**Roza Aceska** (Ball State): *Scalability generated by iterative
actions of operators*.

We study systems of vectors of the form (1) {\(A^n\)*g* :
*g* ∈ *G*, *n* = 0,
1, 2, . . . },
where *A* is a bounded operator on a Hilbert space *H* and
*G* is a discrete set of vectors in *H*.
Recent results have made it clear if (1) is a frame, complete Bessel system
etc.
This opens the question of frame scalability; some of the recent results and
open questions
are discussed. The problem originates from a sampling theory problem,
where the unknown signal is recovered from its spatio-temporal samples

**Ryan Berndt** (Otterbein): *The weighted Fourier inequality,
polarity, and reverse Hölder inequality*.

We examine the problem of the Fourier transform mapping one weighted
Lebesgue space into another, by studying necessary conditions and sufficient
conditions which expose an underlying geometry. In the necessary conditions,
this geometry is connected to an old result of Mahler concerning the the
measure of a convex set and its geometric polar being essentially
reciprocal. An additional assumption, that the weights must belong to a
reverse Hölder class, is used to formulate the sufficient condition.

**Bob Booth** (North Carolina): *Localized energy for wave equations
with degenerate trapping, Part II*.

When studying wave equations on differentiable manifolds, it is known that
geodesic trapping necessitates a loss in standard local energy estimates.
For non-degenerate hyperbolic trapping, the loss is logarithmic. For
elliptic trapping, everything is lost except a logarithm. In our work, we
consider a manifold with degenerate hyperbolic trapping, attaining an
algebraic loss. This talk will utilize a first weaker estimate that is
interesting in its own right. We will focus on the use of a WKB inspired
analysis to attain the sharp result and prove sharpness via a quasimode
construction. This is joint work with Hans Christianson, Jason Metcalfe,
and Jacob Perry.

**Josh Brummer** (Kansas State): *Bilinear operators with
homogeneous symbols, smooth molecules, and Kato-Ponce inequalities*.

We present a unifying approach to establish mapping properties for bilinear
pseudodifferential operators with homogeneous symbols in the settings of
function spaces that admit a discrete transform and molecular
decompositions in the sense of Frazier and Jawerth. As an application, we
obtain related Kato-Ponce inequalities. This is a joint collaboration with
Virginia Naibo.

**Erin Compaan** (Illinois): * Well-posedness and nonlinear smoothing
for the "good" Boussinesq equation on the half-line*.

In this talk we consider the regularity properties of the "good"
Boussinesq equation on the half line. We discuss local existence, uniqueness
and continuous dependence on initial data in low-regularity spaces.
Moreover, we show that the nonlinear part of the solution is up to smoother
than the initial data. The method involves extending the problem to the real
line and using a contraction argument there. A forcing term is used to
enforce the boundary conditions. This approach allows us to apply powerful
harmonic analysis methods, which a priori do not seem useful in the
half-line setting, to the problem. The result improves that in
[Himonas-Mantzavinos 2015], being the first result that constructs solutions
below the *L*^{2} space. Our theorems are sharp within the framework of the
restricted norm method that we use and match the known results on the full
line. This is joint work with Nikolaos Tzirakis.

**Stephen Deterding** (Kentucky): *Bounded point derivations on
R*^{p}(X) and approximate derivatives.

Let *X* be a compact subset of the complex plane and let
*R*_{0}(X) denote
the set of all rational functions whose poles lie off *X*.
The function space
*R(X)* is the uniform closure of *R*_{0}(X). The
space *R*^{p}(X) is an important function space that is
closely related to *R(X)*. For 1 < *p* < ∞,
*R*^{p}(X) is the closure of *R*_{0}(X) in the
*L*^{p} norm.

