**Ryan Alvarado** (Missouri): *A sharp theory of Hardy spaces in
spaces of homogeneous type*.

One significant step in the development of the theory of
Hardy spaces (H^{p} spaces) was the consideration of
H^{p} defined on an
environment which, structurally speaking, is much more general than
the Euclidean ambient. In this talk I will present some recent progress with
M. Mitrea detailing a sharp H^{p}-theory in the abstract
setting of
spaces of homogeneous type. More specifically, in the above context we will
introduce Hardy spaces defined via a grand maximal function and prove that a
satisfactory H^{p}-theory exists for an optimal range of
p's, which depends
on both the geometric and measure theoretic aspects of the ambient.
Many facets of this theory will be discussed including sharp versions of
several tools used in the area of analysis on spaces of homogeneous type such
as a sharp Lebesgue differentiation theorem.

**Miguel Caicedo** (Cincinnati): *Well-posedness of a nonlinear
boundary value problem for the KdV equation on a finite domain*.

In this talk we consider an initial boundary value problem for the
Korteweg-de Vries equation posed on a finite interval, I=(0,L), with a
non-linear boundary condition at the left end point of the interval.
It will be shown that this initial boundary value problem is locally
well-posed in the L^{2}-based Sobolev space H^{s}(0,L)
for any s≥0.

**Dewey Estep** (Cincinnati): *Geometry of the prime end boundary and
the Dirichlet problem for bounded domains in metric measure spaces*.

First introduced in the complex plane by Caratheodory, prime ends provide a
way to define the boundary of a bounded domain such that its closure
retains many properties intrinsic to the structure of the domain itself,
rather than its ambient space. For example, the prime end closure of the
slit disk in \(\mathbb{C}\) retains the structure imposed by the 'slit', while
the normal metric closure ignores it. Using a slight modification of the
definition given by Adamowicz, Björn, Björn, and Shanmugalingam,
we may study prime ends in more general metric spaces. Here we define and
study the dirichlet problem with prime end boundary data on bounded domains,
showing that under certain assumptions we may construct solutions using the
Perron method.

**Lila Greco** (Kenyon): *Brownian motion in the complex plane*.

This project explores Brownian motion, a model of random motion,
in the plane. Given a domain in the complex plane and a basepoint in the
domain, start a Brownian traveler at that basepoint. The h-function of the
domain gives information about where the Brownian traveler is likely to
first hit the boundary of the domain. I will give examples of h-functions I
computed for several families of domains. Next I will describe a connection
between the geometry of domains and their h-functions. I will present
results about the convergence of a sequence h-functions. Finally, I will
end with approximating h-functions using simulations of Brownian motion.

**John Helms** (UC Santa Barbara): *Global solutions to 2-D quasilinear
wave equations*.

This work is in joint collaboration with Professor Thomas Sideris (UCSB).
We consider small-data solutions to equations of the form
where □ = ∂_{t}^{2} - Δ and the
nonlinearity Q is allowed
to depend on ∂u and ∂^{2}u at the quadratic level and
higher. We also assume that Q is linear in ∂^{2}u and that Q
satisfies a null condition, which is due to Christodoulou and Klainerman.
Alinhac proved global existence of small-data solutions with smooth,
compactly supported data (f,g) by using a "Ghost weight" in his main
energy estimate. Our proof extends Alinhac's result by allowing for a
weaker hypothesis on the initial data: (f,g) are only required to have a
certain amount of weighted Sobolev regularity with no restrictions on the
support. Our proof also eliminates the use of the Lorentz boosts
x_{i}∂_{t} + t∂_{i} (i = 1,2)
from the existence argument.

**Svetlana Jitomirskaya** (UC Irvine):
*Quasiperiodic operators with monotone potentials:
sharp arithmetic spectral transitions and small coupling
localization*.

