Gizem Karaali

Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.

On Hopf Algebras and Their Generalizations

We survey Hopf algebras and their generalizations. In particular, we compare and contrast three well-studied generalizations (quasi-Hopf algebras, weak Hopf algebras, and Hopf algebroids), and two newer ones (Hopf monads and hopfish algebras). Each of these notions was originally introduced for a specific purpose within a particular context; our discussion favors applicability to the theory of dynamical quantum groups. Throughout the note, we provide several definitions and examples in order to make this exposition accessible to readers with differing backgrounds.

An Evaluative Calculus Project: Applying Bloom's Taxonomy to the Calculus Classroom.

Abstract In education theory, Blooms taxonomy is a well-known paradigm to describe domains of learning and levels of competency. In this article I propose a calculus capstone project that is meant to utilize the sixth and arguably the highest level in the cognitive domain, according to Bloom et al.: evaluation. Although one may assume that mathematics is a value-free discipline, and thus the mathematics classroom should be exempt from focusing on the evaluative aspect of higher-level cognitive processing, I surmise that we as mathematics instructors should consider incorporating such components into our courses. The article also includes a brief summary of my observations and a discussion of my experience during the Fall 2008 semester, when I used the project described here in my Calculus I course.

Metacognition in the Classroom: Motivation and Self-Awareness of Mathematics Learners.

Abstract Metacognition as a dimension of learning ranks highest in the revised Bloom’s Taxonomy of cognitive tasks. In this paper we illustrate how it can be incorporated into a repeated exercise in the mathematics classroom, through a specific case study in the context of a liberal arts mathematics course. Through the semester, students were asked weekly to evaluate their own progress and review their development in light of their personal goals. We observed positive affective changes (including engagement levels) in the students through the course of the semester. We argue that the weekly metacognitive and self-reflective activities helped students keep their focus on learning deeply and allowed them to remain engaged and motivated through the semester. A brief theoretical discussion is included, and other possible contexts suitable for the described activities are suggested.

Envisioning a Quantitative Studies Center: A Liberal Arts Perspective

Several academic institutions are searching for ways to help students develop their quantitative reasoning abilities and become more adept at higher-level tasks that involve quantitative skills. In this note we study the particular way Pomona College has framed this issue within its own context and what it plans to do about it. To this end we describe our efforts as members of a campus-wide committee that was assigned the duty of investigating the feasibility of founding a quantitative studies center on our campus. These efforts involved analysis of data collected through a faculty questionnaire, discipline-specific input obtained from each departmental representative, and a survey of what some of our peer institutions are doing to tackle these issues. In our studies, we identified three critical needs where quantitative support would be most useful in our case: tutoring and mentoring for entry-level courses; support for various specialized and analytic software tools for upper-level courses; and a uniform basic training for student tutors and mentors. We surmise that our challenges can be mitigated effectively via the formation of a well-focused and -planned quantitative studies center. We believe our process, findings and final proposal will be helpful to others who are looking to resolve similar issues on their own campuses.

Super solutions of the dynamical Yang-Baxter equation

Solutions of the classical dynamical Yang-Baxter equation on a Lie superalgebra are called super dynamical r-matrices. A super dynamical r-matrix r satisfies the zero weight condition if [h⊗1+1⊗h,r(λ)] = 0 for all h ∈ h, λ ∈ In this paper we classify super dynamical r-matrices with zero weight.

Coalitions and Cliques in the School Choice Problem

The school choice problem (SCP) looks at assignment mechanisms matching students in a public school district to seats in district schools. The Gale‐Shapley deferred acceptance mechanism applied to the SCP, known as the student optimal stable matching (SOSM), is the most efficient among stable mechanisms yielding a solution to the SCP. A more recent mechanism, the efficiency adjusted deferred acceptance mechanism (EADAM), aims to address the well-documented tension between efficiency and stability illustrated by SOSM. We introduce two alternative efficiency adjustments to SOSM, both of which necessarily sacrifice stability. Our discussion focuses on the mathematical novelty of new efficiency modifications rather than any practical superiority of implementation or outcome. That is, our contribution lies in process rather than outcome. Yet we argue that the demonstration of multiple processes yielding common outcomes is, in itself, a measure of the quality of that outcome. More specifically the consistency of outcome from different processes strengthens the argument that Pareto dominations of SOSM can be supported as “fair” despite the resulting priority violations.

Unnatural disasters: Two calculus projects for instructors teaching mathematics for social justice

Abstract We provide context and motivation for an instructor to use real-life examples in the calculus classroom. To this end we describe two specific project ideas, one related to the devastating impact of methylmercury fungicide in a grain seed supply and the other to a catastrophic methane leak. By using calculus in contexts that have social justice implications, we hope to empower students to reason for themselves, to use mathematics as a powerful tool to deepen their understanding of the world, and ultimately, to effectively confront the challenges society faces.

On Grades and Instructor Identity: How Formative Assessment Saved me from a Midlife Crisis

Abstract In recent years, I have cultivated an almost pathological resistance to grading. Here I explore the reasons why and describe how I eventually recovered. In particular, I propose that although grading, or more explicitly, effective assessment of student learning, is a challenging component of a mathematics instructor’s job description, reflective use of formative assessment can substantially relieve the pressure, as it allows the instructor to focus on what matters most: student learning and growth. To this end, I describe my experiences with formative assessment in a diverse selection of courses (ranging from calculus to introduction to proofs to mathematics for liberal arts). I conclude that formative assessment can help an instructor move toward a more intentional pedagogical stance, and a more constructive professional identity.

The Problems of Contemporariness and Voice: Review of Literacy & Mathematics: A Contemporary Approach to Quantitative Literacy by Jay P. Abramson and Matthew A. Isom (2005)

Jay P. Abramson and Matthew A. Isom. Literacy & Mathematics: A Contemporary Approach to Quantitative Literacy. (Dubuque IA: Kendell Hunt Publishing Co., 2005). 297 pp. ISBN 0-7575-2165-7. The book under review covers the traditional content of a typical mathematical literacy text. After a brief overview of the book contents, the review then focuses on two specific challenges that QL textbooks have to meet: the timeliness of the contexts used and the subjective author voice that inevitably colors any contextualized discussion. Both issues noticeably arise in the text reviewed. Nonetheless instructors may find it a helpful resource.