Mathematics

What Will Studying Mathematics Look Like in Fall 2020?

In line with the rest of the College, courses in Mathematics will be conducted online in the fall semester. Because courses will likely have a mixture of students who are living on-campus, commuting, and participating online, the class meetings will be conducted on line via MySLC, Zoom, or another platform. Where possible, individual conferences for students will be conducted in-person. Group conferences for lectures will be conducted online for small cohorts of remote students and in-person where possible for on-campus and commuting students. The topics of discussion covered in classes and conferences will remain the same, regardless of whether students are in person or online.

Here’s some specific information mathematics in fall 2020:

  • All Mathematics courses will follow the guidelines described by the College for online courses and in-person conferences where possible.
  • The Mathematics Resource Center will be equally available in both online and in-person formats to offer free tutoring for students that require help with mathematical concepts in any course. Peer tutors will be available on a daily basis. Students having difficulty with the mathematical skills that are necessary to complete successful seminar or conference work are invited to join an online meeting and have their questions answered. No appointment is necessary.

As of now, the Summer Science Program for 2021 is expected to proceed as normal. Information and applications regarding internship and externship opportunities for research in science and mathematics will be made available towards the end of the fall and beginning of the spring semester. Advising for our pre-professional programs (pre-health and Columbia combined Engineering plan) will continue to be provided to students in-person or online.

Further questions regarding any of the 2020-21 academic year plans should be directed to the appropriate faculty.


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.

2020-2021 Courses

Mathematics

An Introduction to Statistical Methods and Analysis

Open, Lecture—Fall

Prerequisites: basic high-school algebra and plane-coordinate geometry

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? 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 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. Given that this is a presidential election year, we will also be closely watching the national polls and discussing the difficulties of projecting future results with accuracy (and why pollsters got it wrong in 2016). Conference work, 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 graduate work and/or research in the natural sciences or social sciences.

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Game Theory: The Study of Conflict and Strategy

Open, Lecture—Spring

Prerequisite: one-year each of high-school algebra and geometry

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 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? Although the principles of game theory have been widely applied throughout the social and natural sciences, their greatest impact has been felt in the fields of economics, political science, 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.

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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. 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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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, 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 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. The theory of limits, differentiation, and integration will be briefly reviewed at the beginning of the term.

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Mathematics in Theory and Practice: Probability, Risk Analysis, and Optimization

Intermediate, Seminar—Year

Prerequisite: successful completion of two semesters of college-level calculus (or its equivalent)

What is chance, and how do we measure it? Do we measure the probability of winning the Mega Millions lottery in the same way that we assess the likelihood of a volcanic eruption in Hawaii? What tools are available to understand and measure uncertainty and risk? How can an understanding of probabilities better inform the decisions that we make in our personal and professional lives? How can we make the very best choice(s) amidst an enormous number of available options? How can individuals and businesses make critical decisions with confidence despite incomplete information and considerable uncertainty of future states? This calculus-based introduction to advanced probability theory, risk analysis, and operations research (optimization theory) engages these topics with an eye on diverse applications in the natural sciences, business, economics, and the social sciences. Topics of exploration will include the essential preliminaries of discrete mathematics (symbolic logic, proof technique, and set theory), combinatorial probabilities, distributions of prominent discrete and continuous random variables (Gaussian normal, binomial, Poisson, etc.), conditional probability and independence, joint distributions, expectation, variance, covariance, laws of large numbers, the Central Limit Theorem, Bayes Theorem, Markov chains, stochastic processes, linear programming and the powerful simplex method, sensitivity of optimized solutions to slight shifts in input parameters, duality theory, integer programming, nonlinear optimization, stochastic programming, and the four classic examples of optimization theory (the transportation/assignment problem, the network-flow problem, the diet problem, and the traveling-salesman problem). Using mathematical software, students will gain practical experience in the art of computer simulation and optimal solution identification.

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

Intermediate, Seminar—Year

This yearlong course will cover the central ideas of linear algebra, vector calculus, and differential equations from both a theoretical and a computational perspective. These three topics typically comprise the intermediate series of courses that students study after integral calculus but before more advanced topics in mathematics and the sciences. This course will be especially meaningful for students interested in pure or applied mathematics, the natural sciences, economics, and engineering but would also be a great choice for students who have completed the calculus sequence and are simply curious to see how deep the rabbit hole goes. While our focus will be primarily on the mathematics itself, the tools that we will develop are useful for modeling the natural world; and we will look at some of those applications. Conference work will revolve around pursuing the theory or application of those topics on a deeper level, according to students' personal interests.

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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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Introduction to Functional Programming

Open, Seminar—Spring

This course will explore the functional programming paradigm, a style of programming in which all computation is performed by applying functions to arguments. Our study of functional programming will make ample use of recursion—one of the most powerful concepts in computer science—and recursively-defined data structures such as lists, using the strangely beautiful programming language Scheme. Scheme and other functional languages provide powerful abstraction techniques for controlling the complexity of programs and a simple mathematical basis for reasoning about their correctness. Many of the most important concepts and features of modern programming languages were first introduced and studied within the framework of functional programming. Today, functional languages remain at the cutting edge of new programming-language design, and key ideas from functional programming continue to play a central role in the development of highly parallel, scalable, and distributed computing systems. Topics to be covered include recursion and its close cousin, mathematical induction; induction proofs and program correctness; mapping, filtering, and reducing lists; lambda expressions and higher-order functions; data abstraction and representation independence; infinite lists; continuations and continuation-passing style; and other topics as time permits. No prior knowledge of computer programming is necessary, though students should be comfortable with algebra and the fundamentals of mathematical thinking.

