Abstracts

• Ken Richardson (Texas Christian University) - "Eigenspace isomorphisms"

We will survey some known results about cohomology and harmonic forms, and we will also show relationships among eigenforms of the Laplacian of different degrees. We will specifically apply them to general Riemannian manifolds, K\"ahler manifolds, and Sasaki manifolds, obtaining specific obstructions to the existence of special types of Riemannian metrics. The talk contains joint work with Robert Wolak and Georges Habib.

• Catherine Searle (Wichita State University) - TBD

• Alessandro Minuzzo (Università di Parma) - "Rational Hyperbolicity Problem"

We find counterexamples to a conjecture by Grove, Wilking and Yeager. The conjecture states that if $M$ is a closed, simply connected $G$-manifold, whose quotient $M/G$ is a hyperbolic polygon, then $M$ is rationally hyperbolic. Furthermore, we are able to prove that, under additional hypothesis, the conjecture holds true. More in general, we consider a Singular Riemannian Foliation $(M,\mathcal{F})$ of codimension two on a closed simply connected manifold $M$, such that the leaf space $M/\mathcal{F}$ supports a metric of hyperbolic type. If we also require that the multiplicities of the sides of $M/\mathcal{F}$, which count the dimensional drop of the leaves, are strictly greater than one, then $M$ is rationally hyperbolic.

This is a joint work with Professor M. Radeschi and Professor R. Mendes.

• Bach Tran (University of Oklahoma) - "A lower bound of the first nonzero basic eigenvalue on a singular Riemannian foliation."

In this talk, we provide a lower bound of the first basic eigenvalue on a closed singular Riemannian manifold $(M,\mathcal{F})$ with basic mean curvature that depends on the given nonnegative lower bound of the Ricci curvature of M and the diameter of the leaf space $M/\mathcal{F}$, which are  generalizations of Zhong-Yang's estimate and Shi-Yang's estimate in the case of singular Riemannian foliations.  We also provide the rigidity result of the generalized Zhong-Yang's estimate, which is so-called generalized Hang-Wang's rigidity.

• Tomoya Tatsuno (University of Oklahoma) - "Sectional Curvature Pinching of Two-Step Nilmanifolds"

Nilmanifolds are homogeneous Riemannian manifolds admitting a transitive nilpotent Lie group of isometries. By classical results (Wolf, Milnor), nilmanifolds are always of mixed curvature. Two-step nilmanifolds are particularly important, as they play a crucial role in the classification of quarter-pinched homogeneous manifolds of negative curvature by Eberlein and Heber. Given a two-step nilmanifold, we study its pinching constant, which is the ratio of the minimum and maximum of sectional curvature.