It is well known that spectral properties of quasiperiodic
operators depend rather delicately on the arithmetics of the parameters
involved. Consequently, obtaining results for all parameters often
requires considerably more difficult arguments than for a.e. parameter,
and does offer a deeper insight. In the first part of the talk we will
report the first result of this kind in regard to the spectral
decomposition: full description of spectral types of the Maryland model
for *all* (in contrast with almost every, known for ~30 years)
values of frequency,
phase, and coupling (with nontrivial dependence on the arithmetics). In
the second part of the talk we show that for (a large class of) bounded
monotone potentials there is Anderson localization for all non-zero
couplings.

**Kay Kirkpatrick** (Illinois):
*Bose-Einstein condensation: from many quantum
particles to a quantum "superparticle."*

Near absolute zero, a gas of quantum particles can condense into an
unusual state of matter, called Bose-Einstein condensation (BEC), that
behaves like a giant quantum particle. The rigorous connection has
recently been made between the physics of the microscopic many-body
dynamics and the mathematics of the macroscopic model, the cubic
nonlinear Schrodinger equation (NLS). I'll discuss progress
with Gerard Ben Arous and Benjamin Schlein on a central limit theorem
for the quantum many-body dynamics, a step towards large deviations for
Bose-Einstein condensation.

**Phi Le** (Missouri): *Carleson measure estimate and Dirichlet
problems for degenerate elliptic equations*.

In this project, we were interested in Carleson measure estimates and the
solvability of Dirichlet problem for degenerate elliptic equations.
More precisely, we proved that if u is a bounded solution of the elliptic
equation Lu= - div A∇u = 0 in the domain
\({\textbf R}^{n+1}_+\)
where the elliptic matrix A is
t-independent, not necessarily symmetric,
and satisfies the weighted ellipticity condition

( joint work with Steve Hofmann and Andrew Morris).

**Jungang Li** (Wayne State): *Best constants for Moser's inequality on
noncompact Riemannian manifolds*.

We will consider the sharp Moser-Trudinger inequality on complete noncompact
Riemannian manifolds. Namely,
\[
\sup_{u\in W^{1,n}(M), ||u||_{1,\tau}\leq 1}\int_M
\phi(\alpha_n|u|^{\frac{n}{n-1}})dV_g\leq C(n,\tau)
\]
Where \( \phi(t)=\sum_{k=n-1}^{\infty}\frac{t^k}{k!}\),
\(\alpha_n=n\omega_{n-1}^{\frac{1}{n-1}}\), where ω_{n-1}
is the area of the unit sphere in **R**^{n},
\(||u||_{1,\tau}=(\int_M \tau|u|^n+|\nabla u|^n)^{\frac{1}{n}}\).
The inequality is sharp in the sense that for α > α_{n},
the above inequality fails.

**Xining Li** (Cincinnati): * Preservation of bounded geometry under
sphericalization and flattening: quasiconvexity and ∞-Poincare
inequality*.

This is a joint work with Estibalitz Durand-Cartagena.
In this work we explore the preservation of quasiconvexity and
∞-Poincare inequality under sphericalization and flattening in the
metric setting. The results developed in our previous work show that the
Ahlfors regularity, doubling property, and the p-Poincare
inequality for p < ∞ are preserved under the sphericalization and
flattening
transformations if one assumes the underlying metric space has annular
quasicovexity. In this work, we propose a weaker assumption to still
preserve quasiconvexity and ∞-Poincare inequality, called radial
starlike quasiconvexity and meridian starlike quasiconvexity, extending in
particular a result by Buckley, Herron and Xie to a wider class of metric
spaces and covering the case p = ∞ in our previous work.

**Alex Misiats** (Purdue): *Invariant measures for stochastic
reaction-diffusion equations*.

We study the long-time behavior of systems governed by nonlinear
reaction-diffusion type equations
du = (Au + f(u))dt + σ(u) dW(t), where A is an elliptic operator,
f and σ are nonlinear maps and W is an infinite dimensional
nuclear Wiener process. This equation is known to have a uniformly bounded
(in time) solution provided f(u) possesses certain dissipative
properties. The existence of a bounded solution implies, in turn, the
existence of an invariant measure for this equation, which is an important
step in establishing the ergodic behavior of the underlying physical
system.

