The Topology Seminar will be following Bott & Tu's Differential Forms in Algebraic Topology. This week we will define the de Rham Complex of a smooth manifold. We'll also look at a Mayer-Vietoris sequence and integration of a differential form.

The recent (surprising(?)) success of the numerical binary
black hole collision codes poses a number of challenges
for mathematical relativists. Following a quick overview
of numerical relativity and the binary black hole
problem, I will give my explanation as to why the codes
worked. I will then focus on what I see as the key
barriers that need be overcome to permit further
development, and what contribution mathematical
relativity can make to this exciting field. This talk
will be aimed at the non-expert.

In this talk I will give an overview of the recent development of boundary integral
methods for solving Poisson-Boltzmann equations (PBE) for the electrostatic
potential in a solvation system of biomolecules. Electrostatic forces are crucial
in determining the structure and dynamics of biomolecules and their interactions
with solvents. As a mean-field approximation, the PBE has proved to be a very useful
model of such electrostatics. However, numerically solving the PBE in a very efficient
and accurate way is challenging.

I will first describe the method and present some of my computational results. I will
then compare different numerical methods for solving the PBE with complicated geometry.
In particular, I will present the newly developed fast multipole method for the boundary
integral discretization of the PBE. I will finally mentions some open questions.

The results presented in talk are mainly from the work done in McCammon's group at UCSD.

This talk focuses on asymptotic properties
of geometric branching processes on hyperbolic spaces
and manifolds. (In certain aspects, processes on hyperbolic
spaces are simpler than on Euclidean spaces.)
The first paper in this direction was
by Lalley and Sellke (1997) and dealt with a homogenous
branching diffusion on a hyperbolic (Lobachevsky) plane).
Afterwards, Karpelevich, Pechersky and Suhov (1998) extended
it to general homogeneous branching processes on
hyperbolic spaces of any dimension. Later on, Kelbert and Suhov
(2006, 2007) proceeded to include non-homogeneous branching
processes. One of the main questions here is to calculate
the Hausdorff dimension of the limiting set on the absolute.
I will not assume any preliminary knowledge of hyperbolic
geometry.

In 1956 Hans Lewy shocked the world. For decades, people had believed that just like ODEs, well-behaved PDEs should have a general existence theorem, and that it was only a matter of time before it was proven. However, Lewy constructed a very simple example to prove them all wrong. He was in as much shock and awe as they were.

We'll talk about this equation and the proof, which coincidentally uses a lot or CR geometry. The talk should be accessible to anyone who has ever taken a partial derivative.

Traditional Genome annotation involves the enumeration of open
reading frames and their functional assignment. Currently, there are
on-going efforts to identify all the interactions between these
components. The resulting map of interactions effectively represents
a 2D annotation. It takes the form of a stoichiometric matrix, if
the interactions are described with chemical equations. The
formulation and properties of this matrix are detailed and how it can
be used as the basis for computing allowable phenotypic functions.
The issues associated with the packing of the bacterial genome and
the function of the interaction map in 3D will also be discussed.
Finally, we will go over the issue of genomes changing in space and
time (4D) through adaptive evolution and describe the full re-
sequencing of bacterial genomes to map all genetic changes that occur
during adaptation. All of these efforts represent mathematical
challenges.