Mathematics

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.

Mathematics 2021-2022 Courses

An Introduction to Statistical Methods and Analysis

Open, Lecture—Spring | 5 credits | Hybrid Remote/In-Person

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.

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

Intermediate, Seminar—Year | 10 credits

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.

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Mathematics in Theory and Practice: Real Analysis and Topology

Intermediate, Seminar—Year | 10 credits

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.

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Calculus I: The Study of Motion and Change

Open, Seminar—Fall and Spring | 5 credits

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.

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Mathematics and Jorge Luis Borges

Open, Seminar—Fall | 5 credits

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.

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Pattern

Open, Seminar—Fall | 5 credits

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.

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Calculus II: Further Study of Motion and Change

Open, Seminar—Spring | 5 credits

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 science, for those preparing for careers in the health sciences or engineering, or for any simply wishing to broaden and enrich the life of the mind.

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Introduction to Computer Science: The Way of the Program

Open, Small Lecture—Fall

This lecture course is a rigorous introduction to computer science and the art of computer programming, using the elegant, eminently practical, yet easy-to-learn programming language Python. We will learn the principles of problem-solving with a computer while gaining the programming skills necessary for further study in the discipline. We will emphasize the power of abstraction and the benefits of clearly written, well-structured programs, beginning with imperative programming and working our way up to object-oriented concepts such as classes, methods, and inheritance. Along the way, we will explore the fundamental idea of an algorithm; how computers represent and manipulate numbers, text, and other data (such as images and sound) in binary; Boolean logic; conditional, iterative, and recursive programming; functional abstraction; file processing; and basic data structures such as lists and dictionaries. We will also learn introductory computer graphics, how to process simple user interactions via mouse and keyboard, and some principles of game design and implementation. All students will complete a final programming project of their own design. Weekly hands-on laboratory sessions will reinforce the concepts covered in class through extensive practice at the computer.

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Programming the Web: An Introduction

Open, Small Lecture—Spring

This seminar introduces the fundamental principles of computer science, via the use of HTML and JavaScript, to create interactive web pages. Examples of the kinds of web applications that we will build include: a virtual art gallery; a password generator and validator; and an old-school, arcade-style game. We will learn JavaScript programming from the ground up and demonstrate how it can be used as a general-purpose, problem-solving tool. Throughout the course, we will emphasize the power of abstraction and the benefits of clearly written, well-structured code. We will cover variables, conditionals, loops, functions, arrays, objects, and event handling. We will also discuss how JavaScript communicates with hypertext markup language (HTML) via the document object model (DOM) and the relationship between HTML, JavaScript, and cascading style sheets (CSS). Along the way, we will discuss the history of the web, the challenge of establishing standards, and the evolution of tools and techniques that drive the web’s success. We will learn about client-server architectures and the differences between client-side and server-side web programming. We will consider when it makes sense to design from the ground up and when it might be more prudent to make use of existing libraries and frameworks rather than reinventing the wheel. We will also discuss the aesthetics of web design: Why are some pages elegant (even art) when others are loud, awkward to use, or—worse yet—boring. Weekly hands-on laboratory sessions will reinforce the programming concepts covered in class. No prior experience with programming or Web design is necessary (nor expected nor even desirable).

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Principles of Programming Languages

Intermediate, Seminar—Fall

This course explores the principles of programming-language design through the study and implementation of computer programs called interpreters, which are programs that process other programs as input. A famous computer scientist once remarked that if you don't understand interpreters, you can still write programs—and you can even be a competent programmer—but you can’t be a master. We will begin by studying functional programming, using the strangely beautiful and recursive programming language Scheme. After getting comfortable with Scheme and recursion, we will develop an interpreter for a Scheme-like language of our own design, gradually expanding its power in a step-by-step fashion. Along the way, we will become acquainted with the lambda calculus (the basis of modern programming-language theory), scoping mechanisms, continuations, lazy evaluation, nondeterministic programming, and other topics if time permits. We will use Scheme as our "meta-language" for exploring those issues in a precise, analytical way—similar to the way in which mathematics is used to describe phenomena in the natural sciences. Our great advantage over mathematics, however, is that we can test our ideas about languages, expressed in the form of interpreters, by directly executing them on the computer.

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It’s About Time

Open, Small Lecture—Fall

This seminar will explore the topic of time from a wide variety of viewpoints—from the physical to the metaphysical to the practical. We will seek the answers to questions such as: What is time? How do we perceive time? Why does time appear to flow only in one direction? Is time travel possible? How is time relative? We will explore the perception of time across cultures and eras, construct an appreciation of the arrow of time by designing and building a Rube Goldberg machine, and discuss scientific articles and science-inspired works of fiction to make sense of this fascinating topic. Time stops for no one, but let’s take some time to appreciate its uniqueness.

