Andrea Young

Asymptotic stability of the cross curvature flow at a hyperbolic metric

We show that for any hyperbolic metric on a closed 3-manifold, there exists a neighborhood such that every solution of a normalized cross curvature flow with initial data in this neighborhood exists for all time and converges to a constant-curvature metric. We demonstrate that the same technique proves an analogous result for Ricci flow. Additionally, we prove short-time existence and uniqueness of cross curvature flow under slightly weaker regularity hypotheses than were previously known.

Flipping the Calculus Classroom: A Cost-Effective Approach

Abstract This article discusses a cost-effective approach to flipping the calculus classroom. In particular, the emphasis is on low-cost choices, both monetarily and with regards to faculty time, that make the daunting task of flipping a course manageable for a single instructor. Student feedback and overall impressions are also presented.

Cross curvature flow on a negatively curved solid torus

The classic 2 ‐Theorem of Gromov and Thurston constructs a negatively curved metric on certain 3‐manifolds obtained by Dehn filling. By Geometrization, any such manifold admits a hyperbolic metric. We outline a program using cross curvature flow to construct a smooth one-parameter family of metrics between the “2 ‐metric” and the hyperbolic metric. We make partial progress in the program, proving long-time existence, preservation of negative sectional curvature, curvature bounds and integral convergence to hyperbolic for the metrics under consideration. 53C44; 57M50, 58J35, 58J32 In this note, we outline a program that uses cross curvature flow to answer certain questions inspired by the 2 ‐Theorem of Gromov and Thurston. We begin and make partial progress toward completing this program. More specifically, we consider cross curvature flow on a class of negatively curved metrics on the solid torus, the simplest nontrivial handlebody. This is motivated by the “Dehn surgery” construction in 3‐manifold topology, of which we give a brief account below. We apply the flow to a negatively curved metric described by Gromov‐Thurston on a solid torus with prescribed Dirichlet boundary conditions. Perelman’s use of Ricci flow with surgery to prove the Geometrization Conjecture demonstrates the considerable power of geometric flows to address questions about 3‐manifolds [25; 26]. Subsequent work, that of Agol‐Storm‐Thurston [3], for example, shows that Ricci flow may give information even about 3‐manifolds known a priori to admit a metric of constant curvature. The results of [3] are obtained by using the monotonicity of volume under Ricci flow with surgery. Ricci flow is nonetheless an imperfect tool for analyzing hyperbolic 3‐manifolds ‐ those admitting metrics with constant negative sectional curvatures. According to Geometrization, such manifolds form the largest class of prime 3‐manifolds. Paradoxically, they are also the least understood class. In particular, the facts that Ricci flow

Strange Spinners and Diversity of Dice in Chutes and Ladders

We mine the popular board game Chutes and Ladders for a mathematical question regarding shortest average expected game length. By allowing for nonuniform probability distributions in the form of strange spinners and diverse dice combinations, we are able to dramatically improve on previous results. We employ generating functions, combinatorial theorems, Markov chains, evolutionary algorithms, and an appropriate touch of child-like exploration to find a number of possibilities that decrease the average expected game length.

Chalk It up to Experience: Using Chalkboard Paint to Create Mathematical Manipulatives.

In this article, we give two examples of creating portable chalkboards using chalkboard paint for students to use during cooperative learning. This provides a creative method for professors to facilitate active learning in the undergraduate mathematics classroom.

Improvisation in the Mathematics Classroom

Abstract This article discusses ways in which improvisational comedy games and exercises can be used in college mathematics classrooms to obtain a democratic and supportive environment for students. Using improv can help students learn to think creatively, take risks, support classmates, and solve problems. Both theoretical and practical applications are presented.