BA, PhD, Columbia University. Research interests in geometry, topology, and the intersection of mathematics with the arts. Mathematical consultant to New York-based artists since 2003. Currently writing a compendium of mathematical style to be published by Princeton University Press. SLC, 2014–

## Undergraduate Courses 2017-2018

### Mathematics

#### Discrete Mathematics: A Bridge to Advanced Mathematics

##### Intermediate , Seminar—Fall

*Some prior study of calculus is highly recommended.*

Your voice will produce a mostly continuous sound signal when you read this sentence out loud. As it appears on the page, however, the previous sentence is composed of 79 distinct characters, including letters and a punctuation mark. Measuring patterns—whether continuous or discrete—is the *raison d'être* of mathematics, and different branches of mathematics have developed to address the two sorts of patterns. Thus, a course in calculus treats motion and other continuously changing functions. In contrast, discrete mathematics addresses problems of counting, order, computation, and logic. We will explore these topics and their implications for mathematical philosophy and computer science. The form of this seminar will be that of a (mathematical) writing workshop. We will work collaboratively to identify and reproduce the key formal elements of mathematical exposition and proof as they appear in both mathematical literature and each other's writing. This seminar is designed for students interested in advanced mathematical study and highly recommended for students with an interest in computer science, law, logic, or philosophy.

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###### Related Disciplines

#### Mathematical Modeling II: Differential Equations and Linear Algebra

##### Intermediate , Seminar—Spring

*Prerequisite: Mathematical Modeling I or the equivalent (college-level course in multivariable calculus).*

At the center of many mathematical models, one often finds a differential equation. Newton’s laws of motion, the logistic model for population growth, and the Black-Scholes model in finance are all examples of models defined by a differential equation; that is, an equation in terms of an unknown function and its derivatives. Most differential equations are unsolvable; however, there is much to learn from the tractable examples, including first-order equations and second order linear equations. Since derivatives are themselves linear approximations, an important approach to differential equations involves the algebra of linear transformations, or linear algebra. Building on the study of vectors begun in Mathematical Modeling I, linear algebra will occupy a central role in the course, with topics that include linear independence, Gaussian elimination, eigenvectors, and eigenvalues. Students will gain experience developing mathematical models through conference work, which will culminate in an in-depth application of seminar ideas to a mathematical model in the natural, formal, or social sciences, based on student interest.

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###### Related Disciplines

#### Mathematical Modeling I: Multivariable Calculus

##### Intermediate , Seminar—Fall

*Prerequisite: successful completion of Calculus II or the equivalent (a score of 4 or 5 on the Calculus BC Advanced Placement exam).*

It is difficult to overstate the importance of mathematics for the sciences. Twentieth century polymath John von Neumann even declared that the “sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which…describes observed phenomena.” This two-semester sequence will introduce students to the basic mathematical ingredients that constitute models in the natural and social sciences. This first course in the sequence will concentrate on extending the concepts and tools developed in single-variable calculus to work with multiple variables. Multivariable calculus is a natural setting for studying physical phenomena in two or three spatial dimensions. We begin with the notion of a vector, a useful device that combines quantity and direction, and proceed to vector functions, their derivatives (gradient, divergence, and curl), and their integrals (line integrals, surface integrals, and volume integrals). The inverse relationship between derivative and integral appearing in single-variable calculus takes on new meaning and depth in the multivariable context, and a goal of the course is to articulate this through the theorems of Green, Gauss, and Stokes. These results will be of particular interest to students pursuing physics, engineering, or economics, where they are widely applicable. Students will gain experience developing mathematical models through conference work, which will culminate in an in-depth application of seminar ideas to a mathematical model in the natural, formal, or social sciences, based on student interest.

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###### Related Disciplines

#### The New Elements: Mathematics and the Arts

##### Open , Lecture—Spring

This lecture will explore the bearing of modern mathematical ideas on 20th-century Western creative and performing arts. Euclid’s collection of geometric propositions and proofs, entitled *The Elements,* is an archetype of logical reasoning that, since antiquity, has had a broad influence beyond mathematics. The non-Euclidean revolution in the 19th century initiated a radical reconception of not only geometry but also mathematics as a whole. We will investigate, on the one hand, mathematical content as a source of new forms of expression, including non-Euclidean geometry, the fourth dimension, set theory, functions, networks, topology, and probability. On the other hand, we will study mathematical practice and the artists and writers who, intentionally or not, reflect modern mathematical attitudes in an attempt to break with the past. While this lecture does not aim for a comprehensive survey of the entire last century, we will investigate a sequence of case studies, including: Russian Suprematist art; the Bauhaus school in Western European architecture and design; Serialism in Western music; OuLiPo, “a secret laboratory of literary structures” in post-war French literature; and the origins of postmodern dance in 1960-70s North America, among others. This course assumes no particular expertise with mathematics or cultural history. Course readings and a program of art and performance viewings, both in lecture and off campus, will establish a basis for investigating the relevance of fundamental mathematical concepts to modern literature and the arts. Group conferences will provide practice for students, working with such mathematical concepts as they relate to particular artistic practices.

