Experimental measurements of transdermal vapor alcohol concentration are used to estimate alcohol concentration in the body using an inverse problem approach. First we propose a model for the transport of alcohol from blood compartments to the skin surface and use the transdermal measurements to estimate the signal obtained by a breathalyzer which is the standard for blood alcohol concentration. Later we couple our skin model to a body model of the human body. The human body is divided in several compartments to facilitate the description of the transport of alcohol in the human body from ingestion to elimination. The adjoint method is used for the computation of the least squares functional gradient. Parameters of the model are estimated using real breathalyzer and a transdermal alcohol skin device data applied to individuals in a hospital. The parameter values obtained are used to predict the evolution of alcohol concentration for patients in the field. Kalman filtering techniques can be used to correct predictions in real time.

We discuss a refinement of the Rubin-Stark Conjecture for abelian L-functions of arbitrary order of vanishing at s=0. This generalizes Gross's v-adic refinement of the abelian, order of vanishing 1, integral Stark Conjecture and predicts a link between special values of derivatives of p-adic and global L-functions. Time permitting, we will also show how our refinement relates to a recent strengthening of Gross's Conjecture due to Tate.

We study the following model for the spread of a rumor or infection: There is a ``gas'' of so-called $A$-particles, each of which performs a continuous time simple random walk on $\Bbb Z^d$, with jumprate $D_A$. We assume that ``just before the start'' the number of $A$-particles at $x$, $N_A(x,0-)$, has a mean $\mu_A$ Poisson distribution and that the $N_A(x,0-), \, x \in \Bbb Z^d$, are independent.
In addition, there are $B$-particles which perform continuous time simple random walks with jumprate $D_B$. We start with a finite number of $B$-particles in the system at time 0. The positions of these initial $B$-particles are arbitrary, but they are non-random. The $B$-particles move independently of each other. The only interaction is that when a $B$-particle and an $A$-particle coincide, the latter instantaneously turns into a $B$-particle. \cite {KSb} gave some basic estimates for
the growth of the set $\wt B(t):= \{x \in \Bbb Z^d:$ a $B$-particle visits $x$ during $[0,t]$\}. In this article we show that if $D_A=D_B$, then $B(t) = \wt B(t) + [-\frac 12, \frac 12]^d$ grows linearly in time with an asymptotic shape, i.e., there
exists a non-random set $B_0$ such that $(1/t)B(t) \to B_0$, in a sensewhich will be made precise. Joint work with H. Kesten.

In Calculus and PDE, people study rather complicated functions, equations on
relatively simple spaces (real line or n-dim Euclidean spaces). On the other hand,
in topology, people study complicated spaces with relatively simple function theory
on them. We are going to introduce a kind of calculus that takes the underlying
topological space into account. Thus we can see how topology interacts with calculus
naturally. The kind of new Calculus is called differential geometry. From this point
of view, Calculus and topology are finally unitfied into differential geometry.