## AWM at JMM 2010

### Special Session and Workshop Abstracts

### Contents

- 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

### AMS-AWM Special Session on Spectral Problems on Compact Riemannian Manifolds, I

### 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 exp_{p} : T_{p}M → M denote the exponential map. Between 1932 and 1965, papers of Littauer-Morse, Savage, and Warner established that a nonzero vector v ∈ T_{p}M is a singular point of expp if and only if exp_{p} 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

### AMS-AWM Special Session on Spectral Problems on Compact Riemannian Manifolds, II

### 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,

*Can one hear the shape of a triangle with a “real” ear?*

–Chang, DeTurck, Lu

–Chang, DeTurck, Lu

##### In conclusion, we will discuss some open extremal spectral problems for polygons and offer ideas.

### 3:30 p.m. Break

### 4:00 p.m. Eigenvalues and Isoperimetric Inequalities

#### Jesse Ratzkin*, University College Cork (1056-35-283)

##### I will discuss some bounds for the first eigenvalue of the Laplacian with Dirichlet boundary conditions, particularly for domains in a cone. These bounds arise from weighted isoperimetic inequalities, which are interesting themselves. Time allowing, I will also discuss some applications.

### 4:30 p.m. Hearing Delzant polygons from the equivariant spectrum

#### Emily B. Dryden*, Bucknell University

#### Victor Guillemin, Massachusetts Institute of Technology

Rosa Sena-Dias, Instituto Superior Técnico, Lisbon (1056-58-1299)

##### Can one hear the moment polytope of a toric manifold? Motivated by this question of Miguel Abreu, we ask whether the equivariant spectrum of the Laplacian acting on smooth functions on a toric manifold determines the moment polytope associated to the manifold. Heat invariant techniques, combined with combinatorial and geometric arguments, allow us to prove the desired result for Delzant polygons.

### 5:00 p.m. The fundamental domain of Random Riemann surfaces

#### Eran Makover*, Central Connecticut State University

Jeff McGowan, Central Connecticut State University (1056-58-527)

##### We investigate relations between the cubic graphs and Riemann surfaces that are constructed from a random choice of a graph and orientation. Our goal is to describe that global geometry of a ”typical” Riemann Surfaces. This model of constructing surfaces from graphs enables us to study properties like the Cheeger constant, systole length, and the size of embedded balls in large genus surfaces by examining random cubic graphs.

### 5:30 p.m. The fundamental domain of Random Riemann surfaces

#### Jeffrey McGowan*, Central Connecticut State University

Eran Makover, Central Connecticut State University (1056-58-525)

##### We investigate relations between cubic graphs and Riemann surfaces that are constructed from a random choice of a graph and orientation. Our goal is to describe the global geometry of such a ”typical” Riemann Surface. This model of constructing surfaces from graphs enables us to study properties like the Cheeger constant, systole length, and the size of embedded balls in large genus surfaces by examining random cubic graphs.

#### Saturday January 16, 2010, 8:30 a.m.-10:20 a.m., Room 2007, 2nd Floor, Moscone

### AWM Workshop: Research Presentations by Recent Ph.D.’s, I

### 8:30 a.m. Families over special base manifolds and a conjecture of Campana

#### Kelly Jabbusch*, University of Freiburg

Stefan Kebekus, University of Freiburg (1056-14-169)

##### Campana defined the class of special log varieties (Y, D), characterized by the fact that if ? ⊆ Ω_{p}^{Y} (log D) is an invertible subsheaf for some p, then κ(A) < p. Generalizing classical Shafarevich Hyperbolicity he conjectured that any smooth projective family of canonically polarized manifolds over a special base variety is necessarily isotrivial. I will report on joint work with Stefan Kebekus in which we prove Campana’s conjecture for quasi-projective base manifolds Y^{◦} of dim Y^{◦} ≤ 3.

### 9:00 a.m. Shift Automorphism Varieties Are Not Residually Finite

#### Kate S Owens*, College Station, TX (1056-08-213)

##### An algebra is a nonempty set equipped with some finitary operations. The equational theory of an algebra is the set of all equations true in that algebra. If we can deduce all of an algebra’s true equations from a finite set of equations, we say that the algebra’s equational theory is finitely axiomatizable. In this talk we examine a property shared by some finite algebras whose equational theories are not finitely axiomatizable.

### 9:30 a.m. Oscillation Criteria for Second Order Linear Delay Dynamic Equations

#### Raegan Higgins*, Texas Tech University (1056-39-118)

##### In this talk we consider the second-order linear delay dynamic equation ?

^{∆}(t)

^{∆}()∆ + q(t)y(τ(t)) = 0