Abstracts

Abstracts of all lectures are needed for our records. Several participants have also expressed interest in having them available by the beginning of the meeting. Please email us your abstract as soon as possible, not to arrive later than May 10.

Romina Arroyo, Universidad Nacional de Córdoba
Homogeneous Ricci solitons in low dimensions
One of the most important open problems on Einstein homogeneous manifolds is "Alekseevskii's conjecture". This conjecture says that any homogeneous Einstein space of negative scalar curvature is diffeomorphic to a Euclidean space. Due to recent results of Lafuente - Lauret and Jablonski, this conjecture is equivalent to the analogous statement for algebraic solitons, which we call "Generalized Alekseevskii's conjecture".
The aim of this talk is to study the classification of expanding algebraic solitons in low dimensions and use these results to check that the Generalized Alekseevskii conjecture holds in these dimensions. This is a joint work with Ramiro Lafuente.

Werner Ballmann, Universität Bonn
Small eigenvalues on closed surfaces
Eigenvalues of the Laplacian on closed hyperbolic surfaces are called small, if they lie below $1/4$, the bottom of the spectrum of the Laplacian on the hyperbolic plane. Buser showed that, for any $\varepsilon>0$, the surface $S$ of genus $g\ge2$ carries a hyperbolic metric such that $\lambda_{2g-3}<\varepsilon$, where $$\lambda_0=0<\lambda_1\le\lambda_2\le\dots$$ denote the eigenvalues of the Laplacian. He also showed that $\lambda_{2g-2}\ge c>0$, where $c$ is independent of genus and hyperbolic metric, and conjectured that $c=1/4$ is the best constant. I will discuss this conjecture, its recent solution by Otal and Rosas, and some related problems and results.

Peter Buser, École Polytechnique Fédérale de Lausanne
Isometric actions on hyperbolic polyhedra and trigonometry
The lecture is about a class of hyperbolic three manifolds that are obtained by gluing together polyhedral three dimensional analogs of geodesic hexagons. By moving the polyhedra around we shall obtain numerous trigonometric relations in a funny way. These relations may be used to compute the volume of the manifolds.

Tom Farrell, Binghamton University (SUNY)
Negatively curved and Anosov bundles
I will report on joint work with Pedro Ontaneda on negatively curved bundles which leads to more recent joint work with Andrey Gogolev on Anosov bundles. A smooth bundle M-->E-->X is negatively curved when each of its fibers is equipped with a negatively curved Riemannian metric. It is called an Anovov bundle if each of its fibers is equipped with either an Anosov diffeomorphism or an Anosov flow. (Here the abstract fiber M is a closed smooth manifold and the structure group is Diff(M).)
In the case that the base space is a sphere and the abstract fiber M supports a negatively curved Riemannian metric, Pedro and I showed that many smooth M-bundles cannot be a negatively curved bundle. On the other hand we constructed many that are and in fact some that are in more than one way. These results led us to the conjecture that every negatively curved bundle is topologically trivial provided X is simply connected. By considering Anosov bundles, Andrey and I obtain some positive information on this conjecture.

Carolyn Gordon, Dartmouth College
Classical and quantum equivalence of metrics and magnetic fields on nil manifolds
We will address various questions concerning the classical dynamics (geodesics) and ``quantum'' behavior (Laplace spectra) in the setting of small-step Riemannian nilmanifolds.   For flat tori, we will also consider questions of classical and quantum rigidity in the presence of a translation invariant non-degenerate magnetic field $\omega$ (i.e., a translation invariant symplectic structure). If $\omega$ represents an integral cohomology class, geometric quantization associates to the flat metric $h$ and magnetic field $\omega$ a Laplace operator acting on sections of the complex line bundle $L$ with Chern class $[\omega]$ and on the higher tensor powers of $L$. We ask, for example, whether the eigenvalue spectra of these operators determine $\omega$ up to symplectomorphism and whether one can tell from the spectra whether $(h,\omega)$, is a K\"ahler structure.

Francisco Gozzi, University of Pennsylvania
Low Dimensional Polar actions
Polar manifolds are Riemannian G-manifolds admitting a section, i.e., a complete submanifold passing through every orbit and doing so orthogonally. We consider compact simply-connected polar manifolds and achieve an equivariant diffeomorphism classification in dimension 5 or less.

