Mathematics

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Whether they had any interest in mathematics in high school, students often discover a new appreciation for the field at Sarah Lawrence College. In our courses—which reveal the inherent elegance of mathematics as a reflection of the world and how it works—abstract concepts literally come to life. That vitality further emerges as faculty members adapt course content to fit student needs, emphasizing the historical context and philosophical underpinnings behind ideas and theories.

By practicing rigorous logic, creative problem solving, and abstract thought in small seminar discussions, students cultivate habits of mind that they can apply to every interest. With well-developed, rational thinking and problem-solving skills, many students continue their studies in mathematics, computer science, philosophy, medicine, law, or business; others go into a range of careers in fields such as insurance, technology, defense, and industry.

2017-2018 Courses

Mathematics

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

An Introduction to Statistical Methods and Analysis

Open , Lecture—Fall

Mathematical prerequisite: basic high school algebra and geometry.

Correlation, regression, 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 important? And what exactly fueled the failure of statistical polls and projections leading up to the 2016 US presidential election? 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, 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. Conference work will serve as a complete practicum of the theory learned in lecture: Students working closely in small teams will conceive, design, and fully execute a small-scale research study. This lecture is recommended for anybody wishing to be a better-informed consumer of data and strongly recommended for those planning to pursue graduate work and/or research in the natural sciences or social sciences.

Faculty
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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.

Faculty
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Calculus I: The Study of Motion and Change

Open , Seminar—Fall

Prerequisites: successful completion of trigonometry and pre-calculus. Students concerned about meeting the course prerequisites are encouraged to contact the instructor as soon as possible.

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 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

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.

Faculty
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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.

Faculty
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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.

Faculty
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Abstract Algebra: Theory and Applications

Advanced , Seminar—Spring

Prerequisite: Calculus I and Discrete Mathematics or other evidence of successful preparation for advanced study in mathematics; permission of the instructor is required.

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
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