A bounded point derivation on *R(X)* at a point
*x*_{0} is a bounded linear functional *D* on *R(X)*
such that
*D(fg) = D(f)g(x*_{0}) + D(g)f(x_{0}) for
all functions *f*, *g* belonging to *R(X)*. It is known if
there is a bounded point
derivation on *R(X)* at *x*_{0}, then every function in
*R(X)* has an
approximate derivative at *x*_{0}. An approximate
derivative is defined in the same way as the usual derivative except that
instead of taking the limit in the difference quotient over all of *X*,
the limit is taken over a subset of *X* with full area
density at *x*_{0}.
One can also define bounded point derivations for
*R*^{p}(X),
although a slightly different definition must be made.

In this talk we will introduce bounded point derivations on
*R*^{p}(X) and discuss when the existence of a bounded point
derivation on *R*^{p}(X) implies that
every function in *R*^{p}(X) has an approximate derivative at
*x*_{0}.

**Maxim Gilula** (Michigan State): *An analytic perspective on
stability for oscillatory integrals in higher dimensions*.

Given real analytic functions \(\phi,
\psi:\mathbb{R}^d\to\mathbb{R}\), we consider scalar oscillatory integrals
with phase \(\phi+s*\psi\) for small real *s*. It has been known since
1976 that the estimate of such an integral is not uppersemicontinuous in the
real variable *s* in general
(with respect to the oscillatory variable, due to Varchenko). However,
the techniques used to develop such a conclusion
were not robust enough to examine how exactly the oscillatory integral
estimate changes as \(s\to 0\), but only determined that the estimates may be
different asymptotically in the oscillatory variable.

Under the same
nondegeneracy condition of the phase assumed by Varchenko when he showed
stability does not hold in general, we examine some simple cases where we
can predict exactly how the stability does not hold by considering any
finite oscillatory variable instead of just considering the asymptotic
behavior. We argue that in some sense stability *does* hold if we consider
this perspective instead of the classical one. The Newton polyhedron plays
an important role in determining the estimates of such integrals.

**Fazel Hadaifard** (Kansas): *The global regularity of the 2D
critical Boussinesq system with α > 2/3*.

We examine the question for global regularity for the Boussinesq equation with
critical fractional dissipation (α, β): α + β = 1.
The main result states that the system admits global regular
solutions for all (reasonably) smooth and decaying sata, as long as
α > 2/3. This improves upon some recent works.

The main new idea is the introduction of a new, second generation
Hmidi-Keraani-Rousset type change of variables which further improves the
linear derivative in temperature term in the vorticity equation.
This approach is then complemented by a new set of commutator estimates
(in both negative and positive index Sobolev spaces).

**Keaton Hamm** (Vanderbilt): *Regular families of kernels for
nonlinear approximation*.

Best *N*-term wavelet approximations for Triebel-Lizorkin smoothness
spaces have been studied by DeVore, Jawerth, and Popov (Amer. J. Math. 1992)
for many wavelet systems. Analogously, best *N*-term approximations from
the space of Gaussian approximants given by
\[
\mathcal{G}_N := \bigg\{ \sum_{j=1}^N a_j e^{-\frac{|x-x_j|^2}{\sigma_j^2}}:
(a_j) \subset \mathbb{C}, (x_j) \subset \mathbb{R}^d, (\sigma_j) \subset
(0,\infty) \bigg\}
\]
were considered by Hangelbroek and Ron (J. Funct. Anal. 2010). In each case,
optimal rates of \(O(N^{-s/d})\) are obtained where *s* is the
smoothness parameter of the Triebel-Lizorkin space and *d* is the
dimension. We give sufficient conditions on a family of approximating
kernels \((\phi_\alpha)_{\alpha \in A}\) such that the space
\[
\Phi_N := \bigg\{ \sum_{j=1}^N a_j \phi_{\alpha_j}(x-x_j):
(a_j) \subset \mathbb{C}, (x_j) \subset \mathbb{R}^d, (\sigma_j) \subset
(0,\infty) \bigg\}
\]
yields the same approximation orders.