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Classical Mechanics (Calculus-Based General Physics)

Open, Seminar—Fall

Calculus-based general physics is a standard course at most institutions; as such, this course will prepare you for more advanced work in the physical science, engineering, or health fields. This course will cover introductory classical mechanics, including kinematics, dynamics, momentum, energy, and gravity. Emphasis will be placed on scientific skills, including: problem-solving, development of physical intuition, scientific communication, use of technology, and development and execution of experiments. The best way to develop scientific skills is to practice the scientific process. We will focus on learning physics through discovering, testing, analyzing, and applying fundamental physics concepts in an interactive classroom, as well as in weekly laboratory meetings.

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Introduction to Mechanics (General Physics Without Calculus)

Open, Seminar—Fall

This course covers introductory classical mechanics, including dynamics, kinematics, momentum, energy, and gravity. Students considering careers in architecture or the health sciences, as well as those interested in physics for physics’ sake, should take either this course or Classical Mechanics. Emphasis will be placed on scientific skills, including problem-solving, development of physical intuition, scientific communication, use of technology, and development and execution of experiments. Seminars will incorporate discussion, exploratory activities, and problem-solving activities. In addition, the class will meet weekly to conduct laboratory work. A background in calculus is not required.

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Classical and Quantum Waves

Intermediate, Seminar—Fall

This course, which will provide an introduction to both classical and quantum waves, is a required prerequisite course for those interested in pursuing the Columbia Combined Plan program in applied mathematics, applied physics, biomedical engineering, electrical engineering, and materials science and engineering. Topics will include: classical waves and the wave equation, oscillations and normal modes, Fourier series and Fourier transforms; quantum waves and the Schrödinger equation; topics from quantum physics, including quantization of energy levels and reflection and transmission off barriers; and various applications of waves corresponding to student interests.

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Electromagnetism & Light (Calculus-Based General Physics)

Open, Seminar—Spring

Calculus-based general physics is a standard course at most institutions; as such, this course will prepare you for more advanced work in the physical science, engineering, or health fields. This course will cover waves, geometric and wave optics, electrostatics, magnetostatics, and electrodynamics. We will use the exploration of the particle and wave properties of light to bookend our discussions and ultimately finish our exploration of classical physics with the hints of its incompleteness. Emphasis will be placed on scientific skills, including: problem-solving, development of physical intuition, scientific communication, use of technology, and development and execution of experiments. The best way to develop scientific skills is to practice the scientific process. We will focus on learning physics through discovering, testing, analyzing, and applying fundamental physics concepts in an interactive classroom, as well as in weekly laboratory meetings.

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20th-Century Physics

Open, Seminar—Spring

This course will provide an overview of the pivotal developments in 20th-century physics that dramatically overturned the centuries-old scientific understanding of the fundamental laws of our universe. In this seminar-style class, we will discuss readings, walk through thought experiments, and unravel paradoxes to understand the concepts behind Einstein’s theories of special and general relativity, debate various interpretations of quantum mechanics, and explore the open questions that are motivating theoretical physics research in the 21st century.

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Theories of Agency and Action in Science Studies

Open, Lecture—Fall

This course surveys a rich historical debate in science, technology, and society studies on the nature of agency—or the motivation behind, and responsibility for, action. The lecture course begins with an exploration of the nature of scientific fact, including how discoveries are made and how they become accepted in society. We will pay special attention to the concepts of co-production, the idea that humans and technologies work together, and situated action, the reality that actions are rooted in social context, to study how technologies become central to social interaction. This grounding theory will lay a foundation for students to consider an ongoing debate on the distinction between human and nonhuman action. The course culminates with an exploration of three contemporary discussions on the nature of agency with respect to automated weapons systems, assistive technologies for people with disabilities, and the use of algorithms to order social life. For each topic, we will consider how technologies influence social interaction and who or what is responsible when things go wrong. In group conference, students will practice analyzing how technologies shape social interaction through a series of “object readings,” short analyses of a single technological object. These assignments are designed to prepare students for a final group analysis of a technology of their choice.

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Measuring Difference: Constructing Race, Gender, and Ability

Open, Seminar—Spring

In this seminar, we will explore the sociology of classification, a subfield that critiques the ways in which society measures differences like race, gender, ability, and other social categories that communicate social worth. Three questions guide our inquiry: How does society construct and understand categories of difference? How do people experience and resist categories of social difference in themselves? How does social difference shape institutions like the family, education, employment, and government? Each week, students will engage a selection of texts that put theory, substantive research on social categories, and critical responses to them in conversation with one another. For a final class project, students will explore one area of social difference through individual and group writing assignments. Those assignments will provide training in documentary analysis, a qualitative method often used in historical and ethnographic research. Students will leave the course with the ability to identify areas of social difference, the practices through which these are produced, and a systematic critique of the ways in which measurement creates inequality in the social world.

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