Regularization via Cheeger Deformations

We show that Cheeger deformations regularize $G$--invariant metrics in a very strong sense.Comment: 9 page

Transforming Maine’s Economy: Innovation and Entrepreneurship Policy

Innovation and entrepreneurship are major drivers of economic growth. Support for them is a primary role of state government in order to increase the well being of its citizens through the provision of well-paying jobs that cannot be easily exported. Today, the state’s role is described as “enhancing the innovation ecosystem,” with the goal of increased productivity, innovation and competitiveness. Policies to build the ecosystem include: (1) building and supporting a state’s research and development capacity; (2) encouraging a state’s entrepreneurial community, (3) increasing the productivity of a state’s economy though the commercialization of new products, services, processes, business models and marketing strategies and (4) supporting sectors and/or clusters. Each of these is part of an overall policy environment that will support innovation and entrepreneurship

Almost non-negatively curved 4-manifolds with torus symmetry

We prove that if a closed, smooth, simply-connected 4-manifold with a circle action admits an almost non-negatively curved sequence of invariant Riemannian metrics, then it also admits a non-negatively curved Riemannian metric invariant with respect to the same action. The same is shown for torus actions of higher rank, giving a classification of closed, smooth, simply-connected 4-manifolds of almost non-negative curvature under the assumption of torus symmetry.Comment: 15 pages. As suggested by a referee for Proc. Amer. Math. Soc., the result is expanded in this version to include all torus actions, with the title changing accordingl

Positively curved Alexandrov spaces with circle symmetry in dimension 4

Positively curved Alexandrov spaces of dimension 4 with an isometric circle action are classified up to equivariant homeomorphism, subject to a certain additional condition on the infinitesimal geometry near fixed points which we conjecture is always satisfied. As a corollary, positively curved Riemannian orbifolds of dimension 4 with an isometric circle action are also classified