BS, Lafayette College. MS, PhD, University of Virginia. Special interests in mathematics education, game theory, history and philosophy of mathematics, and the outreach of mathematics to the social sciences and the humanities. Author of research papers in the areas of nonassociative algebra, fair-division theory, and mathematics education; governor of the Metropolitan New York Section of the Mathematical Association of America; member, board of editors, *The College Mathematics Journal.* SLC, 1997–

#### Research Interests

Special interests in mathematics education, history and philosophy of mathematics, game theory, fair division theory, social choice theory, abstract algebra, applied statistics, and the outreach of mathematics to areas in the social sciences; author of research papers in the areas of Jordan theory, nonassociative superalgebras, fair division theory, mathematics education and mathematical literature.

## Undergraduate Courses 2021-2022

### Mathematics

#### An Introduction to Statistical Methods and Analysis

##### Open, Lecture—Spring

Variance, correlation coefficient, regression analysis, statistical significance, and margin of error...you’ve heard these terms and other statistical phrases bantered about before, and you’ve seen them interspersed in news reports and research articles. But what do they mean? And why are they so important? Serving as an introduction to the concepts, techniques, and reasoning central to the understanding of data, this lecture course focuses on the fundamental methods of statistical analysis used to gain insight into diverse areas of human interest. The use, misuse, and abuse of statistics will be the central focus of the course; and specific topics of exploration will be drawn from experimental design theory, sampling theory, data analysis, and statistical inference. Applications will be considered in current events, business, psychology, politics, medicine, and other areas of the natural and social sciences. Statistical (spreadsheet) software will be introduced and used extensively in this course, but no prior experience with the technology is assumed. Group conferences, conducted in workshop mode, will serve to reinforce student understanding of the course material. This lecture is recommended for anybody wishing to be a better-informed consumer of data and strongly recommended for those planning to pursue advanced undergraduate or graduate research in the natural sciences or social sciences.

###### Faculty

#### Mathematics and Jorge Luis Borges

##### Open, Seminar—Fall

The works of Jorge Luis Borges, the highly influential 20th-century Argentine writer, feature imaginatively intelligent and deeply provocative use of mathematical ideas and imagery. Borges’s writings—primarily short stories, essays, and poetry—describe fictitious worlds that warp standard notions of time, space, and existence and reveal the unavoidable friction between competing notions at the heart of modern mathematics: the infinite versus the finite versus the infinitesimal (set theory); the discrete versus the continuous (calculus); the reasonable versus the paradoxical (logic); the Euclidean versus the otherworldly (geometry); the symmetric versus the distorted (fractals, chaos); the convergent versus the divergent (limits, series); the improbable versus the impossible (combinatorics, probability). In short, this seminar will explore various fundamental and foundational topics in mathematics from a Borgesian perspective. Student conference projects for this seminar may focus upon the mathematical themes in the works of other writers or explore any mathematically-themed subject.

###### Faculty

#### Multivariable Mathematics: Linear Algebra, Vector Calculus, and Differential Equations

##### Intermediate, Seminar—Year

Rarely is a quantity of interest—tomorrow’s temperature, unemployment rates across Europe, the cost of a spring-break flight to Fort Lauderdale—a simple function of just one primary variable. Reality, for better or worse, is mathematically multivariable. This course introduces an array of topics and tools used in the mathematical analysis of multivariable functions. The intertwined theories of vectors, matrices, and differential equations and their applications will be the central themes of exploration in this yearlong course. Specific topics to be covered include the algebra and geometry of vectors in two, three, and higher dimensions; dot and cross products and their applications; equations of lines and planes in higher dimensions; solutions to systems of linear equations, using Gaussian elimination, theory and applications of determinants, inverses and eigenvectors, volumes of three-dimensional solids via integration, spherical and cylindrical coordinate systems, and methods of visualizing and constructing solutions to differential equations of various types. Conference work will involve an investigation of some mathematically-themed subject of the student’s choosing.

###### Faculty

## Previous Courses

### Mathematics

#### Abstract Algebra: Theory and Applications

##### Advanced, Seminar—Spring

In pre-college mathematics courses, we studied the underlying methodology, concepts, and applications of basic algebra. We appointed letters of the alphabet to abstractly represent unknown quantities and translated real world (and often complicated) problems into simple equations whose solutions, if they could be found, held the key to greater understanding of the situation at hand. Fine, but algebra does not end there. Advanced algebra examines sets of various types of objects (matrices, polynomials, functions, rigid motions, etc.) and the operations that exist on these sets. The approach is axiomatic: One assumes a small number of basic properties, or axioms, and attempts to deduce all other properties of the mathematical system from these few properties. Such abstraction allows us to study, simultaneously, all of the various structures satisfying a given set of axioms and identify both their commonalties and their differences. Specific topics to be covered include groups, actions, isomorphisms, symmetries, permutations, rings, and fields and their various applications.

