Abstracts for Plenary Talks

of the

Smoky Great Plains Geometry Conference

The abstracts for the 2015 Smoky Great Plains Geometry Conference will be posted as they become available.

Lydia Bieri

Spacetime Geometries and Radiation
Abstract: In General Relativity, geometry is interwoven with the physical theory of gravity through the Einstein equations. Solutions of the latter are Lorentzian manifolds. A major goal in the study of these equations is to investigate the geometry of the solution spacetimes. The Cauchy problem for these nonlinear, hyperbolic partial differential equations involves interesting questions in pdes as well as a rich geometry. How do matter fields change the geometry? And how does the Weyl curvature as well as the matter fields 'shape' the spacetime? How does gravitational radiation manifest itself in the geometric structure? These questions will in particular be addressed for the Einstein equations in the presence of neutrinos.

Carla Cederbaum

On the center of mass in general relativity
Abstract: In many situations in Newtonian gravity, understanding the motion of the center of mass of a system is key to understanding the general "trend" of the motion of the system. It is thus desirable to also devise a notion of center of mass with similar properties in general relativity. However, while the definition of the center of mass via the mass density is straightforward in Newtonian gravity, there is a priori no definitive corresponding notion in general relativity. Instead, there are several alternative approaches to defining the center of mass of a system. We will discuss these different approaches for both asymptotically Euclidean ("isolated") and asymptotically hyperbolic ("collapsing") systems and present some explicit (counter-)examples.

Ruth Gornet

The Covering Spectrum vs. The Laplace Spectrum
Abstract: The covering spectrum, which roughly measures the size of the 1-dimensional holes in the space, was introduced by C. Sormani and G. Wei in 2004. In their paper, they observed that certain isospectral manifolds constructed using Sunada's method share the same covering spectrum. We present a group theoretic condition under which Sunada's method gives manifolds with identical covering spectra. When the group theoretic condition of our method is not met, we are able to construct isospectral manifolds with distinct covering spectra in dimensions 3 and higher. Using isospectral graphs, we are able to use other means to construct Sunada isospectral manifolds with distinct covering spectra in dimension 2. A key ingredient in the higher-dimensional proof has an interpretation as the minimum-marked-length analogue of Colin de Verdier's classic result on constructing metrics where the first k eigenvalues of the Laplace spectrum have been prescribed. This is joint work with Bart DeSmit and Craig Sutton.

Karsten Grove

Reflection groups in non-negative curvature
Abstract: We will discuss the equivariant description classification of all complete (compact or not) non-negatively curved manifolds M together with a co-compact action by a reflection group W. In particular, we will show that the building blocks consist of the classical constant curvature models and generalized open books with non negatively curved bundle pages, and derive a corresponding splitting theorem for the universal cover. This is joint work with Fuquan Fang.

Ursula Hamenstadt

Almost totally geodesic surfaces in locally symmetric spaces
Abstract: Given a closed rank one locally symmetric space M, we use ideas from metric geometry to construct immersed surfaces whose fundamental groups embed into the fundamental group of M and which are arbitrarily close geometrically to totally geodesic surfaces.

Michael Jablonski

Non-compact, homogeneous Einstein spaces.
Abstract: In the 1970s, D. Alekseevskii conjectured that if G/K is a non-compact, homogeneous Einstein space with negative scalar curvature, then K is a maximal compact subgroup of G. In this talk, we will present the current state of knowledge, with evidence both for and against the conjecture.

Vitali Kapovitch

Higher homotopy groups of spaces of nonnegatively curved metrics
Abstract: We construct examples of noncompact manifolds M for which the space of nonnegatively curved metrics on M has some nontrivial higher homotopy groups. This is joint work with Igor Belegradek and Tom Farrell. http://arxiv.org/abs/1501.03475

Lee Kennard

Torus actions in positive curvature
Abstract: The study of positively curved Riemannian manifolds is classical but difficult. One enlightening approach is via symmetry, where one studies the interaction of curvature, topology, and group actions. In this talk, I will focus on the case of torus actions and discuss some recent results with Manuel Amann on Euler characteristic calculations.

Karin Melnick

Geometric Levi Decomposition for Riemannian and Lorentzian manifolds
Abstract: I will survey past results and work in progress on structure theorems for local isometries of closed Riemannian and Lorentzian manifolds. Let M be such a manifold, with universal cover X, and let G0 = Isom0(X). S. Frankel proved (1994) that when G0 is semisimple with finite center and no compact local factors, then M fibers over a locally symmetric space associated to G0. I proved a similar result for M Lorentzian, under the additional assumptions that M is asperical, real-analytic, and geodesically complete (2009). Recently, W. van Limbeek found a Riemannian structure theorem allowing for G0 with nontrivial solvable radical, giving a sort of Levi decomposition of M. We are now working on a similar Levi decomposition for M Lorentzian. These results involve, among other things, Lie theory, harmonic maps, cohomological dimension arguments of Farb and Weinberger, and Lorentzian dynamics.

Peter Petersen

Uniqueness of Warped Product Einstein Metrics
Abstract: The talk will cover aspects of Einstein metrics that are related to warped products. There is a surprisingly large class of examples of such spaces despite their special construction. We will discuss a few interesting examples and also show that examples often have a large degree of symmetry. This leads to a few classification results.

Anna Siffert

Isoparametric Hypersurfaces In Spheres
Abstract: The problem of classifying isoparametric hypersurfaces in spheres is still not completely solved. Recent contributions to the case with six different principal curvatures contain fundamental gaps. My talk consists of two parts: first I summarize a structural approach towards the classification of isoparametric hypersurfaces in spheres I developed in my thesis. Afterwards I present a new method for the classification of isoparametric hypersurfaces of spheres with six different principal curvatures all of multiplicity one.

David Wraith

Positive Ricci curvature on highly connected manifolds.
Abstract: This talk concerns the existence of positive Ricci curvature metrics on compact (2n-2)-connected (4n-1)-manifolds. The focus will be largely topological: we will describe new constructions of these objects to which existing curvature results can be applied. The constructions are based on the technique of plumbing disc bundles. This is joint work with Diarmuid Crowley.

Ling Xiao

Bernstein type thorem in Minkowski space
Abstract: In 2005, Mark A. S. Aarons conjectured that if u is a downward translating solution to the mean curvature flow with forcing term in Minkowski space, then it has to be rotaionally symmetric or flat. In this talk, we will discuss some classical results related to this topic and our results. This is a joint work with J. Spruck.