The cohesive movement of a biological population is
a commonly observed natural phenomenon.
With the advent of platforms of unmanned vehicles, this occurrence
is attracting renewed interest from the engineering community.
This talk will review recent research results
on both modeling and analysis of biological swarms and
also design ideas for efficient algorithms to control groups of
autonomous agents.
For biological models we consider two kinds of systems:
driven particle systems based on force laws
and continuum models based on kinematic rules.
Both models involve long-rage social attaction and short range dispersal
and yield patterns involving clumping, mill vortices,
and surface-tension-like effects.
For artificial platforms we consider the problem of boundary tracking
of an environmental material and consider both computer models
and demonstrations on real platforms of robotic vehicles.
We also consider the motion of vehicles using artificial potentials.

Recent simulations [1,2] of binary fluid convection with a
negative separation ratio reveal the presence of multiple numerically
stable spatially localized steady states we have called 'convectons'.
These states consist of a finite number of convection rolls embedded
in a nonconvecting background and are present at supercritical Rayleigh
numbers. The convecton length decreases with decreasing Rayleigh number;
below a critical Rayleigh number the convectons are replaced by
relaxation oscillations in which the steady state is gradually eroded
until no rolls are present (the slow phase), whereupon a new steady state
regrows from small amplitude (the fast phase) and the process repeats.
Both He3-He4 mixtures [1] and water-ethanol mixtures [2] exhibit
this remarkable behavior. Stability requires that the convectons are
present in the regime where the conduction state is convectively unstable
but absolutely stable. The multiplicity of stable convectons can be attributed
to the presence of a 'pinning' region in parameter space, or equivalently
to a process called homoclinic snaking [3]. In the pinning region the
fronts bounding the convecton are pinned to the underlying roll structure;
outside it the fronts depin and allow the convecton to grow at the
expense of the small amplitude state (large Rayleigh numbers) or shrink
back to the small amplitude state (low Rayleigh numbers). The convectons
may exist beyond the onset of absolute instability but the background
state is then filled with small amplitude traveling waves. A theoretical
understanding of these results will be developed.

[1] O. Batiste and E. Knobloch. Simulations of localized states of
stationary convection in He3-He4 mixtures. Phys. Rev. Lett. 95, 244501 (2005).
[2] O. Batiste, E. Knobloch, A. Alonso, I. Mercader. Spatially localized
binary fluid convection. J. Fluid Mech. 560, 149 (2006).
[3] J. Burke and E. Knobloch. Localized states in the generalized
Swift-Hohenberg equation. Phys. Rev. E 73, 056211 (2006)

The talk will contain a sruvey of mathematical problems and results of the
so called Thermoacoustic Tomography (TAT) and its sibling Photoacoustic
Tomography (PAT). These are among novel methods of medical imaging that
have been emerging recently. The main feature of these new methods is
combining different physical types of waves for creating and for measuring a
signal. In the case of TAT, the signal is triggered by irradiation of an
object by a MW or RF pulse, while the measured signal itself is an
ultrasound wave.

Mathematics of these new methods is very interesting and often hard. The
mathematical model of TAT/PAT boils down to an inverse problem for the wave
equation, or to an equivalent problem of recovering a function from its spherical
means with a restricted set of centers. Significant breakthroughs in this
area have been made very recently, e.g. in the last few months.

No prior knowledge of the subject will be assumed.

It is now believed, but proved only in a few cases, that the distribution
functions
of random matrix theory are universal for a wide class of stochastic
problems in combinatorics,
growth processes, and statistics. These developments will be surveyed.
No prior knowledge
of random matrix theory will be assumed.

In 1938, Alexandroff introduced a class of smooth metrics $g$ in the
unit disc $D\subset \mathbb R^2$ such that the Gauss curvature $K$ satisfies
$K>0$ in $D$, $K=0$ and $dK\neq 0$ on $\partial D$ and the total curvature of
$g$ in $D$ is $4\pi$. Alexandroff proved that such metrics are rigid, in the
sense that the isometric embedding of $(D, g)$ in $\mathbb R^3$ is unique up to
rigid body motions if it exists. It is easy to derive necessary conditions for
such metrics to be isometrically embedded in $\mathbb R^3$, among which the
geodesic curvature of the boundary is negative. We will prove that those
necessary conditions are also sufficient.

The proof is based on a discussion of elliptic Monge-Ampere equations which are
degenerate on the boundary. Because of the rigidity, boundary conditions cannot
be described. In fact, there is only one boundary condition which makes this
Monge-Ampere equation solvable.