**Rodrigo Bañuelos** (Purdue):
*Fourier multipliers arising from Lévy
processes*.

We will discuss recent applications of martingale inequalities to
Fourier multipliers that arise from transformations of Lévy
processes via the Lévy–Khintchine formula.
While these results are of interest on
their own right, they are motivated by applications to some classical
multipliers such as Riesz transforms, the Beurling-Ahlfors operator,
Marcinkiewicz multipliers, and connections to rank-one convexity and
quasiconvexity.

Some background material can be found in the speaker's overview article "The
foundational inequalities of D. L. Burkholder and some of their
ramifications," Illinois J. Math. Volume 54, Number 3 (2010), 789-868.

**Thomas Bieske** (South Florida): *Generalizations of a Laplace-type
equation in the Heisenberg group and class of Grushin-type spaces*.

Beals, Gaveau and Greiner (1996) found the fundamental solution
to a 2-Laplace-type equation in a class of sub-Riemannian spaces. This
solution is related to the well-known fundamental solution to the p-Laplace
equation in Grushin-type spaces (Bieske-Gong, 2006) and the Heisenberg
group (Capogna, Danielli, Garofalo, 1997). We extend the 2-Laplace-type
equation to a p-Laplace-type equation. We show that the obvious
generalization does not have desired properties, but rather, our
generalization does preserve some natural properties.

**Tanya Christiansen** (Missouri):
*Counting resonances for Schrödinger
operators*.

Physically, a *resonance* may correspond to a decaying wave.
This is in contrast with an eigenvalue of a selfadjoint operator, which in
many models corresponds to a periodic wave. Mathematically, resonances are
analogs of discrete spectral data on certain noncompact domains or manifolds.
Their behavior is less well understood than that of eigenvalues of
selfadjoint operators, leaving many questions unanswered.

We give an introduction to resonances and the problem of
counting them, concentrating on the case of
Schrödinger operators on **R**^{d}. A subtext of
the talk is that Schrödinger operators with complex potentials
both provide intriguing examples and, in some settings, can facilitate proofs
about Schrödinger operators with real-valued potentials.

**Robert Hardt** (Rice):
*Some homology and cohomology theories for a metric
space*.

Various classes of chains and cochains may reveal geometric as
well as topological properties of metric spaces. In 1957, Whitney introduced
a geometric "flat norm" on polyhedral chains in Euclidean space, completed
to get flat chains, and defined flat cochains as the dual space. Federer and
Fleming also considered these in the sixties and seventies, for homology and
cohomology of Euclidean Lipschitz neighborhood retracts. These include
smooth manifolds and polyhedra, but not algebraic varieties or subspaces of
some Banach spaces. In works with Thierry De Pauw and Washek Pfeffer, we
find generalizations and alternate topologies for flat chains and cochains
in general metric spaces. With these, we homologically characterize
Lipschitz path connectedness and obtain several facts about spaces that
satisfy local linear isoperimetric inequalities.

**Nathan Pennington** (Kansas State): * Local and global existence for
the Lagrangian Averaged Navier-Stokes equations in Besov spaces*.

Through the use of a non-standard Leibniz rule estimate, we prove
the existence of unique local and global solutions to the incompressible
isotropic Lagrangian Averaged Navier-Stokes equation with initial data in
various categories of Besov spaces. Specifically, for
p > n, we get local existence with initial data u_{0} ∈
${\mathrm{B}}_{\mathrm{p},\mathrm{q}}^{\mathrm{r}}$(**R**^{n}) for r > 0.
For p = 2, we get local existence with
initial data
u_{0} ∈
${\mathrm{B}}_{2,\mathrm{q}}^{\mathrm{n}/2-1}$(**R**^{n}) and the local solution
can be extended to a global solution for n = 3, 4.

**Armin Schikorra** (Planck Institute): *Knot-energies and fractional
harmonic maps*.

We will present a proof that curves (knots) which are stationary for
the Moebius energy are smooth in the critical dimension.
This energy was introduced by O'Hara (*Topology*, 30(2):241–247,
1991),
and it was shown by Freedman, He and Wang (*Ann. of Math.* (2),
139(1):1–50, 1994), and He (*Comm. Pure Appl. Math.*,
53(4):399–431,
2000) that minimizers of this energy are smooth, using a geometric
argument employing crucially the Moebius invariance of this energy.

Our approach, however, which shows regularity even for critical
points, does not rely on this geometric invariance, but rather uses
analytic methods from Harmonic Analysis and Potential Theory inspired
by the regularity arguments of fractional harmonic maps by Da
Lio-Riviere (*APDE*, 4(1):149–190, 2011;
*Advances in Mathematics*,
227:1300–1348, 2011) and the speaker
(*J. Differential Equations*, 252:1862–1911, 2012;
Preprint, arXiv:1103.5203, 2011).

This is joint work with S. Blatt and P. Reiter.

**Alex Stokolos** (Georgia Southern): *On the rate of a.e. convergence of
certain classic integral means*.

We are going to present several nearly optimal results concerning
the rate of almost everywhere convergence of the Fourier integral means
T_{t} f. A typical result for these means is the following:

If the function f belongs to the Besov space
${\mathrm{B}}_{\mathrm{p},\mathrm{p}}^{\mathrm{s}}$
in the range 1 < p < ∞, 0 < s < 1, then
T_{t} f(x) − f(x) = o_{x}(t^{s})
a.e. as t→0^{+}.

The above statement is valid for the Abel-Poisson, Gauss-Weierstrass,
Bochner-Riesz means in particular. This is joint work with Walter Trebels.

We will demonstrate that the initial-boundary-value problem (IBVP)
of the nonlinear Schrödinger equation, when posed on the half line
**R**^{+}, is well-posed in the space
H^{s}(**R**^{+}) for any s≥0 with
φ ∈ H^{s}(**R**^{+}),
h ∈
${\mathrm{H}}_{\mathrm{loc}}^{\frac{2s+1}{4}}$(**R**^{+}).

The IBVP of the equation when posed on the finite interval (0,L) is
well-posed in the space H^{s}(0,L) for any s≥0 with
φ ∈ H^{s}(**R**^{+}),
h_{1}, h_{2} ∈
${\mathrm{H}}_{\mathrm{loc}}^{\frac{s+1}{2}}$(**R**^{+}).

The results reported in this talk are joint work with Jerry L. Bona of
University of Illinois at Chicago and Shuming Sun of Virginia Tech.