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 2021-2022

### Mathematics

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

##### Open, Seminar—Fall and Spring

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 change that we see. 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 topics 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 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.

###### Faculty

#### Mathematics in Theory and Practice: Real Analysis and Topology

##### Intermediate, Seminar—Year

The calculus of Newton and Leibniz was so successful that science forgave the logical shortcomings of its “fluxions” and “evanescent quantities.” In the 19th century, however, calculus evolved into the study of functions of a real variable—real analysis—which is a model of the foundational rigor that has come to define mathematics as a discipline. In the 20th century, the search for axioms of the real numbers uncovered subtle assumptions about spatial properties of the real line. These properties—such as continuity, separability, and dimension—do not depend on magnitude but on more general notions of position. The geometry of position, or topology as it is called today, is the study of exactly such properties. This yearlong seminar will begin with preliminaries of discrete mathematics, including symbolic logic, proof technique, and set theory. We will study these topics in the context of networks and surfaces, which are some of the most intuitive topological objects. This will be followed by an in-depth study of the real numbers, sequences and series, limits, continuity, the derivative, and the integral. To motivate our revision of these familiar calculus terms, the seminar will read and discuss important counterexamples, such as nowhere-differentiable continuous functions, rearrangements of infinite series, and the Cantor set. At the end of the year, we will return to topology. This will give us the opportunity to see how many of the geometric properties of curves, surfaces, and maps between them find a unified expression in terms of relations among point sets. Conference work will clarify seminar ideas and possibly their application to mathematical models in the natural sciences, computer science, or economics.

###### Faculty

#### Pattern

##### Open, Seminar—Fall

This seminar will study patterns in nature and design from the mathematical point of view. Examples will be primarily visual, including beadwork, braids, tilings, trees, waves, and crystals, among others. The workshop format of the class will give students the opportunity to discover, collaboratively, the structures that govern patterns. Students can expect to use both visual and logical reasoning to answer open-ended problems that involve hands-on experimentation and creative problem-solving. By the end of the semester, students will know how to reproduce a given pattern in one, two, or three dimensions; how to identify its symmetries; and how to compare it to related structures. For conference, there is a possibility of service-learning placements in community-based organizations, depending on availability. No particular math background is required. This course is recommended for any students interested in mathematics as the science of patterns and strongly recommended for those studying education.

###### Faculty

## Previous Courses

### Mathematics

#### 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.

###### Faculty

#### An Introduction to Real Analysis

##### Advanced, Seminar—Spring

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.

###### 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. 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, 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 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 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: A Bridge 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 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. Some prior study of calculus is highly recommended.

###### Faculty

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

##### Open, FYS—Year

The development of linear perspective in Renaissance painting presents one of the clearest examples of the intersection of mathematics and the arts. To paraphrase art historian Erwin Panofsky, perspective recasts perceptual space as a uniform, infinite, abstract space with its own logical and aesthetic properties. The mathematics needed in perspectival constructions was worked out by Euclid in antiquity. What novel aesthetic and logical forms are made possible by the mathematics beyond Euclid’s *Elements*? This seminar will explore the bearing of modern mathematical ideas on 20th-century Western creative and performing arts. While we will not aim for a comprehensive survey of the entire last century, we will investigate a sequence of case studies, including: De Stijl and the painting of Piet Mondrian; serialism and the music of Arnold Schoenberg; the Bauhaus in Germany and its legacy; OuLiPo, “a secret laboratory of literary structures” in postwar French literature; American postmodern dance; and structural film, among others. Mathematical topics will include sets, logic, non-Euclidean geometry, topology, and chance. A central goal of the seminar is to assess the meaning of structure as it pertains to artistic and mathematical practices. This course assumes no particular expertise with mathematics or cultural history. Seminar readings and a program of art viewings will establish a basis for investigating the relevance of fundamental mathematical concepts to modern literature and the arts. Outside the seminar, students will attend both individual and group conferences. Weekly individual conference meetings for the first six weeks of the fall semester will give students the opportunity to develop their first individualized conference projects, focusing on a particular mathematical structure. Individual conferences after the first six weeks will be held on a weekly or biweekly basis, depending on student progress. During the fall semester, a series of group conferences will afford students time for art viewings and collaborative writing and problem solving.

###### Faculty

#### Introduction to Real Analysis

##### Advanced, Seminar—Spring

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 the “fluxions” and “evanescent quantities” of calculus. 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 to 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.

###### Faculty

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

##### Intermediate, Seminar—Fall

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.

###### Faculty

#### Mathematical Modeling I: Multivariable Calculus

##### Intermediate, Seminar—Fall

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.

###### Faculty

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

##### Intermediate, Seminar—Spring

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.

###### Faculty

#### Mathematical Modeling II: Multivariable Calculus

##### Intermediate, Seminar—Spring

###### Faculty

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

##### Open, Seminar—Fall

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 extend 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 19th century. These include hyperbolic, spherical, and 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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#### 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.