This talk will discuss results on singularity formation for two different fluid flow problems,
and the physical significance of these results.

The first problem is the so-called Muskat problem, which describes the evolution
of the interface between two immiscible fluids in a porous medium or Hele-Shaw cell. In contrast to the Hele-Shaw problem (the one phase version of the Muskat problem), there are few nontrivial exact solutions or analytic results for the Muskat problem.

For the stable Muskat problem, in which the higher viscosity fluid expands into the lower-viscosity
fluid, it is shown that smooth solutions exist for all t>0, even if the initial data contains weak (e.g., curvature) singularities.

For the unstable problem, in which the higher viscosity fluid contracts, solutions are constructed that start
off smooth but develop a singularity in finite time.

The second example is the unsteady Prandtl equations for flow in an incompressible boundary layer.
A semi-analytic method for constructing singular solutions is suggested, and some preliminary results
toward this construction are presented.

This is joint work with Russ Caflisch and Sam Howison.

We investigate certain operator-valued symbols that arise
from elliptic or parabolic equations on wedge domains, and from free boundary problems with moving contact lines in the context of phase transitions and fluid flows. We show that the associated symbols lead to well-posed evolution problems. The tools involve recent results on maximal regularity for
non-commuting operators. (Joint work with J. Pruess.)

I will present a statistical theory of reactive trajectories between two pre-specified sets A and B, i.e. the portions of the path of a
Markov process during which the path makes a transition from A to B. This
problem is relevant e.g. in the context of metastability, in which case
the two sets A and B are metastable sets, though the formalism we propose is independent of any such assumptions on A and B. Various probability distributions on the reactive trajectories can be expressed in terms of the equilibrium distribution of the process and the so-called committor functions which give the probability that the process reaches first B before reaching A, either backward or forward in time. Using these objects, the following can be obtained: (i) the distribution of reactive trajectories, which gives the proportion of time reactive trajectories spend in sets outside of A and B; (ii) the hitting point distribution of the reactive trajectories the integral of which on a surface gives the net average flux
of reactive trajectories across this surface; and (iv) the average frequency of reactive trajectories, which gives the average number of transitions between A and B per unit of time. Time permitting, I will also illustrated how this theoretical framework can be used to design a numerical algorithm, termed the string method, to identify the committor function in high
dimensional systems.

One of the central questions in Geometric Analysis is the
interplay between the curvature of the manifold and the spectrum
of an operator.

In this talk, we will be considering the Hodge Laplacian on
differential forms of any order $k$ in the Banach Space $L^p$. In
particular, under sufficient curvature conditions, it will be
demonstrated that the $L^p\,$ spectrum is independent of $p$ for
$1\!\leq\!p\!\leq\! \infty.$ The underlying space is a
$C^{\infty}$-smooth non-compact manifold $M^n$ with a lower bound
on its Ricci Curvature and the Weitzenb\"ock Tensor. The further
assumption on subexponential growth of the manifold is also
necessary. We will see that in the case of Hyperbolic space the
$L^p$ spectrum does in fact depend on $p.$

As an application, we will show that the spectrum of the Laplacian
on one-forms has no gaps on certain manifolds with a pole and on
manifolds that are in a warped product form. This will be done
under weaker curvature restrictions than what have been used
previously; it will be achieved by finding the $L^1$ spectrum of
the Laplacian.

Time permitting, we will take a short look at an alternative
method for finding the Gaussian Heat kernel bounds for the Hodge
Laplacian via Logarithmic Sobolev Inequalities. Such bounds are
necessary in the proof of the $L^p$ independence.

Let $M$ be a compact manifold of dimension $n \geq 3$. Along the > Yamabe flow, a Riemannian metric on $M$ is deformed according to the > equation $\frac{\partial g}{\partial t} = -(R_g - r_g) \, g$, where $R_g$ > is the scalar curvature associated with the metric $g$ and $r_g$ denotes > the mean value of $R_g$. > > It is known that the Yamabe flow exists for all time. Moreover, if $3 \leq > n \leq 5$ or $M$ is locally conformally flat, then the solution approaches > a metric of constant scalar curvature as $t \to \infty$. I will describe > how this result can be generalized to dimensions $6$ and higher under a > technical condition on the Weyl tensor. The proof requires the > construction of a suitable family of test functions.