Nonnegatively curved homogeneous metrics in low dimensions

We consider invariant Riemannian metrics on compact homogeneous spaces $G/H$ where an intermediate subgroup $K$ between $G$ and $H$ exists. In this case, the homogeneous space $G/H$ is the total space of a Riemannian submersion. The metrics constructed by shrinking the fibers in this way can be interpreted as metrics obtained from a Cheeger deformation and are thus well known to be nonnegatively curved. On the other hand, if the fibers are homothetically enlarged, it depends on the triple of groups $(H,K,G)$ whether nonnegative curvature is maintained for small deformations. Building on the work of L. Schwachh\"ofer and K. Tapp \cite{ST}, we examine all $G$-invariant fibration metrics on $G/H$ for $G$ a compact simple Lie group of dimension up to 15. An analysis of the low dimensional examples provides insight into the algebraic criteria that yield continuous families of nonnegative sectional curvature.Comment: 14 pages, to appear in Annals of Global Analysis and Geometr

Low Cohomogeneity and Polar Actions on Exceptional Compact Lie Groups

We study isometric Lie group actions on the compact exceptional groups E6, E7, E8, F4 and G2 endowed with a biinvariant metric. We classify polar actions on these groups. We determine all isometric actions of cohomogeneity less than three on E6, E7, F4 and all isometric actions of cohomogeneity less than 20 on E8. Moreover we determine the principal isotropy algebras for all isometric actions on G2.Comment: 27 pages; introduction rewritten; references updated; final version; to appear in Transformation Group

Together We Rise: Reaching Inclusivity for Student Excellence

This presentation outlines the BIONIC (Believe It Or Not I Care) Program at Mattoon High School. For the past 10 years, Dr. Larson and a team of counseling interns have partnered with Mattoon High School to implement BIONIC (Believe It Or Not I Care), a school-wide peer mentoring program

Cohomogeneity one manifolds and selfmaps of nontrivial degree

We construct natural selfmaps of compact cohomgeneity one manifolds with finite Weyl group and compute their degrees and Lefschetz numbers. On manifolds with simple cohomology rings this yields in certain cases relations between the order of the Weyl group and the Euler characteristic of a principal orbit. We apply our construction to the compact Lie group SU(3) where we extend identity and transposition to an infinite family of selfmaps of every odd degree. The compositions of these selfmaps with the power maps realize all possible degrees of selfmaps of SU(3).Comment: v2, v3: minor improvement

Polar actions with a fixed point

AbstractWe prove a criterion for an isometric action of a Lie group on a Riemannian manifold to be polar. From this criterion, it follows that an action with a fixed point is polar if and only if the slice representation at the fixed point is polar and the section is the tangent space of an embedded totally geodesic submanifold. We apply this to obtain a classification of polar actions with a fixed point on symmetric spaces