Jens Heber, Universität Kiel
Helical submanifolds of Euclidean space
We report on joint work with M. Scheffel. Helical submanifolds of Euclidean space are characterized by either of the following
equivalent conditions: Geodesics are normal sections; geodesics are pairwise congruent screw lines ("helical"); the Riemannian distance is a function of the Euclidean distance. Compact helical submanifolds are Blaschke manifolds. The only known examples are "nice embeddings" of harmonic manifolds (as introduced by A. Besse) and are therefore compact rank-one-symmetric spaces, as was proved by Z. Szabo. We exhibit structural properties of Jacobi tensors along geodesics in compact helical submanifolds.

Ernst Heintze, Universität Augsburg
Symmetric spaces, affine Kac-Moody algebras, and submanifolds
It is conjectured that isoparametric submanifolds in Hilbert space of codimension at least 2 arise as isotropy orbits of symmetric
spaces G/K where G is an affine Kac-Moody group and K the fixed point set of an involution of the second kind. If time permits we indicate a new classification of these involutions (joint work with W. Freyn). It turns out that they are in close connection to pairs of involutions on finite dimensional simple Lie algebras. As an application we construct a surjective mapping from the set of isoparametric submanifolds to such pairs.

Ramiro Lafuente, Universidad Nacional de Córdoba
On homogeneous warped product Einstein metrics
There is a close link between algebraic solitons and noncompact homogeneous Einstein manifolds, which relies in the fact that any algebraic soliton admits a one-dimensional extension which is a homogeneous Einstein manifold, and the converse also holds provided the Einstein metric is invariant under a non-unimodular group.
It was recently shown by C. He, P. Petersen and W. Wylie that any algebraic soliton also admits a one-dimensional homogeneous extension which is a $(\lambda,n+m)$-Einstein manifold (a $(\lambda,n+m)$-Einstein manifold is the base of an $n+m$-dimensional Riemannian manifold with a warped product Einstein metric), leaving open the question of whether the converse construction can always be carried out. In this talk we will explain both previously mentioned connections, and show that the converse statement for the $(\lambda,n+m)$ case always hold: namely, that any homogeneous $(\lambda,n+m)$-Einstein manifold is a one-dimensional extension of an algebraic soliton. The proof is based on the structural results for homogeneous $(\lambda,n+m)$-Einstein manifolds, but it also makes use of a number of tools from the theory of homogeneous Ricci solitons.

Jorge Lauret, Universidad Nacional de Córdoba
On nonsingular two-step nilpotent Lie algebras
A 2-step nilpotent Lie algebra n is called nonsingular if ad(X): n --> [n,n] is onto for any X not in [n,n]. In this talk, we will explore nonsingular algebras in several directions, including the classification problem (isomorphism invariants), the existence of canonical inner products (nilsolitons) and their automorphism groups (maximality properties). Our main tools are the moment map for certain real reductive representations and the Pfaffian form of a 2-step nilpotent Lie algebra, which is a positive homogeneous polynomial in the nonsingular case.

Yuri Nikolayevsky, La Trobe University
Solvable Lie groups of negative Ricci curvature
We consider the question of whether a given solvable Lie group admits a left-invariant metric of strictly negative Ricci curvature. We give necessary and sufficient conditions of the existence of such a metric for the Lie groups the nilradical of whose Lie algebra is either abelian or standard filiform, and sufficient conditions close to necessary ones for the Lie groups the nilradical of whose Lie algebra is the Heisenberg Lie algebra. This is a joint work with Yurii Nikonorov.

Peter Petersen, University of California, Los Angeles
Linear Stability of Algebraic Solitons
We discuss local stability of expanding algebraic solutions under compactly supported perturbations. Koiso’s stability criterion for compact Einstein metrics is shown to also yield stability in this setting. We give the first examples of Einstein metrics that do not satisfy Koiso’s condition and show that all 2-step nilsolitions are linearly stable.

Marco Radeschi, Universität Münster
Clifford algebras, and new singular Riemannian foliations in round spheres
Singular Riemannian foliations are characterized by the property that leaves stay at a constant distance from each other. On a round sphere, almost every (indecomposable) singular Riemannian foliation is homogeneous, i.e. it is given by the orbits of an isometric action. Here we show how Clifford algebras can be used to produce a new class of non homogeneous singular Riemannian foliations, in some sense larger than the class of homogeneous foliations, which contains all the previously known examples.

Allie Ray, University of Texas, Arlington
Graphs, Group Actions, and Isospectrality
I am searching for new examples of isospectral manifolds constructed from pairs of Schreier graphs of a Sunada-Gassmann triple, which are therefore isospectral graphs. I will be discussing one construction introduced by S. Dani and M. Mainkar and another suggested by C. Gordon that associates with each graph a Lie algebra. This latter construction relies on the action of the generators of the overlying group on the graph and resultant Lie algebra.

