I will present a novel multiscale clustering algorithm inspired by algebraic multigrid techniques. Our method begins with assembling
data points according to local similarities. It uses an aggregation process to obtain reliable scale-dependent global properties, which arise from the local similarities. As the aggregation process proceeds, these global properties influence the formation of coherent clusters. The
global features that can be utilized are for example density, shape, intrinsic dimensionality and orientation. The last three features are a
part of the manifold identification process which is performed in parallel to the clustering process. The algorithm detects clusters that
are distinguished by their multiscale nature, separates between clusters with different densities, and identifies and resolves intersections between clusters. The algorithm is tested on synthetic and real data sets, its running time complexity is linear in the size of the data set.

Joint work with: Meirav Galun and Achi Brandt.

We investigate the asymptotic behavior of the principal eigenvalue of an elliptic operator as the coefficient of the advection term approaches infinity. As a biological application, a Lotka-Volterra reaction-diffusion-advection model for two competing species in a heterogeneous environment is studied. The two species are assumed to be identical except their dispersal strategies: both species disperse by random movement and advection along environmental gradients, but one species has stronger biased movement than the other one. It is shown that at least two scenarios can occur: if only one species has a strong tendency to move upward the environmental gradients, the two species will coexist; if both species have such strong biased movements, the species with the stronger biased movement will go to extinct. These results provide a new mechanism for the coexistence of competing species, and they also suggest that an intermediate biased movement rate may be evolutionary stable.

Let $A$ denote a unital algebra of bounded linear operators on an infinite
dimensional real or complex Banach space $X$.
$A$ is said to have the closure property if every densely defined
linear operator on $X$ which commutes with
the closure of A in the weak operator topology
(WOT) is closeable.
$T \in L(X)$ is said to have the closure property if $A_T$ does, where
$A_T$ denotes the algebra of polynomials in $T$.
This new concept is motivated by deep classical work of W.B.~Arveson.
I shall discuss how some of his results may be formulated in terms of the
closure property, give some structural results we have obtained, and present
some examples of operators with the closure property.
The following conjecture, if true, would yield a very broad generalization
of Arveson's main result.

Conjecture.
Let $A$ have the closure property.
Then either $A$ has a non-trivial closed invariant linear subspace, or the WOT closure of $A$ coincides with L(X)$.