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.

## Mathematics 2023-2024 Courses

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

Open, Lecture—Fall | 5 credits

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, natural 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” or behave? 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, psychology, and biology. This course represents a survey of the basic techniques and principles in the field. Of primary interest will be the applications of the theory to real-world conflicts of historical or current interest. Enrolled students are expected to have an understanding of basic high-school algebra and plane coordinate geometry.

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### An Introduction to Statistical Methods and Analysis

Open, Lecture—Spring | 5 credits

Variance, correlation coefficient, regression analysis, statistical significance, 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? How are they used? 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; 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 many 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. Enrolled students are expected to have an understanding of basic high-school algebra and plane coordinate geometry.

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### Modern Mathematics: Logic, Risk, Analytics, and Optimality

Open, Seminar—Year | 10 credits

There is great elegance in the fact that mathematics can be both deeply theoretical and magnificently useful. This course, available to both first-year students (as an FYS) and upper-class students (as an open seminar), explores the theory of optimization and its profound applications. We will study and employ elements in the toolbox of mathematics—specifically logic, probability, game theory, and operations research—for the purpose of optimization. In various and diverse settings, our goal will be to identify the single optimal choice amidst a sea of available options to determine the optimal decision despite a cloud of incomplete information and the mystery of an uncertain future and to select the optimal mode of behavior (strategy) in situations of personal or professional conflict. Specific applications of the mathematical theory will be explored through case-study analysis in business, biology, psychology, sociology, education, politics, law, literature, and art (among others). For example: How should SLC most effectively assign courses to students during Registration Week based on students’ indicated course preferences? How should United Airlines most efficiently route its planes to meet the transportation needs of its customers? How can Rubik’s Cubes be used in mass to most accurately reproduce Leonard da Vinci’s *Mona Lisa*? How can jointly-owned possessions be most fairly divided in an inheritance or divorce settlement? Specific topics of study in this calculus-based course will include topics in the foundation of mathematics (logic, proof technique, set and function theory); probability theory (combinatorics, discrete and continuous random variables, conditional probability, independence, expectation, Bayes Theorem); game theory (zero-sum conflicts, cooperative solutions); and analytics (linear programming, the simplex method, sensitivity analysis, duality theory, decision theory). Students pursuing this course should have prior working knowledge of single-variable differential and integral calculus (one year of high-school study or one semester of college study). Conference work can focus on any topic relating to mathematics. Students taking the course as an FYS seminar will meet with the instructor for individual conferences and donning weekly in fall and biweekly in spring. Enrolled upper-class students will conference with the instructor biweekly across both terms.

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

Intermediate, Seminar—Year | 10 credits

Prerequisite: Calculus II or its equivalent; a score of 4 or 5 on the Calculus BC Advanced Placement Exam

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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### Calculus II

Open, Seminar—Fall | 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, including artistic projects. This seminar is intended for students interested in advanced study in mathematics or science, preparing for careers in the health sciences or engineering, or simply wishing to broaden and enrich the life of the mind.

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### Calculus I

Open, Seminar—Fall | 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 changes that we see. 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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### Mirrors, Labyrinths, and Paradoxes: Mathematics and Jorge Luis Borges

Open, Seminar—Spring | 5 credits

Many of the works of Jorge Luis Borges—the highly influential, 20th-century Argentine writer and oft-cited founder of the magic realism literary genre—mirror mathematical concepts in profoundly intelligent and strikingly imaginative ways. Borges’ writings—primarily short fictions but also essays and poetry—often introduce alternate realities that warp standard notions of time, space, and even existence. Borges' works serve to uncover intriguing frictions between competing notions in the foundations of mathematics: the sensible vs. the paradoxical (logic), the infinite vs. the infinitesimal (set theory), the discrete vs. the continuous (analysis), the symmetric vs. the distorted (fractals and chaos), the convergent vs. the divergent (limits), and the likely vs. the impossible (probability). Not restricting itself to mathematics, this course will also explore themes and images in Borges’ works from philosophical, mythological, historical, scientific, psychological, and literary perspectives. Student conference work may focus upon other explorations at the intersection of literature, magic realism, mathematics, philosophy, etc. This course is intended for the student who is curious and open-minded though had never planned to study mathematics at the college level.

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### Calculus II

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, including artistic projects. This seminar is intended for students interested in advanced study in mathematics or science, preparing for careers in the health sciences or engineering, or simply wishing to broaden and enrich the life of the mind.

