CAREER: Classical Problems in Differential Geometry, Topology, and Convexity

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AbstractAward: DMS-0332333Principal Investigator: Mohammad GhomiThe principal investigator is interested primarily in theinterplay between the geometry and topology of submanifolds,including low dimensional problems in Euclidean space (curves andsurfaces). These investigations often involve some notion ofconvexity, and include the following categories: (i) Shadows (orshades) on illuminated hypersurfaces, and their application togeometric variational problems; (ii) Embeddings of manifolds inEuclidean space without creating parallel or intersecting tangentlines (totally skew embeddings), and their relation to quadrichypersurfaces and nonsingular bilinear maps; (iii) Globalproperties of locally convex hypersurfaces with boundary,including connections with Monge-Ampere equations, and a newconvex hull property which is dual to that of negatively curvedsurfaces; (iv) Certain deformations of space curves (unfoldings),and their application to study of extremals of knot energies anddistortion.Curves and surfaces are to geometry what numbers are toalgebra. They form the basic ingredients of our visualperception, and inspire the development of far reachingmathematical tools. For instance, those aspects of the PI's workdealing with shadows on illuminated surfaces are motivated inpart by a study of soap films, and have connections to computervision (the ``shape from shading" problems). Further, theinvestigations on knot energies may be of interest in studyingDNA. Yet, despite an abundance of potential applications andcenturies of pure study, there are still numerous open problemsin submanifold geometry and topology which are strikinglyintuitive and elementary to state. The PI believes thatadvertising these problems at an early stage is an excellent toolfor sparking the interest of students in mathematicalresearch. With the aid of computer workshops, courses, seminars,and the lecture series proposed in this project, the PI plans tocommunicate the beauty and excitement of geometric problems to aswide an audience as possible.