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###### Related Disciplines

## Previous Courses

#### An Introduction to Real Analysis

##### Advanced , Seminar—Spring

*Prerequisite: Successful completion of a yearlong study of calculus; completion of an intermediate-level course (e.g., Discrete Mathematics, Complex Variables, Modeling I, etc.) is strongly recommended.*

The calculus of Newton and Leibniz is very different from the calculus of modern mathematics. It took more than a century to develop a logically defensible approach to its “fluxions” and “evanescent quantities.” In a sense, the formalization of calculus as the study of functions of a real variable—real analysis—that took place in the 19th century was so successful that it became a model of the foundational rigor that would come to define mathematics as a discipline. This maturation is recapitulated in the typical undergraduate student upon taking the step from the techniques-based calculus course to a proof-based real analysis course. Although our topics will sound familiar from calculus—real numbers, sequences and series, limits, continuity, the derivative, and the integral—their presentation will feature a new level of mathematical rigor. The emphasis on precise definitions and explicit proofs is not merely to develop students’ technical abilities, though that is a desired outcome of the course. Rather, these finer distinctions will be motivated by counterexamples, such as nowhere-differentiable continuous functions and rearrangements of infinite series, which challenge our basic intuitions about numbers and the real number line. Conference work will be allocated to clarifying seminar ideas and exploring additional mathematical topics.

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###### Related Disciplines

#### Mathematical Modeling II: Multivariable Calculus

##### Intermediate , Seminar—Spring

*Prerequisite: Completion of Mathematical Modeling I or the equivalent (college-level courses in linear algebra and differential equations).*

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###### Related Disciplines

#### Mathematical Modeling I: Linear Algebra and Differential Equations

##### Intermediate , Seminar—Fall

*Prerequisite: Successful completion of Calculus II or the equivalent (a score of 4 or 5 on the Calculus BC Advanced Placement exam).*

It is difficult to overstate the importance of mathematics for the sciences. Twentieth-century polymath John von Neumann even declared that “[t]he sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which […] describes observed phenomena.” This two-semester sequence will introduce students to the basic mathematical ingredients that constitute models in the natural and social sciences. At the center of such mathematical models, one often finds a differential equation. Newton’s laws of motion, the logistic model for population growth, and the Black-Scholes model in finance are all examples of models defined by a differential equation; that is, an equation in terms of an unknown function and its derivatives. Since derivatives are themselves linear approximations, an important approach to differential equations involves the algebra of linear transformations, or linear algebra. A subject in its own right, linear algebra will occupy a central role in the course, which seeks an integrated approach to these widely applicable subjects of pure and applied mathematics. Topics will include first- and second-order differential equations, vectors, matrices, determinants, linear independence, Gaussian elimination, eigenvectors, and eigenvalues. Students will gain experience developing mathematical models through conference work, which will culminate—based on student interest—in an in-depth application of seminar ideas to a mathematical model in the natural, formal, or social sciences.

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###### Related Disciplines

#### Topology

##### Advanced , Seminar—Spring

*Successful completion of a yearlong study of calculus is required; completion of an intermediate-level course (e.g., Discrete Mathematics, Linear Algebra, Multivariable Calculus, or Number Theory) is strongly recommended.*

First known as the geometry of position, topology is the study of spatial properties that do not depend on distance or angle measure. Such properties include order, continuity, configuration, boundary, and dimension. Gottfried Leibniz was probably the first mathematician to recognize the need for such methods when he wrote, “I am not content with algebra, in that it yields neither the shortest proofs nor the most beautiful constructions of geometry…we need yet another kind of analysis…which deals directly with position, as algebra deals with magnitude.” The unusual task of measuring space without distance, let alone coordinates, is achieved through the language of sets; and the seminar is primarily concerned with so-called point-set topology. Today, the field of topology is an active area of mathematics research, and we will discuss the questions that motivate its various branches, including geometric, algebraic, and differential topology. Conference work will be allocated to clarifying course ideas and exploring additional mathematical topics.