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### Related Chemistry Courses

#### General Chemistry II

Intermediate, Small Lecture—Spring

This course is a continuation of General Chemistry I. We will begin with a detailed study of both the physical and chemical properties of solutions. This will enable us to consider the factors that affect both the rates and direction of chemical reactions. We will then investigate the properties of acids and bases and the role that electricity plays in chemistry. The course will conclude with introductions to nuclear chemistry and organic chemistry. Weekly laboratory sessions will allow us to demonstrate and test the theories described in the lecture segment of the course.

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### Related Computer Science Courses

#### 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 also 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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#### Digital Disruptions

Open, Seminar—Fall

From autonomous vehicles to ChatGPT and Stable Diffusion, from the rise to the fall of cryptocurrency and NFTs, from YouTube to TikTok, from Instagram and Snapchat to BeReal, from Twitter to Mastodon, from Mr. Robot to M3gan, from Wordle to Elden Ring, from Apple to Zoom...digital technology plays an ever-more “disruptive” role in society. In this seminar, we ponder where this phenomenon may be taking us in the immediate and not-so-immediate future and whether there is (or will be) anything we can (or should) do about it. The miniaturization of electronic computers and the resulting increase in computing power, decrease in short-term cost to harness that power, and ubiquity of computer networks all bring people and places together and make distances formerly thought of as insurmountable ever more trivial. With the advent of gigabit fiber-optic networks, smart phones, and wearable computers, information of all kinds can flow around the world, between people and objects and back again, in an instant. In many ways, the plethora of smaller, cheaper, faster networked devices improves our quality of life. But there is also a dark side of a highly connected society: the more smart phones, the more workaholics; the more text messages exchanged and the easier the access to drones, the less privacy; the greater reach of the internet, the faster the spread of misinformation and the more piracy, spam, and pornography; the more remote-controlled thermostats, the greater the risk of cyberterrorism. This seminar will focus on the relationship between digital networks (the web, social networks, and beyond) to current events, including the economy, politics, and the law. The second half of the course will focus on the cultural impact of digital technology, ranging from video games to science fiction and the rise of artificial intelligence. This is not a technical course, though at times we will discuss some details that lie behind certain crucial technologies—in particular, the internet and the web.

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#### Intermediate Programming in Python

Intermediate, Seminar—Fall

This course is for students with prior programming experience in Python and who want to take their programming skills to the next level. We will explore a variety of advanced programming features of Python, including iterators and generators, list comprehensions, operator overloading, exception handling, context management, first-class functions, introspection and meta-programming, and other topics as time permits. We will also make extensive use of the object-oriented programming paradigm through the development of larger-scale programs organized as collections of classes, with an emphasis on clean, modular design.

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#### Data Structures and Algorithms

Intermediate, Seminar—Spring

In this course, we will study a variety of data structures and algorithms that are important for the design of sophisticated computer programs, along with techniques for managing program complexity. Throughout the course, we will use Java, a strongly typed, object-oriented programming language. Topics covered will include types and polymorphism, arrays, linked lists, stacks, queues, priority queues, heaps, dictionaries, balanced trees, and graphs, as well as several important algorithms for manipulating those structures. We will also study techniques for analyzing the efficiency of algorithms. The central theme tying all of these topics together is the idea of abstraction and the related notions of information hiding and encapsulation, which we will emphasize throughout the course. Weekly lab sessions will reinforce the concepts covered in class through extensive hands-on practice at the computer.

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#### Random and Prime

Advanced, Seminar—Spring

This course is a journey analogous to space exploration. Our infinite cosmos will be the set of natural numbers. Our exploratory rocket ships will be computer programs of our own design. The planets possibly bearing alien life forms are different classes of prime numbers. More literally, this course is a research-driven introduction to elementary number theory, its essential application to computer-network security, and its purported implications for the future of money (think of the buzzwords “crytpo,” “blockchain,” and “bitcoin”). We will write a series of computer programs of increasing sophistication, whose aim will be to identify patterns among prime numbers. We will pose philosophical questions regarding the nature of modern mathematics and computer science; for instance, to what extent can a computer be used to prove theorems? We will investigate what it means to be random: Can true randomness be generated by an algorithmic process? We will see examples of how some problems that appear to be very difficult may be solved quickly using random numbers, with the caveat that the answer we get is only “probably” true. In particular, we will contrast, on the one hand, the ease with which random numbers can be harnessed to discover primes and, on the other, the challenge of finding divisors of composite numbers. We will also consider the web-shaking implications if the latter problem turns out to be less difficult than it appears. Topics in elementary number theory include: primality, unique factorization, modular arithmetic, relative primality, Fermat's Little Theorem, primitive roots, and quadratic residues. Topics in cryptology include: Diffie-Hellman key exchange, RSA encryption, pseudorandom number generators, zero-knowledge proofs, and applications of these to blockchain databases and dreams of digital currency. Algorithmic topics include: modular exponentiation, probabilistic prime testing, factorization and discrete logarithms, and the theory of NP-completeness.