###### Faculty

#### Calculus I: The Study of Motion and Change

##### Open, Seminar—Fall

Our existence lies in a perpetual state of change. An apple falls from a tree; clouds move across expansive farmland, blocking out the sun for days; meanwhile, satellites zip around the Earth transmitting and receiving signals to our cell phones. The calculus was invented to develop a language to accurately describe and study the motion and change happening around us. The Ancient Greeks began a detailed study of change but were scared to wrestle with the infinite; so it was not until the 17th century that Isaac Newton and Gottfried Leibniz, among others, tamed the infinite and gave birth to this extremely successful branch of mathematics. Though just a few hundred years old, the calculus has become an indispensable research tool in both the natural and social sciences. Our study begins with the central concept of the limit and proceeds to explore the dual processes of differentiation and integration. Numerous applications of the theory will be examined. For conference work, students may choose to undertake a deeper investigation of a single topic or application of the calculus or conduct a study of some other mathematically-related topic. This seminar is intended for students interested in advanced study in mathematics or sciences, students preparing for careers in the health sciences or engineering, and any student wishing to broaden and enrich the life of the mind.

###### Faculty

#### Calculus II: Further Study of Motion and Change

##### Open, Seminar—Spring

This course continues the thread of mathematical inquiry following an initial study of the dual topics of differentiation and integration (see Calculus I course description). Topics to be explored in this course include the calculus of exponential and logarithmic functions, applications of integration theory to geometry, alternative coordinate systems, infinite series, and power series representations of functions. For conference work, students may choose to undertake a deeper investigation of a single topic or application of the calculus or conduct a study of some other mathematically-related topic. This seminar is intended for students interested in advanced study in mathematics or sciences, students preparing for careers in the health sciences or engineering, and any student wishing to broaden and enrich the life of the mind. The theory of limits, differentiation, and integration will be briefly reviewed at the beginning of the term.

###### Faculty

#### Discrete Mathematics: Gateway to Higher Mathematics

##### Intermediate, Seminar—Fall

There is an enormous, vivid world of mathematics beyond what students encounter in high-school algebra, geometry, and calculus courses. This seminar provides an introduction to this realm of elegant and powerful mathematical ideas. With an explicit goal of improving students’ mathematical reasoning and problem-solving skills, this seminar provides the ultimate intellectual workout. Five important themes are interwoven in the course: logic, proof, combinatorial analysis, discrete structures, and philosophy. For conference work, students may design and execute any appropriate project involving mathematics. A must for students interested in pursuing advanced mathematical study, this course is also highly recommended for students with a passion for computer science, engineering, law, logic, and/or philosophy.

###### Faculty

#### First-Year Studies: Everything (and Nearly Nothing) About Infinity

##### Open, FYS—Year

“There is a concept that corrupts and upsets all others. I speak not of Evil, whose limited realm is that of ethics; I refer to The Infinite.” So wrote Jorge Luis Borges, the highly influential, 20th-century Argentine writer, though Borges was not alone in his fascination with the subject matter. Indeed, the concept of infinity has been a virtual leitmotif in the history of intellectual thought. The pre-Socratic philosopher Zeno voiced concern over paradoxes involving infinity as related to physical motion, paradoxes that would not be fully resolved until the advent of “the calculus.” In the later Greek era, Euclid provided an elegant proof of the infinitude of prime numbers; and Archimedes, the greatest applied mathematician of antiquity, recognized infinity as a natural extension of the finite through limiting processes. Italian friar, poet, physicist, and mathematician Giordano Bruno, of early modernity, was burned at the stake by the Inquisition for his “antireligious” interest in the infinite and “unholy” belief in a heliocentric solar system. Galileo nearly suffered the same outcome. Newton and Leibniz simultaneously, yet independently, invented calculus, bridging the mathematical divide between the discrete and the continuous and harnessing the power imbedded in the concept of the infinitesimally small. The 19th-century German scholar Georg Cantor was the first to study Infinity with all of the usual rigor associated with other mathematical inquiries, though most of his contemporaries discredited his visionary efforts. Over the ages, writers, painters, musicians, and other artists have taken their turn in an effort to understand and depict infinity in its diverse forms. Though the approach of this first-year studies seminar will be decidedly mathematical, we will not hesitate to explore the notion of infinity from all of its multidisciplinary perspectives. Prior study of the calculus or more advanced mathematics is not a prerequisite for this course, but a willingness to explore and enjoy such hefty concepts is expected.

###### Faculty

#### Game Theory: The Study of Conflict and Strategy

##### Open, Lecture—Spring

Warfare, elections, auctions, labor-management negotiations, inheritance disputes, even divorce—these and many other conflicts can be successfully understood and studied as games. A game, in the parlance of social scientists and mathematicians, is any situation involving two or more participants (players) capable of rationally choosing among a set of possible actions (strategies) that lead to some final result (outcome) of typically unequal value (payoff or utility) to the players. Game theory is the interdisciplinary study of conflict, whose primary goal is the answer to the single, simply-stated, but surprisingly complex question: What is the best way to “play”? Although the principles of game theory have been widely applied throughout the social and natural sciences, their greatest impact has been felt in the fields of economics, political science, and biology. This course represents a survey of the basic techniques and principles in the field. Of primary interest will be the applications of the theory to real-world conflicts of historical or current interest.

