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2001 Mt. Baldy Conference on Differential Geometry: Abstracts

Rick Schoen (Stanford): Special Lagrangian Geometry

Calibrated submanifolds include complex submanifolds of Kaehler manifolds as well as special Lagrangian submanifolds. This latter class of submanifolds is of substantial interest in string theory. This talk will give a general introduction to the subject with particular emphasis on existence questions special Lagrangian and minimal Lagrangian submanifolds. We will describe deformation results, gluing methods, heat equation methods, and variational methods for constructing them.

Herman Gluck (U. Pennsylvania): Geometry, Topology and Plasma Physics

In this talk, I'll report on the work of a number of people at Penn in recent years involving mathematical methods in plasma physics. From the physics point of view, our goal is to determine and study the persistent plasma states observed in astrophysical, solar and laboratory settings. From the mathematics point of view, our goal is to develop the tools to carry this out, and to work on a number of mathematical problems suggested by this enterprise. Key roles in the story are played by the notion of "helicity" of a vector field, which measures the extent to which the field lines wrap and coil around one another, and spectral problems for the curl operator. Because helicity of vector fields is the analogue of "writhing number" of knots, the methods we use also provide an upper bound for the writhing number of a given strand of DNA.

Peter Li (UC Irvine): Complete Manifolds with Positive Spectrum"

In this talk, we will discuss how the Laplace operator is used to determine certain topological information of a complete non-compact manifold. Hodge theory asserts that when a manifold is compact, then its homology group can be computed using harmonic forms. Unfortunately, a similar theory does not exist for non-compact manifolds. However, for certain situations, one can still control some topological aspect using the analytic property of the Laplace operator. An example of this is a generalization of a theorem of Witten-Yau.

Peter Petersen (UCLA): Pinching and Compactness

The talk will be a survey of what types of results one can expect when curvatures of a manifold are pinched. Pinching of curvatures traditionally means that curvatures lie in some small interval on the real line. More recently it has also been generalized to include situations where certain L^p norms of curvatures are small. We will discuss which types of pinching conditions imply compactness in certain topologies and which don't and also how this affects what we can say about the underlying structure of the manifold.