AWM at JMM 2010
Special Session and Workshop Abstracts
- AMS-AWM Special Session on Spectral Problems on Compact Riemannian Manifolds, I
- AMS-AWM Special Session on Spectral Problems on Compact Riemannian Manifolds, II
- AWM Workshop: Research Presentations by Recent Ph.D.’s, I
- AWM Workshop: Research Presentations by Recent Ph.D.’s, II
Friday January 15, 2010, 8:00 a.m.-10:50 a.m., Room 3014, 3rd Floor, Moscone
8:00 a.m. Spectral bounds on 2-orbifold diffeomorphism type
Emily Proctor, Middlebury College ;
Elizabeth Stanhope*, Lewis & Clark College (1056-53-1590)
We show that any collection of isospectral 2-orbifolds sharing a lower bound on sectional curvature contains orbifolds of only finitely many diffeomorphism types.
8:30 a.m. Spectral and geometric bounds on orbifold homotopy type
Emily Proctor*, Middlebury College;
Elizabeth Stanhope, Lewis & Clark College (1056-58-919)
Consider an isospectral set of Riemannian orbifolds with sectional curvature bounded below. In this talk, I will describe our work to prove that such a set contains orbifolds of only finitely many orbifold category homotopy types. A previous result by Stanhope shows that the spectral bounds can be replaced with bounds on diameter and volume. From there our work is to generalize a similar result for manifolds by Grove and Petersen.
9:00 a.m. Some applications of the trace formula on orbifolds to the inverse spectral problem
Alejandro Uribe*, University of Michigan, Ann Arbor (1056-58-1289)
I will discuss some applications of the trace formula on orbifolds to the inverse spectral problem.
9:30 a.m. On the singularities of the exponential map
Benjamin Schmidt*, Michigan State University
Keith Burns, Northwestern University (1056-51-1041)
Let p be a point in a complete Riemannian manifold M and let expp : TpM → M denote the exponential map. Between 1932 and 1965, papers of Littauer-Morse, Savage, and Warner established that a nonzero vector v ∈ TpM is a singular point of expp if and only if expp fails to be locally injective at the vector v. In recent joint work with Keith Burns, we show that local injectivity fails in radial directions. I’ll discuss this work and an application that characterizes constant curvature projective spaces.
10:00 a.m. Quantum ergodic restriction theorems
Steve Zelditch*, Johns Hopkins Univ and Northwestern Univ (1056-35-748)
Quantum ergodicity is a property of almost the full sequence of eigenfunctions of a Riemannian manifold with ergodic geodesic flow: the eigenfunctions becomes uniformly distributed in phase space. By a quantum ergodic resctriction theorem we mean the question whether the restrictions of eigenfunctions to a hypersurface are quantum ergodic along the hypersurface. Simple examples show that this is sometimes not the case. In my talk, sufficient conditions will be given for restrictions of eigenfunctions to be quantum ergodic. Based on joint work with John Toth and work in progress with John Toth, Hamid Hezari and Hans Christianson.
10:30 a.m. Spectral rigidity of analytic plane domains with one mirror symmetry
Hamid Hezari*, MIT;
Steve Zelditch, Johns Hopkins University (1056-35-1012)
This is a report of a recent joint work with Steve Zleditch on isospectral deformations of plane domains. It is known that there exist non-isometric isospectral plane domains, but all of the known examples have corners and in particular are not smooth. The original problem of Kac, ”Can one hear the shape of a drum?” is therefore open if one interprets ”drum” to mean a smooth drum. The only domain known to be determined by its spectrum among all plane domains is the standard disc. It is not known if ellipses are spectrally determined, even among smooth plane domains. The purpose of this short talk is to prove that bounded real analytic plane domains with one symmetry are spectrally rigid among all real analytic domains, including those without any symmetries. The proofs are based on the calculation and study of variational derivatives of the ”wave trace invariants” associated to a bouncing ball orbit.
Friday January 15, 2010, 1:00 p.m.-5:50 p.m., Room 3003, 3rd Floor, Moscone
1:00 p.m. Inverse scattering results for manifolds hyperbolic at infinity
David Borthwick, Emory University
Peter A. Perry*, University of Kentucky (1056-58-1112)
This is joint work with David Borthwick. We study the inverse resonance problem for conformally compact manifolds which are hyperbolic outside a compact set. Our results include compactness of isoresonant metrics in dimension two and of isophasal negatively curved metrics in dimension three. In dimensions four or higher we prove topological finiteness theorems under the negative curvature assumption.
1:30 p.m. The genus spectrum of a hyperbolic 3-manifold
D. B. McReynolds*, University of Chicago
Alan W. Reid, University of Texas in Austin (1056-53-1227)
In this talk, I will discuss recent work with Alan Reid on higher dimensional spectra. The focus will be on totally geodesic surfaces in hyperbolic 3-manifolds.
2:00 p.m. Length-spectral rigidity for flat metrics
Moon Duchin*, University of Michigan
Christopher J Leininger, University of Illinois
Kasra Rafi, University of Oklahoma (1056-51-1425)
Fix a surface S with a negatively curved metric and consider the marked length spectrum of all closed curves. These length data uniquely determine the metric among all negatively curved metrics on S, by a theorem of Otal. For metrics of constant negative curvature, the situation is much more rigid: it suffices to record the lengths of simple closed curves (and in fact, just 6g − 5 curves will do for the surface of genus g). In joint work with Leininger and Rafi, we consider the rigidity of the length spectrum for singular flat metrics (semi-translation structures) on S, and give a complete solution describing which simple curve sets are rigid.
2:30 p.m. The Cut-off Covering Spectrum
Christina Sormani, CUNY Graduate Center and Lehman College
Guofang Wei*, UC Santa Barbara (1056-53-529)
We introduce the R cut-off covering spectrum and the cut-off covering spectrum of a metric space or Riemannian manifold. The spectra measure the sizes of localized holes in the space and are defined using covering spaces called δ covers and R cut-off δ covers. They are investigated using δ homotopies which are homotopies via grids whose squares are mapped into balls of radius δ.
On locally compact spaces, we prove that these new spectra are subsets of the closure of the length spectrum. We prove the R cut-off covering spectrum is almost continuous with respect to the pointed Gromov-Hausdorff convergence of spaces and that the cut-off covering spectrum is also relatively well behaved. This is not true of the covering spectrum defined in our earlier work which was shown to be well behaved on compact spaces. We close by analyzing these spectra on Riemannian manifolds with lower bounds on their sectional and Ricci curvature and their limit spaces.
3:00 p.m. Spectral problems for polygons
Julie Rowlett*, Hausdorff Center for Mathematics (1056-58-869)
This talk focuses on some spectral problems for polygons, and triangles in particular. We will discuss recent progress with Z. Lu on the fundamental gap conjecture for triangles and related open questions whose statements are pleasantly simple. For example,
–Chang, DeTurck, Lu