Deformations and embeddings of three-dimensional strictly pseudoconvex CR manifolds

Abstract deformations of the CR structure of a compact strictly pseudoconvex hypersurface $M$ in $\mathbb{C}^2$ are encoded by complex functions on $M$. In sharp contrast with the higher dimensional case, the natural integrability condition for $3$-dimensional CR structures is vacuous, and generic deformations of a compact strictly pseudoconvex hypersurface $M\subseteq \mathbb{C}^2$ are not embeddable even in $\mathbb{C}^N$ for any $N$. A fundamental (and difficult) problem is to characterize when a complex function on $M \subseteq \mathbb{C}^2$ gives rise to an actual deformation of $M$ inside $\mathbb{C}^2$. In this paper we study the embeddability of families of deformations of a given embedded CR $3$-manifold, and the structure of the space of embeddable CR structures on $S^3$. We show that the space of embeddable deformations of the standard CR $3$-sphere is a Frechet submanifold of $C^{\infty}(S^3,\mathbb{C})$ near the origin. We establish a modified version of the Cheng-Lee slice theorem in which we are able to characterize precisely the embeddable deformations in the slice (in terms of spherical harmonics). We also introduce a canonical family of embeddable deformations and corresponding embeddings starting with any infinitesimally embeddable deformation of the unit sphere in $\mathbb{C}^2$.Comment: 42 page

Brain Based Learning in Interpretive Exhibit Design: A Field Project

A Field Project Presented in Partial Fulfillment of the Requirements for the Master of Education with an Environmental Education Concentration in the College of Education and Human Service Professions By Sean Curry, B.S., December 2015. Advisor: Ken Gilbertson. This item has been modified from the original to redact the signatures present.This project seeks to use brain based learning as the predominant theory to design interpretive exhibits.University of Minnesota, Duluth. College of Education and Human Service Professions

Conformal submanifolds, distinguished submanifolds, and integrability

For conformal geometries of Riemannian signature, we provide a comprehensive and explicit treatment of the core local theory for embedded submanifolds of arbitrary dimension. This is based in the conformal tractor calculus and includes a conformally invariant Gauss formula leading to conformal versions of the Gauss, Codazzi, and Ricci equations. It provides the tools for proliferating submanifold conformal invariants, as well for extending to conformally singular Riemannian manifolds the notions of mean curvature and of minimal and CMC submanifolds. A notion of distinguished submanifold is defined by asking the tractor second fundamental form to vanish. We show that for the case of curves this exactly characterises conformal geodesics (a.k.a. conformal circles) while for hypersurfaces it is the totally umbilic condition. So, for other codimensions, this unifying notion interpolates between these extremes, and we prove that in all dimensions this coincides with the submanifold being weakly conformally circular, meaning that ambient conformal circles remain in the submanifold. Stronger notions of conformal circularity are then characterised similarly. Next we provide a very general theory and construction of quantities that are necessarily conserved along distinguished submanifolds. This first integral theory thus vastly generalises the results available for conformal circles in [56]. We prove that any normal solution to an equation from the class of first BGG equations can yield such a conserved quantity, and show that it is easy to provide explicit formulae for these. Finally we prove that the property of being distinguished is also captured by a type of moving incidence relation. This second characterisation is used to show that, for suitable solutions of conformal Killing-Yano equations, a certain zero locus of the solution is necessarily a distinguished submanifold.Comment: 87 page

School board governance in changing times: A school’s transition to policy governance

The expansion of the school choice movement and greater flexibility allowed by Every Student Succeeds Act (ESSA) means that education governance is emerging as an important issue for school effectiveness. This longitudinal case study sought to gain an understanding of the implementation of a new governance structure, Policy Governance, in a private, independent school with deeply entrenched culture and patterns of behavior. Findings suggest an immediate positive influence on leadership and culture in the district, including a “trickle down” effect on shared leadership. However, challenges to sustainability indicate that even strict adherence to the model, unanimous support among board members, and strong board and administrator leadership may not be enough to support sustainability

On the Lichnerowicz conjecture for CR manifolds with mixed signature

We construct examples of nondegenerate CR manifolds with Levi form of signature $(p,q)$, $2\leq p\leq q$, which are compact, not locally CR flat, and admit essential CR vector fields. We also construct an example of a noncompact nondegenerate CR manifold with signature $(1,n-1)$ which is not locally CR flat and admits an essential CR vector fields. These provide counterexamples to the analogue of the Lichnerowicz conjecture for CR manifolds with mixed signature.Comment: 7 page