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The aim of the group is to
do research in a wide range of topics. Whenever this leads us outside the
expertise of the core members (which is wide ranging), outside collaborations
will be sought (see below). In addition to stimulating research at PSU,
the groups will open research opportunities for graduate students to write
PhD Thesis of high quality in relevant subjects, and to obtain funding
through grants to support research in a diversity of subjects.
Ongoing Projects:
Flocking
Behavior and Formation Forming.
Movement in formation is very
common, both in nature and in technology (think of a flock a geese in flight
or a squadron of planes). The question arises how to do this efficiently.
That is: How can you fly or move in formation without analyzing the motion
relative to you of EVERY other bird in the flock. Ideally, you would want
to process only the relative motion of your nearest neighbors and let your
self be guided by them. Under what circumstances is this a good approach
and how exactly can it be done?
Models for the Dynamics of Granular
Flows.
Outside Collaborator: Prof
G. Vasconcelos (Physics, Recife, Brazil) and others.
Granular flows form an active
and well-funded research area, since there are many proposals to store
or transport granular foodstuff such as rice or grains. In this project
aspects of the motion of granular flows are investigated in a variety of
contexts. We concentrate on understanding the motion of a single grain.
The presence of other granular material is modeled by assuming the grain
is constrained to move in a rugged landscape. Analytical as well as numerical
tools are used, and results can often be directly compared to physical
experiments.
The Mathematics of Separation.
Students: Isaac Erskine, Rob Thompson. Suppose we have a length space X with two marked points A and B. The
mediatrix is the set of points x such that
d(x,A)-d(x,B) = 0 . We are interested in finding the properties of
mediatrices. First of all, what is the regularity (or: how close are they to
being differentiable), and second, what are their topological properties. These
sets have many applications as they minimally separate the underlying space into
two components, one cob=ntaining A and the oter containing B. Such sets are for
example important in computational geometry where they are related to the
boundaries of the Voronoi cells. Other applications include so-called conflict
sets.
For more information:
Mathematical Physics, Dynamical Systems, Fractal Geometry.
Low-dimensional Dynamics. Existence and characterization of invariant sets.
Modeling physical and biological phenomena as a dynamical system, such as
certain granular flows or infectious diseases. Asymptotic geometry of fractal
sets (their structure at small scales). Applications of geometry and topology.
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