The Southeast Geometry Seminar (SGS) is a semiannual series of one day events sponsored by:
The organizers are: John McCuan (GIT), Vladimir Oliker (Emory), and Gilbert Weinstein (UAB).
Abstract: We consider the mean curvature flow for networks of curves in two dimensions and of surfaces in three dimensions, satisfying the Plateau angle conditions and intersecting a convex boundary orthogonally. I will describe recent progress on local and global existence, and on approximation by `diffuse interface' problems. (Joint work with N. Alikakos and S. Betelu, both at the University of North Texas.)
Abstract: The Skyrme and Faddeev models are nonlinear σ-models of the classical field theory. The configuration space of the Skyrme model consists of the maps from a three-dimensional manifold, M, into a compact Lie group. In Faddeev's model, the maps are from a three-manifold into the two-sphere. In this talk I will discuss the case of a closed manifold M and describe analytically the homotopy classes of such maps. It turns out that the homotopy invariants can be adequately interpreted for the maps with finite energy. Also, within each sector with fixed invariants, the energy functional attains its minimum. Both models can be reformulated in terms of certain classes of flat connections; this leads to new mathematical and physical problems. The talk is based on my joint work with D. Auckly.
Abstract: As is well-known, all Riemannian metrics on the 2-sphere with the same constant Gauss curvature are isometric. The notion of Gauss curvature for Riemannian surfaces generalized to the 'Finsler-Gauss curvature' for Finsler surfaces. It was shown by Akbar-Zadeh in 1988 that any compact Finsler surface with constant negative Finsler-Gauss curvature is Riemannian of constant Gauss curvature, but it has been known for almost 10 years that the same is not true for constant positive Finsler-Gauss curvature. In fact, no complete classification is known, although there are now several methods of constructing large families.
In this talk, I will recall the basics of Finsler surface geometry, survey the above-described results and conclude with the proof that a reversible Finsler surface with constant positive Finsler-Gauss curvature is necessarily the round 2-sphere. This proof is surprisingly delicate and uses recent classification results by Lebrun and Mason of the Zoll projective structures on the 2-sphere and the real projective plane.
Abstract: Gravitational lensing is gravity's action on light. A central geometric result in the subject is the Local Magnification Theorem, which characterizes the generic asymptotic behavior of the Gaussian curvature at time-delay critical points near caustic singularities. However, there is a growing number of lens systems whose observed image-flux-ratios violate the Local Magnification Theorem. Our talk will present the mathematical and physical aspects of this theorem, including the observational evidence for its failure.
In an attempt to reconcile theory and observations, we present a generalization of the local theorem to a physical Global Magnification Theorem that incorporates the reasonable causes for the observed failure of the local theorem. As an application, we show how violation of the global theorem predicts the presence of small-scale structures in the halos of galactic lenses. The fraction of such small-scale structures in the form of dark matter clumps can in turn be used to test the Cold Dark Matter theory on galactic scales.
The talk concludes with some open mathematical issues invoked by observational violation of the Global Magnification Theorem.
This talk is based on joint work with S. Gaudi and C. Keeton. It is aimed at a diverse audience of mathematicians and physicists.
Abstract: Following Felix Klein's program, several special hypersurface theories were developed, according to given transformation groups projective, equiaffine, centroaffine etc.), in the first half of the last century. For a given group of transformations, it was the aim of systematic investigations to find an invariant normalization and its induced geometric structures. We restrict to non-degenerate hypersurfaces in real affine space and modify the view point: recall the fact that, for a given hypersurface, there are infinitely many possible normalizations, each inducing geometric structures. But only in relatively few cases the invariance group, as a subgroup of the general affine transformation group, is known. This fact suggests the study of such invariants and structures with respect to the general affine transformation group which are independent of a particular normalization.
In this talk we give a description of the geometry of non-degenerate hypersurfaces in affine space in terms of such invariants and structures. Besides two well-known structures, namely the conformal and the projectively flat structure, one of our essential tools is a Weyl structure together with its gauge transformations; the gauge transformations are equivalent to a change of the normalization, and thus gauge invariants are independent of a normalization. We present many old and new invariants of this type and give a geometric test of our approach, discussing gauge invariance in the context of well-known problems and results from affine hypersurface theory.
Abstract: For a Riemannian manifold whose sectional curvature is pinched (i.e. bounded) between two negative constants, the ratio of the constants is called pinching. I will discuss optimal pinching estimates for manifolds with virtually nilpotent fundamental group. This is joint work with Vitali Kapovitch and to appear in GAFA.