SGS XVIII: Sunday, April 10, 2011
Emory University

Abstract: The Funk transform takes a function $f$ on the unit sphere in the 3-dimensional Euclidean space to a function on the set of great circles as an integral of $f$ over the corresponding great circle. This transform can be regarded as a member of the analytic family of the cosine transforms on the sphere. We extend basic facts about the Funk transform and the relevant analytic families of cosine and sine transforms to the more general context for Stiefel or Grassmann manifolds. Recent results on Radon transforms on matrix spaces and affine Grassmannians are also presented.

Abstract: Integral geometry comes in a variety of flavors. One of the most delicious arises from geometric probability, e.g. the Buffon needle problem of computing the probability that a needle dropped at random on a wood floor crosses one of the cracks between the boards. The solution may be viewed as an instance of a "kinematic formula", expressing the average, over all possible relative positions of two shapes A, B, of some geometric measurement applied to the intersection of A and B, in terms of measurements of A and B separately. The basic understanding of such formulas is due to Blaschke in the 1930s.

In this century it has been understood that the kinematic formulas may be viewed as features of a certain natural algebraic structure, discovered by S. Alesker. In fact one may associate such a structure with any Lie group G that acts transitively on the sphere, where the classical case above corresponds to G = SO(n). In joint work with A. Bernig we have successfully applied this algebraic perspective to compute the kinematic formulas associated to G= U(n). In particular this yields solutions to Buffon-type problems in hermitian spaces, e.g. if one is given some 5 (real) dimensional submanifold of complex projective 4-space and wishes to compute the expected length of its intersection (generically a real curve) with a 4-dimensional manifold placed in some random position.

Abstract: CR geometry in 3D bears strong resemblence to conformal geometry in 4D. In particular, there are two natural conformally covariant equations one of second order, which is the analogue of the Yamabe equation and the other, a fourth order equation which is the analogue of the Paneitz Q-curvature equation. The positivity of the underlying linear operator gives strong consequences for the underlying structure: the embeddability of the CR structure, and the positivity of the CR mass. I will outline the ideas behind these new results.

Abstract: Flow of Hypersurfaces by Curvatures in Riemannian Manifolds.pdf

Abstract: This talk will be about notions of discrete curvature on surfaces. An abstract triangulation T of an (oriented) surface S induces various geometric structures. Classical is a euclidean cone structure, wherein each face is identified with a unit-sided equilateral triangle, and its associated conformal structure.

Triangulations also induce geometric structures via "circle packings", wherein configurations of circles with tangencies determined by the combinatorics provide the geometry. The results --- termed "discrete conformal" structures --- not only mimic their classical counterparts, but even converge to them under appropriate refinement, meaning that circle packing technology provides a new window on classical geometry. More interesting, perhaps, is that experiments challenge us with new phenomena, new questions, and potentially new applications.

This will be a largely visual talk using my software CirclePack. After illustrating the basics of circle packing and connections to curvature, I will concentrate on manipulations associated with "edge flips" in the combinatorics. Potential applications include the study of the equi-distribution of points on the sphere and the study of the geometry of graphene (single-layer carbon) sheets having dislocations and grain boundaries.