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Abstracts

 

 

Hubert Bray On Dark Matter, Spiral Galaxies, and the Axioms of General Relativity
We define geometric axioms for the metric and the connection of a spacetime where the gravitational influence of the connection may be interpreted as dark matter. We show how these axioms lead to the Einstein-Klein-Gordon equations with a cosmological constant, where the scalar field of the Klein-Gordon equation represents the deviation of the connection from the standard Levi-Civita connection on the tangent bundle and is interpreted as dark matter. In a homogeneous, isotropic universe, this form of dark matter, unlike the WIMP model, is automatically cold and appears to be compatible with the ΛCDM model predictions on the cosmological scale. Furthermore, we describe the evidence for this form of dark matter causing barred spiral patterns in disk galaxies and triaxial shapes with plausible brightness profiles in elliptical galaxies. We compare the results of preliminary computer simulations with photos of actual galaxies.
Benoit Charbonneau Singular monopoles and stable pairs
I will discuss work done with Jacques Hurtubise (McGill) to relate singular monopoles on a circle times a surface to pairs [holomorphic bundle, meromorphic endomorphism] on the surface. The endomorphism is meromorphic, generically bijective, and corresponds to a return map. Its poles and zeros are related to the singularities of the corresponding monopole.
Gregory Galloway On the size and topology of black holes.
We review Hawking's classical result on the topology of black holes in 3+1 dimensions, and discuss a generalization of it to higher dimensions. Gibbons and Woolgar observed that for black hole spacetimes with negative cosmological constant, Hawking's argument yields a lower area bound for the black hole in terms of its genus. We discuss a generalization of this to higher dimensions, as well. As time permits, we will also discuss recent related results about the existence of black holes in 2+1 gravity. The proofs make use of results about marginally trapped surfaces, which are natural spacetime analogues of minimal surfaces.
Mohammad Ghomi Deformations of unbounded convex bodies and hypersurfaces
We study the topology of the space Kn of complete convex hypersurfaces of Rn which are homeomorphic to Rn-1. In particular, using Minkowski sums, we construct a deformation retraction of Kn onto the Grassmannian space of hyperplanes. So every hypersurface in Kn may be flattened in a canonical way. Further, the total curvature of each hypersurface evolves continuously and monotonically under this deformation. We also show that, modulo proper rotations, the subspaces of Kn consisting of smooth, strictly convex, or positively curved hypersurfaces are each contractible, which settles a question of H. Rosenberg.
John Olsen Three Dimensional Manifolds All of Whose Geodesics Are Closed
The existence of closed geodesics and the geometry of manifolds all of whose geodesics are closed are among the classical problems in geometry. One famous problem is the Berger Conjecture, which states that, on a simply connected manifold all of whose geodesics are closed, the geodesics have the same least period. I will give an introduction to the topic and mention known results. I will explain a possible approach via Morse Theory on the free loop space, and present some results on the Morse Theory in dimension three, where the conjecture is still open. If time permits, I will sketch a proof of the main theorem, which states that the energy function is perfect for S1-equivariant Morse Theory with rational coefficients for the cohomology.
Colleen Robles A system of PDE for calibrated submanifolds.
A calibration on a Riemannian manifold (M,g) is a closed p-form φ with the property that, when restricted to a tangent p-plane, φ ≤ volg. A p-dimensional submanifold N of M is calibrated if equality holds on the tangent spaces of N. The closed calibrated submanifolds are global minimizers of volume in their homology class. For this reason, calibrations are a valuable tool in the study of minimal submanifolds. More generally, one may consider the φ-critical submanifolds of M. These submanifolds include the calibrated submanifolds, and are minimal submanifolds of M. In the case that φ is parallel, I will present an exterior differential system whose integral manifolds are precisely the φ-critical submanifolds. The system generalizes the familiar characterizations of some of our favorite calibrations, including the special Lagrangian, (co)associative and Cayley calibrations.
Gil Solanes Total curvature of complete surfaces in hyperbolic space.
We present a Gauss-Bonnet formula for the integral of the extrinsic curvature of complete surfaces in hyperbolic space. Besides total curvature and Euler characteristic, this formula contains two terms, both of them described in the language of integral geometry. The first term can be interpreted as a truncated area of the surface. More precisely, it is the measure of geodesic lines intersecting the surface twice. The second term, which we call ideal defect, depends on the Möbius geometry of the ideal boundary curve. It can be described as the renormalized volume of the set of spheres linked with the curve. This is related to Banchoff-Pohl's definition of the area enclosed by space curves. Moreover, this ideal defect can be represented by a double integral over the curve, showing some connections with several known knot energies.
Jeff Viaclovsky Yamabe invariants and limits of self-dual hyperbolic monopole metrics
Consider the self-dual conformal classes on n # CP2 discovered by LeBrun. These depend upon a choice of n points in hyperbolic 3-space, called monopole points. I will discuss the limiting behavior of various constant scalar curvature metrics in these conformal classes as the points approach each other, or as the points tend to the boundary of hyperbolic space. There is a close connection to the orbifold Yamabe problem (which I will show is not always solvable, in contrast with the case for compact manifolds).
Jeremy Wong Applications of Almost-Convexity.
Almost-convex subsets of a metric space are defined in terms of intrinsic distance functions, generalizing bounded second fundamental form in the smooth setting. I will discuss how they arise when considering limits of smooth Riemannian submanifolds and some of their applications that I have found.

 

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The SEGC is supported by the National Science Foundation and by the Department of Mathematics at the College of Charleston.

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