac{r\log n}{-\log V_{n, r}(\mu)}, \]
if the limit exists. One sees that the quantization dimension is
actually a function $r\mapsto D_r$ which measures the asymptotic
rate at which $e_{n, r}$ goes to zero. Determination of the
quantization dimension for a probability measure generated by an
infinite iterated function system associated with a probability
vector is a long-time open problem. Recently, I have determined the
quantization dimension for a probability measure generated by an
infinite contractive similarity mappings $S_1, S_2, \cdots$
associated with a probability vector $(p_1, p_2, \cdots)$, that is,
the probability measure $\mu$ satisfies
$\mu=\sum_{j=1}^\infty p_j \mu\circ S_j^{-1}.$
In this talk, I will discuss about it.
\bigskip


\begin{center}
\textbf{Boris Rubin: on Stiefel and Grassmann Manifolds
}
\end{center}

The Funk transform takes a function $f$ on the unit sphere in $R^3$ to a function   on the set of great circles as an integral of $f$ over the corresponding great circle. This transform can be regarded as a member of the analytic family of normalized cosine transforms. We extend basic facts about the Funk transform   and related analytic families  of cosine and sine transforms to the  more general context for Stiefel or Grassmann manifolds.
\bigskip

\begin{center}
\textbf{Dmitry Ryabogin: A counterexample to a problem of Klee}
\end{center}

A question of Klee asks whether a convex body is uniquely determined (up to a translation and reflection in the origin) by its inner section function. We construct a counterexample showing that this is not true.
This is a joint work with R.~J.~Gardner,  V.~Yaskin, A.~Zvavitch.
\bigskip

\begin{center}
\textbf{Myung-Sin Song: A Sampling Theory for Infinite Weighted Graphs (A preliminary talk)}\\
Joint work with P. E. T. Jorgensen and C. M. Traub.
\end{center}

Two sampling theorems are discussed for infinite (countable discrete) weighted graphs G; one example being ``large grids of resistors,"
i.e., networks and systems of resistors. Starting with a weighted graph G, and motivated by applications, there is natural ambient continuum $X$ containing $G$, and there are Hilbert spaces of functions on $X$ that allow interpolation by sampling values of the functions restricted only on the vertices in $G$ will be shown. This is introduced systematically: sampling functions on $X$ from their discrete values picked in the vertex-subset $G$. This indeed is sampling in the sense of Shannon and yields two theorems that allow for such realistic ambient spaces $X$ for a fixed graph $G$, and for interpolation kernels in function Hilbert spaces on $X$, sampling only from points in the subset of vertices in $G$.
\bigskip

\begin{center}
\textbf{
Tara D. Taylor:  Using Cantor Sets to Study the Connectivity of Sierpi\'{n}ski Relatives.
}
\end{center}

\noindent
This paper presents an exploration of the connectivity of the class of fractals known as the Sierpi\'{n}ski relatives.  The Sierpi\'{n}ski gasket (or triangle) is the most well-known relative.  The relatives are attractors of iterated function systems that involve the same contractive mappings as for the gasket, combined with symmetries of the square.  These relatives all have the same fractal dimension, but different topologies.  Some are completely disconnected, some are simply-connected, and some are multiply-connected.  For some of the relatives, one can determine the connectivity by considering certain Cantor sets that are subsets.  These Cantor sets are variations of the usual middle thirds Cantor set, and can be viewed in binary or quaternary instead of ternary.






\begin{center}
\textbf{
Jeehyeon See: Bi-Lipschitz embeddability of the Grushin plane into  Euclidean space.
}
\end{center}

\noindent
Many sub-Riemannian manifolds like the Heisenberg group do not admit bi-Lipschitz embedding into any Euclidean space. In contrast, the Grushin plane admits a bi-Lipschitz embedding into some Euclidean space. This is done by extending a bi-Lipschitz embedding of the singular line, using a Whitney decomposition of its complement.
\bigskip


\begin{center}
\textbf{Yang Wang:  Peano curves for fractals}
\end{center}
\bigskip

\begin{center}
\textbf{
Shijun Zheng: Semilinear Schroedinger Equation with Magnetic Potentials
}
\end{center}

In quantum physics the magnetic Schroedinger equation provides an ideal model
when a  Bose-Einstein Condensate is confined or released. As we will understand, the attractive or repulsive
property of the potential has implications in geometry as well as in physics.

 We will present the analytic approach to show the existence of the solution within its lifespan.  The techniques from Harmonic Analysis include
Strichartz and smoothing estimates, Besov and Sobolev embeddings for a magnetic perturbation,  and, bilinear estimates for fractional
derivatives. Time permitting we will also briefly review recent development for semilinear Schroedinger and wave equations
on Riemannian symmetric spaces.

\end{document} 