**Alex Iosevich** (Rochester):
*On a bilinear analog of the spherical averaging
operator*.

The classical spherical averaging operator \(Af(x)=\int_{S^{d-1}}
f(x-y) d\sigma(y)\), where σ is the surface measure on \(S^{d-1}\), maps
\(L^p({\Bbb R}^d) \to L^q({\Bbb R}^d)\) if (1/*p*,1/*q*) is in the
closed
triangle with the endpoints (0,0), (1,1) and (*d/d*+1, 1/*d*+1).
This is
a classical result due to Littman and Strichartz. We are to consider a
family of bilinear operators
\[ B_{\theta}(f,g)(x)=\int f(x-y)g(x-\theta y) d\sigma(y),\]
where \(\theta \in O_d({\Bbb R})\). We are going to determine a sharp range of
exponents
\(p,q,r \ge 1\) such that
\(B_{\theta}: L^p({\Bbb R}^d) \times L^q({\Bbb R}^d) \to L^r({\Bbb R}^d)\).
This operator is motivated by problems in Erdos
combinatorics related to counting the number of times a given triangle
congruence class may arise in a finite point set in \({\Bbb R}^d\). We shall
also discuss applications to combinatorial geometry.

**Ramesh Karki** (Indiana - East): *Optimal reconstruction of initial
data in some evolutionary PDEs via finite discrete samplings*.

Developing an approach introduced in a recent paper by DeVore and
Zuazua [1], we show how to reconstruct optimally initial data, in a suitable
Sobolev class, for some evolutionary PDEs, using only finite discrete
measurements. In particular, we show that for our classes of PDEs the
optimal sampling does not depend on the spectrum of the operators involved,
but just on the order of the PDE, thus answering a question posed in [1]. We
also tackle the same problem in the case in which the coefficients of the
PDEs depend explicitly on time, thus generating a non-autonomous dynamical
system. We finally comment on the possibility of using a variation of this
approach to deal with some nonlinear integro-differential equations or
non-linear PDEs that are C-integrable.

This is joint work with R. Aceska and A. Arsie.

**Rajinder Mavi** (Michigan State): *Anderson localization with
degenerate spectrum*.

Random Schrodinger operators in strong disorder are known to obtain a
phase, known as Anderson Localization, of pure point dense spectrum
with exponentially localized eigenfunctions. An important
characteristic of this phase is that in a properly renormalized
setting the eigenvalue process approximates a Poisson process. We will
investigate models with strong disorder where eigenvalues will
approach a finite or infinite degeneracy. In such models we
demonstrate a partial result in Anderson localization, and discuss the
effect of tunneling in delocalizing the remaning portion of the
wavepacket.

This talk is based on joint work with Jeffery Schenker.

**Darío Mena Arias** (Georgia Tech): *Sparse operators and the
sparse T1 theorem*.

We impose standard *T*1-type assumptions on a
Calderón-Zygmund operator *T*, and deduce that
for bounded compactly supported functions *f*, *g* there is a sparse
bilinear form \(\Lambda\) so that
\[
\lvert \langle T f, g \rangle\rvert \lesssim \Lambda (f,g).
\]
The proof is short and elementary. The sparse bound quickly implies all the
standard mapping
properties of a Calderón-Zygmund on a (weighted) \(L ^{p}\) space.

**Alex Mesiats** (Courant): *Convex duality in nonconvex variational
problems*.

We consider the minimization problem which models martensitic
(diffusionless) phase transitions in rectangular domains. Physical
experiments suggest that if opposite phases are present at opposite sides,
the transition has a form of a zig-zag wall. My presentation addresses the
mathematical model of this phenomenon, which involves the minimization of
singularly perturbed 2D and 3D elastic energies with phase constraints. By
means of sharp upper and lower bounds, we show that the experimentally
observed zig-zag structure provides optimal energy scaling law. Despite the
fact that the problem is highly nonconvex due to the presence of nonconvex
phase constraints and the singular perturbation, in my talk, I will
describe a relaxation method which allows to use the convex duality
technique for the purpose of obtaining a sharp lower bound.

