Arthur has given rather precise conjectures on the
decomposition of the regular representation L^2(G(F)\\G(A)),
where G is a simple Lie group over a number field F,
with adele ring A. In particular, the irreducible constituents are
partitioned into classes called Arthur packets. I will discuss the
construction of some of these packets when G is the exceptional group
$G_2$ and how one can justify that the constructed packets are the right
ones.

We consider tensor products $V_{\\lambda}\\otimes V_{\\mu}$ of
irreducible representations of a classical group $G$.
In general, such a tensor product decomposes in irreducible
components. It is a fundamental question how the components
are embedded in the tensor product.
Of special interest is the so-called Cartan component
$V_{\\lambda+\\mu}$. It appears exactly once in the decomposition.

On the other hand, one can look at decomposable tensors
(tensors of the form $v\\otimes w$) in the tensor product.

A natural question arising here is the following: are the
decomposable
tensors in the Cartan component given as the closure of
the minimal orbit in $V_{\\lambda+\\mu}$? If this is the
case we say that the Cartan component is small.

We give a characterization and a combinatorial description
of tensor products with small Cartan components. In particular,
we show that for general $\\lambda$, $\\mu$, Cartan components
are small.

Jointly Sponsored by UCI

Jointy Sponsored by UCI - This seminar will be held at UCI

Jointy Sponsored by UCI

The Dirichlet problem (DP) for the Laplace operator can be used to model a
number of different physical situations. For instance, if the surface of
a ball is kept at a given constant temperature f, then the steady state
temperature inside the ball is given by the solution of the Dirichlet
problem in the ball with data f on the sphere. A curious fact is that the
solution of the DP with rational data on the unit disk in the plane is
rational, whereas the corresponding statement is not true in 3-space (or
in any dimension greater than or equal to 3 for that matter). This can be
explained by quite elementary methods.

Refreshments will be served.

I shall start by discussing some statistical problems related to mapping the brain, both the cerebrum (a 3-dimensional object) and the cerebral cortex, or \"brain surface\"" (a 2-dimensional manifold in 3-dimensional space). This problem has motivated recent deep results describing the geometry of Gaussian random fields on manifolds