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.

### Current undergraduate courses

#### An Introduction to Statistical Methods and Analysis

##### Fall

An introduction to the concepts, techniques, and reasoning central to the understanding of data, this lecture course focuses on the fundamental ideas of statistical analysis used to gain insight into diverse areas of human interest. The use, abuse, and misuse of statistics will be the central focus of the course. Topics of exploration will include the core statistical topics in the areas of experimental study design, sampling theory, data analysis, and statistical inference. Applications will be drawn from current events, business, psychology, politics, medicine, and other areas of the natural and social sciences. Spreadsheet statistical software will be introduced and used extensively in this course, but no prior experience with the software is assumed. Conference work will involve working in a small group to conceive and execute a small-scale research study. This seminar is an invaluable course for anybody planning to pursue graduate work and/or research in the natural sciences or social sciences. No college-level mathematical knowledge is required.

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###### Related Cross-Discipline Paths

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

##### 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, the greatest impact has been felt in the fields of economics, political science and biology. This course represents a survey of basic techniques and principles. Of primary interest will be the applications of the theory to real-world conflicts of historical or current interest.

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###### Related Cross-Discipline Paths

#### Multivariable Modeling I: Vectors, Functions, and Matrices

##### Fall

Rarely is a quantity of interest (tomorrow’s temperature, unemployment rates across Europe, the cost of a spring break flight to Denver) a simple function of just one other variable. Reality, for better or worse, is mathematically multivariable. This course provides an introduction to an array of topics and tools used in the mathematical analysis of multivariable functions. The intertwined theories of vectors, functions, and matrices will be the central theme of exploration in this first semester of the Multivariable Modeling two-course sequence. 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 solids in three dimension via integration; and spherical and cylindrical coordinate systems. Conference work will involve a concentrated investigation of one application of multivariable mathematics.

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###### Related Cross-Discipline Paths

#### Multivariable Modeling II: Differential Equations

##### Spring

Many laws governing physical and natural phenomena and, of late, a growing number of theories describing social phenomena are expressed in terms of the rates of change (derivatives) of interrelated variables. Differential equations, the branch of mathematics that explores these important relationships, provides a collection of tools and techniques fundamental to advanced study in engineering, physics, economics, and applied mathematics. The investigation of such equations and their applications will be the focus of this second half of the Multivariable Modeling two-course sequence. Conference work will involve a concentrated investigation of one application of multivariable mathematics.

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###### Related Cross-Discipline Paths

### Previous courses

#### Abstract Algebra: Theory and Applications

##### Spring

In pre-college mathematics courses, we learned the basic methodology and notions of algebra. We appointed letters of the alphabet to abstractly represent unknown or unspecified quantities. We discovered how to translate real-world (and often complicated) problems into simple equations whose solutions, if they could be found, held the key to greater understanding. But algebra does not end there. Advanced algebra examines sets of objects (numbers, matrices, polynomials, functions, ideas) and operations on these sets. The approach is typically axiomatic: One assumes a small number of basic properties, or axioms, and attempts to deduce all other properties of the mathematical system from these. Such abstraction allows us to study, simultaneously, all structures satisfying a given set of axioms and to recognize both their commonalties and their differences. Specific topics to be covered include groups, actions, isomorphism, symmetry, permutations, rings, and fields.

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

##### Spring

The world is animated. Earth spins on its axis, as it rotates around the Sun; stock prices rise and fall; and an apple, acting solely in accordance with the laws of physics, falls onto the head of a modern day Newton. Calculus is the intriguing branch of mathematics whose primary goal is the understanding of the laws governing motion and change. The sum of the calculus—its methods, tools, and techniques—is often cited as one of the greatest intellectual achievements of humanity. 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 notions of differentiation and integration. Numerous applications of the theory will be examined. The pre-calculus topics of trigonometry and analytic geometry will be developed as the need arises. For conference work, students may choose to undertake a deeper investigation of a single topic or application of the calculus or conduct a study in some other branch of mathematics. This seminar is intended for students interested in advanced study in mathematics or science, students preparing for careers in the health sciences or engineering, and any student wishing to broaden and enrich the life of the mind.

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

##### Fall

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, 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 in some other branch of mathematics. This seminar is intended for students interested in advanced study in mathematics or science, 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.

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#### First-Year Studies: Mathematics in Context: Philosophy, Society, Culture, and Conflict

##### FYS

Mathematics has been an undeniably effective tool in humanity’s ongoing effort to understand the nature of the world around us, yet the mantra of high-school students is all too familiar: *What is math good for anyway?* *When am I ever going to use this stuff?* What serves to explain the puzzling incongruity between the indisputable success story of mathematics and students’ sense of the subject’s worthlessness? Part of the explanation resides in the observation that all too many mathematics courses are taught in a manner that entirely removes the subject matter from its proper historical, social, and cultural context—naturally leaving students with the distinct impression that mathematics is a dead subject, one utterly devoid of meaningfulness and beauty. In reality, mathematics is one of the oldest intellectual pursuits, its history a fascinating story filled with great drama, extraordinary individuals, and astounding achievements. This seminar focuses on the role played by mathematics in the emergence of civilization and follows their joint evolution over nearly 5,000 years to the 21st century. We will explore some of the great achievements of mathematics and examine the full story behind those glorious achievements. The ever-evolving role of mathematics in society and the ever-intertwined threads of mathematics, philosophy, religion, and culture provide the leitmotif of the course. Specific topics to be explored include the early history of mathematics, logic and the notion of proof, the production and consumption of data, the analysis of conflict and strategy, and the concept of infinity. Readings will be drawn from a wide variety of sources (textbooks, essays, articles, plays, and fictional writings), connecting us to the thoughts and philosophies of a diverse set of scholars; a partial list includes Pythagoras, Euclid, Galileo, René Descartes, Isaac Newton, Immanuel Kant, Lewis Carroll, John Von Neumann, John Nash, Kurt Gödel, Bertrand Russell, Jorge Luis Borges, Kenneth Arrow, and Tom Stoppard.

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#### Linear Algebra: The Mathematics of Matrices and Vector Spaces

##### Fall

An introduction to the algebra and geometry of vector spaces and matrices, this course stresses important mathematical concepts and tools used in advanced mathematics, computer science, physics, chemistry, and economics. Systematic methods of solving systems of linear equations is the underlying theme, and applications of the theory will be emphasized. Topics of exploration include Gaussian elimination, determinants, linear transformations, linear independence, bases, eigenvectors, and eigenvalues. Conference time will be allocated to clarifying course ideas and exploring additional applications of the theory. This seminar is intended for students interested in advanced mathematics, computer science, the physical sciences, or economics.

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#### Topology: The Nature of Shape and Space

##### Fall

Topology, a modernized version of geometry, is the study of the fundamental, underlying properties of shapes and spaces. In geometry, we ask: How big is it? How long is it? But in topology, we ask: Is it connected? Is it compact? Does it have holes? To a topologist there is no difference between a square and a circle and no difference between a coffee cup and a donut because, in each case, one can be transformed smoothly into the other without breaking or tearing the mathematical essence of the object. This course will serve as an introduction to this fascinating and important branch of mathematics. Conference work will be allocated to clarifying course ideas and exploring additional mathematical topics.

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### Activities

#### SLC Mathematical Resource Center

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

#### Current committees

#### Former committees

#### Professional affiliations

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

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

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

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