**Sergii Myrochnychenko** (Kent State): *On the uniqueness of
polytopes with congruent spherical projections*.

We will discuss the problem of unique determination of convex bodies with
congruent spherical projections. Consider two convex bodies *K*,
*L*
contained in the interior of a unit ball **B**^{n},
n ≥ 3. Assume
that for any point *z* on the unit sphere *S*^{n}
(a source of light), the
spherical shadows *K*_{z} and *L*_{z}
of the bodies are congruent
(here *K*_{z} = {x ∈ *S*^{n-1}:
[*x,z*] ∩ *K* ≠ ∅}). Does it follow that
*K=L*?
We give the affirmative answer to this question in the class of convex
polytopes.

**Andrea Nahmod** (Massachusetts):
*Randomization and dynamics in nonlinear wave and
dispersive equations*.

In this talk we show how certain well-posedness results that
are not available using only deterministic techniques (eg. Fourier and
harmonic analysis) can be obtained when introducing randomization in the
set of initial data and using powerful but still classical tools from
probability as well. These ideas go back to seminal work by J. Bourgain on
the invariance of Gibbs measures associated to dispersive PDE. We will
first explain some of these ideas and review some recent probabilistic
well-posedness results. We will then describe recent work of myself joint
with Chanillo, Czubak, Mendelson and Staffilani in which we
treat probabilistic well-posedness of a geometric wave equation with
randomized supercritical data.

**Bae Jun Park** (Wisconsin): *On the boundedness of
pseudo-differential operators on Triebel-Lizorkin spaces*.

I will talk about some endpoint boundedness properties of pseudo-differential
operators in *OP*(\(\mathcal{S}_{\rho, \rho}^m\)), 0 < ρ < 1, on
Triebel-Lizorkin spaces. It was proved that operators in
*OP*(\(\mathcal{S}_{\rho, \rho}^m\)) are bounded in \(h^p\),
0 < *p* < ∞, and Sobolev spaces with some restriction on *m*.
I extend these results to Triebel-Lizorkin spaces. The result is motivated
by the boundedness of a radial multiplier
*m*(D) = \(e^{-2\pi i|D|^{(1-p)}}(1+|D|^2)^{m/2}\), which is a typical
example of operators in *OP*(\(\mathcal{S}_{\rho, \rho}^m\)).

**Guanying Peng** (Cincinnati): *Regularity of the Eikonal equation
with two vanishing entropies*.

We study regularity of solutions to the Eikonal equation \(|\nabla u|=1\)
a.e. in a bounded simply-connected two dimensional domain.
With the help of two vanishing entropies, we prove that solutions of the
Eikonal equation are locally Lipschitz continuous, except at a locally finite
set of points in the domain. The motivation of our problem comes from the zero
energy state of the Aviles-Giga functional in connection with the theory of
smectic liquid crystals and thin film blisters. Our results for the first time
use only two entropies to characterize regularity properties in this
direction.

This is joint work with Andrew Lorent at the University of Cincinnati.

**Jacob Perry** (North Carolina): *Localized energy for wave equations
with degenerate trapping, Part I*.

When studying wave equations on differentiable manifolds, it is known that
geodesic trapping necessitates a loss in standard local energy estimates.
For non-degenerate hyperbolic trapping the loss is logarithmic, while for
elliptic trapping everything is lost except a logarithm. In our work, we
consider an asymptotically flat manifold with degenerate hyperbolic
trapping. This talk will develop a local energy estimate that holds
everywhere except on the trapped set, via separate analysis of high and low
frequency behavior. This estimate allows for local analysis near the
trapped set, where algebraic loss can be attained. This talk is part of a
joint work with Robert Booth, Hans Christianson, and Jason Metcalfe.

**Dat Pham** (Wayne State): *Lipschizian and Hölderian full
stability for parametric variational systems*.

The paper studies the notions of Lipschitzian and Hölderian full
stability of solutions to general parametric variational systems described via
partial subdifferential and normal cone mappings acting in Hilbert spaces. We
derive sufficient conditions of those full stability notions without assuming
differentiability and computable formula for the modulus of prox-regularity of
lower semicontinuous functions. The obtained results combining with related
results are specified for important classes of variational inequalities and
variational conditions in both finite and infinite dimensions.