Ben Schmidt, Michigan State University
Rank-rigidity with lower sectional curvature bounds
A closed Riemannian manifold M is said to have extremal curvature K if all sectional curvatures are bounded above or bounded below by K. If, in addition, each complete parameterized geodesic c(t) admits a parallel and orthogonal vector field V(t) satisfying sec(V(t),c'(t))=K for all t, then M is said to have positive rank.
When K is an upper bound for sectional curvatures, M is known to be rank-rigid: locally symmetric or locally reducible when K=0 (Ballmann, Burns-Spatzier), locally symmetric when K=-1 (Hamenstadt), and locally symmetric when K=1 (Shankar-Spatzier-Wilking).
Comparatively less is known when K is a lower bound for sectional curvatures. After surveying known results, I'll describe some new rank-rigidity theorems for manifolds with lower sectional curvature bounds. Based on joint works with Bettiol, Wolfson, and Shankar-Spatzier.

Craig Sutton, Dartmouth College
Can you hear the length spectrum of a compact Lie group?
Motivated in part by considerations from quantum mechanics and geometric optics, it is a long-standing folk-conjecture that the spectrum of a manifold determines its length spectrum (i.e., the set consisting of the lengths of smoothly closed geodesics). Using the trace formula of Duistermaat and Guillemin one can see that this conjecture is true for sufficiently ``bumpy'' Riemannian manifolds. However, our understanding of the conjecture in the non-bumpy setting is rather incomplete. In this talk, we will demonstrate that the length spectrum of a bi-invariant metric on a compact simple Lie group can be recovered from its Laplace spectrum by computing the singular support of the trace of the associated wave group; that is, we show that the so-called Poisson relation is an equality for such spaces. More generally, we will see that the preceding statement holds for compact irreducible split-rank symmetric spaces.

Haotian Wu, University of Oregon
Dynamical stability of algebraic Ricci solitons
We consider dynamical stability for a modified Ricci ﬂow equation whose stationary solutions include Einstein and Ricci soliton metrics. We focus on homogeneous metrics on non-compact manifolds. Following the program of Guenther, Isenberg, and Knopf, we deﬁne a class of weighted little Hölder spaces with certain interpolation properties that allow the use of maximal regularity theory and the application of a stability theorem of Simonett. With this, we derive two stability theorems, one for a class of Einstein metrics and one for a class of non-Einstein Ricci solitons. Using linear stability results of Jablonski, Petersen, and Williams, we obtain dynamical stability for many speciﬁc Einstein and Ricci soliton metrics on simply connected solvable Lie groups. (This is joint work with Michael Bradford Williams.)

Will Wylie, Syracuse University
Positive weighted curvature and Symmetry
There are now a number of different notions of curvature for Riemannian manifolds with density. The most well known is the Bakry-Emery Ricci curvature which appears in the definition of the gradient Ricci soliton equation. There are a number of obstructions to a compact manifold admitting a metric with positive Bakry-Emery curvature.   In fact, it is an open question whether every compact manifold that admits a metric with positive Bakry-Emery curvature also admits a metric with positive
Ricci curvature.    In this talk we'll give a simple argument showing this is true in the homogeneous case.    Time permitting, we'll also discuss the case of manifolds with positive weighted sectional curvature and symmetry. (This is joint work with Lee Kennard.)

Wolfgang Ziller, University of Pennsylvania
Non-negatively curved submanifolds of Euclidean space
We discuss known results by Weinstein and Moore on positively curved submanfolds of codimension 2, some conjectures in codimension 3, and some recent joint work with Luis Florit on rigidity results of non-negatively curved submanifolds in codimension 2. The proofs combine methods from submanifold theory with Ricci flow and critical point theory for height functions.

Andrew Zimmer, University of Michigan
Rigidity of convex divisible sets
An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts co-compactly. The classic example of a divisible set is the unit ball, this has projective automorphism group SO(1,d) and hence by a theorem of Borel there exists a discrete group which acts co-compactly. There are many additional examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have $C^1$ boundary, and have word hyperbolic dividing group. In this talk I will discuss a notion of convexity in complex and quaternionic projective space and show that every divisible ``convex'' set with $C^1$ boundary is projectively equivalent to the unit ball. The proof uses an analogue of the Hilbert metric and tools from dynamics, geometric group theory, and algebraic groups.