In my presentation I will talk about expanding the existing class
of nonlinearities f and σ, for which the invariant measure exists.
We also show that the equation has a unique invariant measure if A is a
Schrödinger-type operator A = 1/ρ (div ρ∇u) where
ρ = e^{-|x|2} is the Gaussian weight.
In this case the source of dissipation
comes from the operator A instead of the nonlinearity f. The main idea
is to show that the reaction-diffusion equation has a unique
bounded solution, defined for all t ∈ **R**,
i.e. that can be extended
backwards in time. This solution is an analog of the trivial solution for
the linear heat equation.

**Guanying Peng** (Cincinnati): *Analysis of the Lawrence-Doniach
model for layered superconductors in magnetic fields*.

We analyze minimizers of the Lawrence-Doniach energy for layered
superconductors occupying a bounded generalized cylinder. For an applied
magnetic field H_{ex} = h_{ex}**e**_{3}
that is perpendicular to the layers with |ln ε| ≪
h_{ex} ≪ ε^{-2} as ε → 0,
where ε is the reciprocal of the Ginzburg-Landau parameter, we prove
an asymptotic formula for the minimum Lawrence-Doniach energy as
ε and the interlayer distance *s* tend to zero.
We also discuss some compactness and lower bound estimate results on
minimizers of the Lawrence-Doniach energy with the magnetic field in the
regime h_{ex} = O(|ln ε|). Part of this work is joint
work with P. Bauman at Purdue University.

**Carlos Pérez**
(UPV/EHU and Ikerbasque):
*On commutators of singular integral operators with
BMO functions*.

Commutators of singular integral operators with BMO functions were introduced
in the seventies by Coifman-Rochberg and Weiss. These are very interesting
operators for many reasons and their study became a classical topic in modern
harmonic analysis. One reason of this interest is due to the fact that they
are more singular than Calderón-Zygmund operators. This idea can be
expressed in many ways. In this lecture we plan to give three reasons showing
this "bad" behavior.

One of them is related to a sharp weighted L^{2}
estimate with respect to A_{2} weights. The novelty is that the
bound in term of the A_{2} constant of the weight is quadratic and no
better, while in the case of singular integrals it is simply linear. The second
reason is due to the fact that there is an appropriate local sub-exponential
decay, which in the case of singular integrals is of exponential type instead.
The third reason is related to the fact that commutators are controlled by
iterations of the maximal function with a sharp new A_{∞}
constant.

Pieces of the lecture are part of joint works with D. Chung and
C. Pereyra, with C. Ortiz and E. Rela, with T. Luque and E. Rela
and with T. Hytönen.

**David Smith** (Cincinnati): *Heat on a network*.

The recent Unified Transform Method of Fokas is a powerful tool in
the study of initial-boundary value problems for integrable nonlinear
evolution equations, where the spatial domain is a half-line or finite
interval. Remarkably, a linearization of the method can be used to solve all
such well-posed problems for linear evolution equations, and to determine
well-posedness. In this talk, we detail progress towards implementing the
linearized method for spatial domains made up of several intervals, with
interface conditions governing the behavior at the ends. As a simple
example, we study the diffusion of heat through networks of metal rods with
different material properties.

**Lu Zhang** (Wayne State): *L*^{p} estimate for a trilinear
pseudo-differential operator.

We study the L^{p} estimate for a trilinear pseudo-differential
operator with flag symbols. That is, the symbols are in the form of the
product of two standard symbols from the Hörmander class
${BS}_{1,0}^{0}$.
This operator is an extension from the trilinear operator with flag
singularities, with the symbols in the form of product of two
Marcinkiewcz-Mikhlin-Hörmander symbols. This extends the work of
C. Muscalu on L^{p} estimates for a trilinear operator of Fourier
multipliers of flag singularity to the case of pseudo-differential operator
setting. Our work is based on the use of paraproducts and some careful decay
estimates. This is joint work with Guozhen Lu.