This is joint work with Boris Mordukhovich (Wayne State University)
and Nghia Tran (Oakland University).

**Stefanie Petermichl** (Toulouse):
*On the matrix A*_{2} conjecture.

The condition on the matrix weight W that is necessary and sufficient
for the *L*^{2}(W) boundedness of the Hilbert transform,
the matrix *A*_{2}
condition, is known since 1997 (Treil-Volberg). Good or sharp
quantitative norm control depending on the *A*_{2}
characteristic of the
weight in the 'scalar' case dates to 2007 (P.).
Despite notable improvement and many new techniques that apply in the
scalar setting, the matrix *A*_{2} question for the Hilbert
transform
remains unsolved. We present the best to date estimate
(Nazarov-P.-Treil-Volberg), via the use of certain convex bodies.
Certain maximal functions of Christ-Goldberg type already enjoy best
estimates in the matrix setting. We present the first sharp estimate
of a singular operator in this setting, the matrix weighted square
function (Hytonen-P.-Volberg).

**Matthew Romney** (Illinois): *Geometry of Grushin space*.

The Grushin plane is a standard example of a non-equiregular
sub-Riemannian manifold. This talk will discuss quasiconformal and
bi-Lipschitz mappings from the Grushin plane into Euclidean space, with
emphasis on how these results relate to more classical aspects of
quasiconformal mapping theory. We also discuss generalizations to a broader
class of Grushin-type spaces. Portions of this talk are based on joint work
with V. Vellis, C. Gartland, and D. Jung.

**Luis San Martín Jiménez** (UNAM): *A connection between
two initial Dirichlet-type problems for parabolic equations*.

We study operators \(L=\text{div}(A(x,t)\nabla)-\partial_t\) where
\(A(x,t)\) is a smooth and symmetric matrix satisfying the ellipticity
condition
\[\lambda|\xi|^2\leq\sum_{i,j=1}^{n}A_{ij}(x,t)\xi_i\xi_j
\leq\lambda^{-1}|\xi|^2,\quad\quad\quad (x,t)\in\mathbb{R}^{n+1},
\quad\xi\in\mathbb{R}^n.\]
We consider solutions to \(Lu=0\) in a Lipschitz cylinder
\(\Omega_T\), and define a Regularity problem (R)_{p} in this setting.
Finally, we prove that (R)_{p} implies (D)_{p'},
where *p* and *p'* are conjugate exponents.

**Morgan Schreffler** (Kentucky): *Approximation of solutions to
the mixed problem on Lipschitz domains*.

The mixed problem, or Zaremba's problem, has many physical applcations.
In this talk we consider a method for approximating solutions to the mixed
problem on a Lipschitz domain Ω. Further, we determine a desirable
rate of convergence and attempt to formally differentiate with respect to the
approximating parameter ε.

**Christopher Sogge** (Johns Hopkins):
*On the concentration of eigenfunctions*.

I shall present some results in global harmonic analysis that
concern properties of eigenfunctions on compact Riemannian manifolds. Using
local arguments we can show that *L*^{p} norms of
eigenfunctions over the
entire manifold are saturated if and only if there are small balls (if
*p* is large) or small tubular neighborhoods of geodesics
(if *p* is small) on
which the eigenfunctions have very large *L*^{p} mass.
Neither can occur on
manifolds of nonpositive curvature, or, more generally, on manifolds
without conjugate points.

**Gareth Speight** (Cincinnati): *Porosity and differentiability in
Euclidean spaces and Carnot groups*.

Rademacher's theorem states that Lipschitz functions between
Euclidean spaces are differentiable almost everywhere. Investigating
validity of a converse to Rademacher's theorem leads to the construction of
small universal differentiability sets, which contain points of
differentiability for all Lipschitz functions. Porous sets are sets with
relatively large holes on arbitrarily small scales, and have applications to
the study of differentiability. For instance, a universal differentiability
set cannot be a countable union of porous sets.

