%This document is in LaTeX$\documentstyle[11pt]{article} \begin{document} \begin{center} \LARGE{COURSE ANNOUNCEMENT-FALL 1994}\\ \LARGE{EVOLUTION BY CURVATURE}\\ \large{DEPENDING ON INTEREST} \end{center} Let$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. \vspace {1 cm} Alex Freire \end{document}