Faculty
Research Profile |
 Robert
Ghrist
Associate
Professor of Mathematics
University of
Illinois at Urbana-Champaign
329 Altgeld Hall
1409 W. Green
Street, Urbana, IL 61801
217-244-5857
ghrist@math.uiuc.edu
www.math.uiuc.edu/~ghrist
Research
Summary
My
work focuses on those methods in applied mathematics which are
topological in nature. Such methods have the characteristic of
being very robust: topological results are tolerant of the "noise"
inherent in physical systems. Such techniques are therefore surprisingly
useful in engineering and science.
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Current
Projects
- Geometric
Methods in Robotics & Computer Science
Robotics
is an ideal domain for a mathematician to work in: here, one has
a genuine need for rigor. Imagine trying to verify that a control
system for a robotic brain surgeon works. Would you prefer to
have a successful computer simulation or a theorem guaranteeing
performance? (Answer: get both if you can...) I use methods and
ideas from topology and geometric group theory to prove rigorous
theorems about robot motion-planning and control. In particular,
I have worked with spaces of nonpositive curvature as applied
to metamorphic and reconfigurable robots, and also to robot
coordination problems.
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Contact Topology and Fluid Dynamics
Rigorous
results about fluid dynamics are rare for fully three-dimensional
flows. I use global techniques from contact topology (a
variant of symplectic topology) to determine results about the
most difficult classes of steady inviscid fluid flows. These techniques
utilize cutting-edge ideas from the contact topology world ---
e.g., contact homology --- to answer questions about concrete
physical phenomena such as hydrodynamic instability.
- Knots,
Links, & Braids in Dynamical Systems
One of the
ways in which topological methods most directly impact applications
is via differential equations: much of the history of dynamical
systems theory traces back to topological perspectives. I have
contributed to the relationships between knot theory and
dynamics.
One way in which these fields interact arises whenever
you have a vector field on a three-dimensional domain: periodic
orbits naturally trace out simple closed curves. In what ways
do the knotting and linking data reflect or indeed force dynamical
data? There is a rich theory here, including simple examples of
differential equations for which the most chaotic types of knotting
imaginable are present.
Recent work has focused on applications
of braid theory to scalar parabolic PDEs via a
topological version of the comparison principle.
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