We discuss measure zero
universal differentiability sets in the Heisenberg group and applications of
porosity to differentiability in Carnot groups, which can be used to help
prove Pansu's differentiability theorem for Lipschitz mappings.

**Eric Stachura** (Haverford): *Existence of propagators for
Coulomb-like potentials in density functional theory*.

We prove existence of propagators for a time dependent Schrödinger
equation with a new class of softened Coulomb potentials, which we allow to be
time dependent, in the context of time dependent density functional theory.
We compute explicitly the Fourier transform of these new potentials, and
provide an alternative proof for the Fourier transform of the Coulomb
potential. Finally we show the new potentials are dilatation analytic, and so
the spectrum of the corresponding Hamiltonian can be fully characterized.

**Jeremy Tyson** (Illinois):
*Quasiregular mappings of Euclidean space and the
Heisenberg group*.

I will discuss two lines of research related to quasiregular
mappings. A continuous map *f* is quasiregular (QR) if it lies in a
suitable (local) Sobolev space and satisfies a uniform relative analytic
distortion condition. Roughly speaking, the singular values of the (a.e.
defined) differential matrix *Df* should be comparable up to a fixed
multiplicative constant. We allow the possibility that *Df* vanishes in a
point. Quasiregular mappings are noninjective generalizations of
quasiconformal mappings, and higher-dimensional analogs of holomorphic
functions. Topologically, they are branched covers, in particular, they are
local homeomorphisms off of a small branch set.

Higher-dimensional QR maps exhibit certain types of rigidity not shared by
their planar counterparts. Specifically, QR maps whose distortion is
sufficiently small, or which are sufficiently regular, are necessarily
unbranched. I will survey the literature on such rigidity results in the
Euclidean setting.

Second, I will discuss the QR ellipticity problem. A Riemannian
*n*-manifold is quasiregularly elliptic if it receives a nonconstant
QR map
from \({\mathbb R}^n\). QR ellipticity will be discussed both for mappings
\(f:{\mathbb R}^n \to N\) into a Riemannian *n*-manifold, and mappings
\(f:{\mathbb H}^n \to N\) into a contact sub-Riemannian manifold. Here
\({\mathbb H}^n\) denotes the Heisenberg group, which we equip with its
standard sub-Riemannian metric structure. The theory of QR mappings between
sub-Riemannian manifolds is in its infancy, and the Heisenberg QR
ellipticity problem is a good testbed to try out various definitions of
sub-Riemannian quasiregularity.

**Vladimir Vinogradov** (Ohio University): *On applications of special
functions in probability theory*.

We employ numerous analytical methods and special functions for solving
problems of probability theory.
Some of our results illustrate the widely recognized fact that probability
theory is an excellent source of
new problems for analysis. We demonstrate how such interplay between these
two disciplines is effective in
providing a simple derivation of new assertions of probability theory, as
well as in finding a meaningful
interpretation of particular analytical results, or even in deriving
previously unknown representations for some
special functions.

This is joint work in collaboration with Richard Paris (Abertay University).

**Scott Zimmerman** (Pittsburgh): *Sobolev extensions of Lipschitz
mappings into metric spaces*.

Wenger and Young proved that the pair \((\mathbb{R}^m,\mathbb{H}^n)\) has the
Lipschitz extension property for *m* ≤ *n* where
\(\mathbb{H}^n\) is the
sub-Riemannian Heisenberg group. That is, for some *C* > 0,
any *L*-Lipschitz
map from a subset of \(\mathbb{R}^m\) into \(\mathbb{H}^n\)can be extended to
a *CL*-Lipschitz mapping on \(\mathbb{R}^m\).

In this talk, I construct
Sobolev extensions of such Lipschitz mappings with no restriction on the
dimension *m*. I will show that any Lipschitz mapping from a compact
subset of \(\mathbb{R}^m\) into \(\mathbb{H}^n\) may be extended to a Sobolev
mapping
on any bounded domain containing the set. More generally, I will explain
this result in the case of mappings into any Lipschitz (*n*-1)-connected
metric space.