Boundary slopes of immersed surfaces in 3-manifolds

This paper presents some finiteness results for the number of boundary slopes of immersed essential surfaces of given genus g in a compact 3-manifold with torus boundary. In the case of hyperbolic 3-manifolds we obtain uniform quadratic bounds in g for the number of possible slopes, independent of the 3-manifold. We also look at some related quantities, such as how many times the slopes of two such surfaces of specified genus can intersect

On the dependability and feasibility of layperson ratings of divergent thinking

A new system for subjective rating of responses to divergent thinking tasks was tested using raters recruited from Amazon Mechanical Turk. The rationale for the study was to determine if such raters could provide reliable (aka generalizable) ratings from the perspective of generalizability theory. To promote reliability across the Alternative Uses and Consequence task prompts often used by researchers as measures of Divergent Thinking, two parallel scales were developed to facilitate feasibility and validity of ratings performed by laypeople. Generalizability and dependability studies were conducted separately for two scoring systems: the average-rating system and the snapshot system. Results showed that it is difficult to achieve adequate reliability using the snapshot system, while good reliability can be achieved on both task families using the average-rating system and a specific number of items and raters. Additionally, the construct validity of the average-rating system is generally good, with less validity for certain Consequences items. Recommendations for researchers wishing to adopt the new scales are discussed, along with broader issues of generalizability of subjective creativity ratings. © 2018 Hass, Rivera and Silvia

Computing trisections of 4-manifolds

Algorithms that decompose a manifold into simple pieces reveal the geometric and topological structure of the manifold, showing how complicated structures are constructed from simple building blocks. This note describes a way to algorithmically construct a trisection, which describes a $4$-dimensional manifold as a union of three $4$-dimensional handlebodies. The complexity of the $4$-manifold is captured in a collection of curves on a surface, which guide the gluing of the handelbodies. The algorithm begins with a description of a manifold as a union of pentachora, or $4$-dimensional simplices. It transforms this description into a trisection. This results in the first explicit complexity bounds for the trisection genus of a $4$-manifold in terms of the number of pentachora ($4$-simplices) in a triangulation.Comment: 15 pages, 9 figure

Proof of the Double Bubble Conjecture in R^n

The least-area hypersurface enclosing and separating two given volumes in R^n is the standard double bubble.Comment: 20 pages, 22 figure

Directorial Fiduciary Duties in a Tracking Stock Equity Structure: The Need for a Duty of Fairness

Part I of this article briefly describes the key distinctions between a tracking stock corporation and a conventional corporation. It then touches on the reasons why corporations have adopted tracking stock equity structures. Part II articulates the unique legal challenges presented by a tracking stock equity structure. Part III discusses the disclosure that tracking stock corporations have made with respect to these challenges. Part IV briefly summarizes the fiduciary duties of care and loyalty and explores why these duties are ill-equipped to address these challenges. Part V presents the duty of fairness and discusses the duty\u27s elements in detail. In addition, Part V sets forth practical advice as to what tracking stock boards can do today to help minimize their exposure to litigation that may be commenced by disgruntled tracking stock stockholders in the future