Self-intersection local time $\beta_t$ is a measure of how often
a Brownian motion (or other process) crosses itself. Since Brownian
motion in the plane intersects itself so often, a renormalization
is needed in order to get something finite. LeGall proved that
$E e^{\gamma \beta_1}$ is finite for small $\gamma$ and infinite
for large $\gamma$. It turns out that the critical value is related
to the best constant in a Gagliardo-Nirenberg inequality. I will discuss
this result (joint work with Xia Chen) as well as large deviations
for $\beta_1$ and $-\beta_1$ and LILs for $\beta_t$ and $-\beta_t$.
The range of random walks is closely related to self-intersection
local times, and I will also discuss joint work with Jay Rosen
making this idea precise.

Consider a compact manifold $M$ (e.g. a torus) equipped with
a smooth measure $\mu$ (e.g. Lebesgue measure in the case
of torus) as a probability space $(M,\mathcal M,\mu)$. Consider
an ergodic map $T:M \to M$ along with a smooth function
$p:M \to (0,1)$. Define a random walk along orbits of $T$ as follows:
a point $x$ jumps to $T x$ with probability $p(x)$ and
to $T^{-1} x$ with probability $1-p(x)$.
Is there a limiting distribution of such a random walk for a generic
initial point? Is it absolutely continuous with respect to $\mu$?
We shall present an answer for several essentially different
maps $T$.

This will be a survey of one family of results which
followed the famous question "Can you hear the shape of a drum? (Mark Kac
1966). I will discuss the ways that the spectrum (energy levels) of a
(two
body) Schrodinger operator constrain the possible potentials for the
interaction.

The events of 9/11 in the US changed the way we look at routine activities such as air and mass-transportation travel. We (as a society) are somewhat prepared to respond to natural acts (epidemics, earthquakes, etc.) but have no data or reliable information that would help in the planning or identification of a set of responses if a deliberate act (against unsuspecting population) were to take place. I will highlight some of the challenges that we face and outline the use of mathematical models in our efforts to meet some of them. I will use recent SARS and foot and mouth epidemics to ground some of the ideas. Should we prepare for worst case scenarios? If so, how do we define worst case scenarios mathematically? I will conclude with the use of some of these ideas on the potential impact or consequences associated with the deliberate release of a biological agent in the mass transportation system of a major metropolitan area.

Mathematical Models and Their Application to the Spread and Control of Tuberculosis

Tuberculosis high levels of prevalence in the world have been the norm, particularly in poor and/or developing nations. The impact of travel and immigration as well as the costs associated with the TB treatment and the consequences associated with treatment compliance (antibiotic resistance) will be discussed. The application of mathematical models in the evaluation of epidemiological and sociological factors associated with TB dynamics and its control at the population level will be highlighted.