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\LARGE{COURSE ANNOUNCEMENT-FALL 1994}\\
\LARGE{EVOLUTION BY CURVATURE}\\
\large{DEPENDING ON INTEREST}
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Let $X:M\rightarrow R^n$ be an immersed hypersurface
in $R^n$. Evolve $X$ in time according to the law:
the velocity vector has the direction of the inner normal {\bf N}.
and its magnitude is proportional to the (mean) curvature $H$
of the surface at any given point. This motion is
described by the PDE:
$$\frac {dX}{dt}=H\bf{N}.$$
Although the problem is simple to state and geometrically
natural, the first strong results were only obtained
in the 1980's! Since then several results have been obtained on
existence of classical and generalized solutions, but our
understanding of the singularities that develop (and of
how to continue the evolution past a singularity) is still incomplete.
These problems lie at the interface between differential geometry
and p.d.e's, and closely related problems occur in many areas of
pure and applied mathematics.
TOPICS: Evolution of convex hypersurfaces, evolution
of plane curves, development and classification of singularities,
Harnack inequalities and monotonicity properties, evolution
of entire open hypersurfaces. The emphasis will be on classical
solutions, but if there is time some results on `viscosity solutions'
will be presented.
PREREQUISITES: No familiarity with differential geometry will
be assumed; the course is addressed to graduate students
who have some background in p.d.e.'s at the graduate level,
and are interested indifferential equations or differential geometry.
CREDIT: If confirmed, this course will run as a section of
Math 636 (Advanced p.d.e's).
Graduate students are encouraged to take the course for credit;
the grade will be based on a presentation of a short paper
in the area.
TIME and PLACE- to be announced; the course would meet twice a week,
for a total of 3h/week.
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Alex Freire
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