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### Related Economics Courses

#### Research Methods in Economics

Intermediate, Seminar—Fall

Evidence-based empirical research is an essential tool for an economist’s toolbox, allowing economists to better understand people’s behaviors; to discover underlying mechanisms of some major economic events or phenomena; and, most importantly, to critically examine many foundational economic theories. For instance: Standard economic theories tell us that raising the minimum wage will increase unemployment, but more and more empirical research has been showing us that such an effect is not supported by empirical evidence. Economic theories also tell us that tightening a country’s environmental policies will motivate the country’s businesses to outsource and relocate abroad and cause job loss, yet empirical research had failed to find clear evidence for that. This course will introduce you to the basics of conducting empirical economic research. Empirical research also has been used to support the making of public policies in areas such as health, education, urban and rural development, environment and climate change, food, etc. We will learn about formulating a research question; finding and critically evaluating relevant economics literature; developing a research proposal; finding and processing relevant economic data; analyzing data using appropriate quantitative techniques; clearly and meaningfully presenting, summarizing, and explaining the findings; writing a paper; and preparing a presentation. You will organize and complete a conference research project in stages.

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### Related Mathematics Courses

#### Mirrors, Labyrinths, and Paradoxes: Mathematics and Jorge Luis Borges

Open, Seminar—Spring

Many of the works of Jorge Luis Borges—the highly influential, 20th-century Argentine writer and oft-cited founder of the magic realism literary genre—mirror mathematical concepts in profoundly intelligent and strikingly imaginative ways. Borges’ writings—primarily short fictions but also essays and poetry—often introduce alternate realities that warp standard notions of time, space, and even existence. Borges' works serve to uncover intriguing frictions between competing notions in the foundations of mathematics: the sensible vs. the paradoxical (logic), the infinite vs. the infinitesimal (set theory), the discrete vs. the continuous (analysis), the symmetric vs. the distorted (fractals and chaos), the convergent vs. the divergent (limits), and the likely vs. the impossible (probability). Not restricting itself to mathematics, this course will also explore themes and images in Borges’ works from philosophical, mythological, historical, scientific, psychological, and literary perspectives. Student conference work may focus upon other explorations at the intersection of literature, magic realism, mathematics, philosophy, etc. This course is intended for the student who is curious and open-minded though had never planned to study mathematics at the college level.

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### Related Physics Courses

#### 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. The 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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#### Thermal Physics

Intermediate, Seminar—Fall

Some bears like their porridge very hot. Others like their porridge very cold. And then there are certain bears that like their porridge to have a temperature that is just right. What is temperature, anyway? In this course, we will not be cooking any porridge but will provide an introduction to thermal physics. Topics will include: thermodynamics (energy, temperature, work, heat, ideal gases); statistical mechanics (entropy, partition functions, distributions, chemical potential, non-ideal gases, bosonic gas, fermionic gas); and applications from physics, chemistry, and engineering (engines, refrigerators, Bose-Einstein condensates, maybe black holes). Previous experience with introductory physics (velocity, forces, energy) and chemistry is helpful but not required.

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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. The 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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#### Introduction to Electromagnetism, Light, and Modern Physics (General Physics Without Calculus)

Open, Seminar—Spring

This course covers waves and optics, electricity and magnetism, and overviews the discoveries made that transformed physics during the 20th century. 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, 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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### Related Writing Courses

#### Creative Nonfiction

Intermediate/Advanced, Seminar—Fall

This is a course for creative writers who are interested in exploring nonfiction as an art form. We will focus on reading and interpreting outside work—essays, articles, and journalism by some of our best writers—in order to understand what good nonfiction is and how it is created. During the first part of the semester, writing will be comprised mostly of exercises and short pieces aimed at putting into practice what is being illuminated in the readings; in the second half of the semester, students will create longer, formal essays to be presented in workshop.