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###### Related Disciplines

#### Discrete Mathematics: A Gateway to Advanced Mathematics

##### Intermediate , Seminar—Fall

Your voice will produce a mostly continuous sound signal when you read this sentence out loud. As it appears on the page, however, the previous sentence is composed of 79 distinct characters, including letters and a punctuation mark. Measuring patterns—whether continuous or discrete—is the raison d'être of mathematics, and different branches of mathematics have developed to address the two sorts of patterns. Thus, a course in calculus treats motion and other continuous rates of change. In contrast, discrete mathematics addresses problems of counting, order, computation, and logic. We will explore these topics and their implications for mathematical philosophy and computer science. The form of this seminar will be that of a (mathematical) writing workshop. We will work collaboratively to identify and reproduce the key formal elements of mathematical exposition and proof as they appear in both the discrete mathematics literature and in each other’s writing. This seminar is designed for students interested in advanced mathematical study and highly recommended for students with an interest in computer science, law, logic, or philosophy.

###### Faculty

###### Related Disciplines

#### First-Year Studies: The New Elements: Mathematics and the Arts

##### Open , FYS

This seminar will explore the bearing of modern mathematical ideas on 20th-century creative and performing arts. Euclid’s collection of geometric propositions and proofs, entitled *The Elements,* is an archetype of logical reasoning that, since antiquity, has had a broad influence beyond mathematics. The non-Euclidean revolution in the 19th century initiated a radical re-conception of not only geometry but also mathematics as a whole. We will investigate, on the one hand, the role of “math as muse,” a source of new forms of expression that include, for example, non-Euclidean geometry, the fourth dimension, set theory, functions, networks, topology, and chance. On the other hand, we will study “math as maker” and the artists and writers who, intentionally or unintentionally, adopt modern mathematical attitudes in an attempt to break with the past. While the seminar will not aim for a comprehensive survey of the entire last century, we will investigate a sequence of case studies, including: Russian Suprematist art; the Bauhaus school in Western European architecture and design; Serialism in Western music; OuLiPo, “a secret laboratory of literary structures” in postwar French literature; and postmodern dance in 1960-70s North America, among others. This course assumes no particular expertise with mathematics or cultural history. Seminar readings, guest speakers, and a program of art and performance viewings will establish a basis for investigating the relevance of fundamental mathematical concepts to modern literature and the arts. One of the primary goals of the seminar is to assess the variety of ways that mathematics and the arts pose and address questions. Conference projects in the fall will focus on one of the elements of modern mathematics; in the spring, on an individual artist, composer, writer, or dancer whose work reflects a mathematical imagination.

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#### Abstract Algebra: Theory and Applications

##### Advanced , Seminar—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. Abstract 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, fields, and applications of these algebraic structures to questions outside of mathematics.

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#### Strange Universes: An Introduction to Non-Euclidean Geometries

##### Open , Seminar—Spring

If you draw two straight lines on a piece of paper, it’s not difficult to keep them from crossing. Imagine, however, that the lines extended in both directions off the page and without end. Do these hypothetical lines cross? Surprisingly, this mundane question goes to the heart of our modern conception of space. Your experience might suggest that the lines will cross unless they head off the edge of the page at exactly the same angle. In that case we call the lines parallel; and this is the answer Euclid asserts with his fifth (or “parallel”) postulate of the *Elements*. Roughly 2,000 years later, mathematicians came to the shocking realization that lines need not obey the parallel postulate. The resulting non-Euclidean geometries were so unexpected to the mathematicians who first conceived of them that one, János Bolyai, remarked, “Out of nothing I have created a strange new universe.” This course will explore the alternatives to Euclidean geometry that first appeared in the 18th century. These include hyperbolic, spherical, projective geometry, as well as more idiosyncratic geometries that we will devise together. Our exploration of these strange universes will be aided by visualizations that include drawing, computer graphics animation, and video game technology. Throughout, we will discuss the impact of the non-Euclidean revolution on astronomy, philosophy, and culture.

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#### Beyond Perspective: Mathematics and Visual Art

##### Open , Seminar—Fall

For many, the experience of doing mathematics is dominated by formulas, order, and following rules. It might come as a surprise that some mathematicians (especially in so-called “pure mathematics”) view what they do as more of an art than a science. For example, Georg Cantor, a leading mathematician of the early modern era, claimed that the “essence of mathematics lies entirely in its freedom.” This course will explore similarities between mathematical and contemporary art practices. We will study a variety of ways that mathematics and art pose questions. We will also investigate the intersection of the two disciplines, including selected applications of mathematics to art-making (from the Renaissance on) and the presence of modern mathematical attitudes in contemporary art (from the historical avant-garde through the present). This course assumes no particular expertise with mathematics, studio art, or art history. Seminar readings, guest speakers, and a program of art viewings will establish a basis for investigating the relevance of fundamental mathematical concepts to contemporary art. These concepts will include axiom, proof, structure, and symmetry, among others. Conference work will involve more in-depth study of individual artists, art works, mathematical ideas, or student work in mathematics and/or art.