###### Faculty

#### Mathematics in Theory and Practice: Probability, Risk Analysis, and Optimization

##### Intermediate, Seminar—Year

What is chance, and how do we measure it? Do we measure the probability of winning the Mega Millions lottery in the same way that we assess the likelihood of a volcanic eruption in Hawaii? What tools are available to understand and measure uncertainty and risk? How can an understanding of probabilities better inform the decisions that we make in our personal and professional lives? How can we make the very best choice(s) amidst an enormous number of available options? How can individuals and businesses make critical decisions with confidence despite incomplete information and considerable uncertainty of future states? This calculus-based introduction to advanced probability theory, risk analysis, and operations research (optimization theory) engages these topics with an eye on diverse applications in the natural sciences, business, economics, and the social sciences. Topics of exploration will include the essential preliminaries of discrete mathematics (symbolic logic, proof technique, and set theory), combinatorial probabilities, distributions of prominent discrete and continuous random variables (Gaussian normal, binomial, Poisson, etc.), conditional probability and independence, joint distributions, expectation, variance, covariance, laws of large numbers, the Central Limit Theorem, Bayes Theorem, Markov chains, stochastic processes, linear programming and the powerful simplex method, sensitivity of optimized solutions to slight shifts in input parameters, duality theory, integer programming, nonlinear optimization, stochastic programming, and the four classic examples of optimization theory (the transportation/assignment problem, the network-flow problem, the diet problem, and the traveling-salesman problem). Using mathematical software, students will gain practical experience in the art of computer simulation and optimal solution identification.

###### Faculty

## Additional Information

### Activities

#### SLC Mathematical Resource Center

##### Co-Director

Providing assessment, counseling, and tutoring for students wishing to strengthen their mathematical skills.

#### Current committees

Curriculum Committee, Committee on Academic Preparation of Teachers

#### Former committees

Advisory Committee on Appointments and Tenure, General Committee, Web Advisory Committee, Advisory Committee for the Center for Continuing Education, Substance Use Committee, Admissions Committee, Sports Center Committee, Budget Committee

### Professional affiliations

#### Metropolitan New York Section of the Mathematical Association of America (MAA)

##### Governor

A professional society of over 1,000 mathematicians in the greater New York City area focused on enhancing undergraduate mathematics education.

#### The College Mathematics Journal

##### Member of Board of Editors

A publication of the Mathematics Association of America that publishes articles, short Classroom Capsules, problems, solutions, media reviews and other pieces specifically aimed at the college mathematics curriculum with emphasis on topics taught in the first two years.

#### Section NExT: New Experiences in Teaching

##### Co-founder

A program aimed at supporting new and rising Ph.D's in mathematics or mathematics education. Section NExT is a local version of the highly successful national MAA program Project NExT. Like Project NExT, Section NExT's goal is to support new and pre-tenured faculty who are interested in improving the teaching and learning of undergraduate mathematics. Section NExT aims to provide New York area mathematicians who have recently entered the profession with practical information about, and concrete suggestions for, implementing more effective pedagogical and professional strategies, ranging from new teaching methods to writing grant proposals and balancing teaching and research responsibilities.

### Selected Publications

#### Mathematical Ideas and Images in the Works of Jorge Luis Borges

(In progress with students N. Mendoza, H. Mezzabolta, N. Scott and C. Wolf)

#### An Interesting Application of Multivariable Calculus to Marine Biology Research

(In progress with Ray Clarke)

#### Efficient Fair Division: Helping the Worst Off or Avoiding Envy?

*Rationality and Society*

17, no. 4, November 2005

#### From Calculus to Topology: Teaching Lecture-Free Seminar Courses at all levels of the Undergraduate Mathematics Curriculum

*PRiMUS (Problems, Resources, and issues in Mathematics Undergraduate Studies)*

September, 2002

#### Quadratic Jordan Superalgebras

*Communications in Algebra*

29 (2000), 375-401

#### The Kantor Doubling Process Revisited

*Communications in Algebra*

23 (1995), 357-372 (with Kevin McCrimmon)

#### The Split Kac Superalgebra K10

*Communications in Algebra*

22 (1994), 29-40

#### The Kantor Construction of Jordan Superalgebras

*Communications in Algebra*

20 (1992), 109-126 (with Kevin McCrimmon)

### Lectures, Talks and Presentations

#### “Win, Lose or Draw (But Most Likely Win!): The Mathematics of TV Game Shows”

Science Seminar Series, Sarah Lawrence College, Bronxville, NY, November 2010

#### “The Mathematics of Conflict: Games Against Gods and Criminals”

Mount Saint Mary College, Newburgh, NY, November 2010

#### “What’s Wrong with the Electoral College? Historical and Mathematical Perspectives”

Dutchess Community College, Poughkeepsie, NY, September 2009

#### “Jorge Luis Borges and Mathematics”

Oberlin College, Oberlin, OH, April, 2008