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Privacy, Technology, and the Law

Open, Seminar—Spring

What do remote-conferencing systems, digital currency, self-driving vehicles, and Edward Snowden have in common? The answer lies in this course, which focuses on how a few very specific computer technologies are dramatically altering daily life. In this course, we will develop a series of core principles that explain the rapid change and that help us chart a reasoned path to the future. We begin with a brief history of privacy, private property, and privacy law. Two examples of early 20th-century technologies that required legal thinking to evolve are whether a pilot (and passengers) of a plane are trespassing when the plane flies over someone’s backyard and whether the police can listen to a phone call from a phone booth (remember those?) without a warrant. Quickly, we will arrive in the age of information and update those conundrums: a drone flies by with an infrared camera, a copyrighted video is viewed on YouTube via public WiFi, a hateful comment is posted on reddit, a playful tweet is taken out of context and goes viral for all to see, an illicit transaction involving Bitcoin is made between seemingly anonymous parties via Venmo. To get a better handle on the problem, we will consider the central irony of the internet: It was developed at the height of the Cold War as a way to maintain a robust communication system in the event of a nuclear attack; now, its open nature puts us at risk of 21st-century security threats, such as electronic surveillance, aggregation and mining of personal information, and cyberterrorism. We will contrast doomsday myths popularized by movies such as War Games with more mundane scenarios such as total disruption of electronic commerce. Along the way, we will address questions such as: Does modern technology allow people to communicate secretly and anonymously? Can a few individuals disable the entire internet? Can hackers launch missiles or uncover blueprints for nuclear power plants from remote computers on the other side of the world? We will also investigate other computer security issues, including spam, computer viruses, and identity theft. Meanwhile, with our reliance on smart phones, text messages, and electronic mail, have we unwittingly signed up ourselves to live in an Orwellian society? Or can other technologies keep “1984” at bay? Our goal is to investigate if and how society can strike a balance so as to achieve computer security without substantially curtailing rights to free speech and privacy. Along the way, we will introduce the science of networks and describe the underlying theories that make the internet and its related technologies at once tremendously successful and so challenging to regulate. A substantial portion of the course will be devoted to introductory cryptology— the science (and art) of encoding and decoding information to enable private communication. We will conclude with a discussion of how cutting-edge technologies, such as blockchains, are impacting commerce today and how quantum cryptography and quantum computing may impact the privacy of communications tomorrow.

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Programming the Web: An Introduction

Open, Seminar—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 of 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!

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Intermediate Programming

Intermediate, Seminar—Fall

Permission of the instructor is required. Students should have at least one semester of programming experience.

This course is designed for students who understand the basics of computer programming (whether in Python, JavaScript, or another language) but want to take their skills to the next level. We will use the elegant and sophisticated programming language Haskell to learn about software design, abstract data types, and higher-order functions. We will introduce the basic principles of computational complexity and tree structures. We will emphasize top-down problem-solving, using recursion. We will also learn how to use cloud-based version control; e.g., using git and GitHub. Time permitting, we will learn how to build larger programs that leverage databases and networking protocols.

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Bio-Inspired Artificial Intelligence

Intermediate, Seminar—Spring

Students should have at least one semester of programming experience in a high-level, object-oriented language such as Python, Java, or C++.

The field of artificial intelligence (AI) is concerned with reproducing the abilities of human intelligence in computers. In recent years, exciting new approaches to AI have been developed, inspired by a wide variety of biological processes and structures that are capable of self-organization, adaptation, and learning. This course is a hands-on introduction to the algorithms and techniques of biologically-inspired AI and is intended for students with prior programming experience. Examples of these new approaches include evolutionary computation, artificial neural networks, autonomous robots, and swarm intelligence. We will focus, from both theoretical and practical perspectives, primarily on genetic algorithms, neural networks, deep learning, and reinforcement learning. We will use the Python programming language to implement and experiment with these techniques in detail and to test them out on both simulated and real robots. Students will have many opportunities for extended exploration through open-ended, hands-on lab exercises and conference work.

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First-Year Studies: Economics for Responsible Citizens

Open, FYS—Year

Today’s world is facing multiple economic, social, environmental, and political challenges: increasing income and wealth inequality, growing economic uncertainty, unstopping climate change, artificial intelligence and robotic automation of the workplace, political polarization, growing nationalism and populism, and diverging quality of life between the developed world and the majority of the developing world, to name a few. The challenging world requires every one of us to act as more responsible citizens. Using the economics literature, we will learn: Why do the challenges exist? What is our role in the community and in the bigger economic and political world? How can we transcend the “contemplation of single individuals and of civil society” and take the standpoint of “human society” or “social humanity”? In addition, through in-class practices, discussions, assignments, conference meetings, and conference work, we will work together to prepare you for academic achievement. You will enhance your academic skills, such as finding and reading academic literature, writing literature review, thinking critically, making your own argument with quantitative or qualitative evidence, and formatting a conference paper. Most importantly, you will grow professionally and prepare yourself to be a responsible citizen. During the fall semester, you will meet with me weekly for individual conferences. In the spring semester, we will meet weekly or every other week, depending on your needs and the progress of your conference projects.

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Studies in Ecocriticism: The Idea of Nature in the Western Tradition

Open, Small Lecture—Spring

As the capitalistic and predatory model aggressively promoted by the United States continues to reveal itself as a major threat for biodiversity and the environment in general, it is vital to explore and understand the concept of “nature” at the core of the Western tradition and how it was shaped over the course of more than 2,000 years. This course will create a series of bridges between and among the history of literature, philosophy, and science, with implications for many other disciplines. Most importantly, we will discuss the Western and Judeo-Christian concept of nature in the context of race and ethnicity in America today by confronting it with works and arguments developed by black, indigenous, Latine, and Asian American authors. Among many themes, we will study how antiquity came to develop a concept of “physis,” so different from our modern understanding of physics, but also shaped our aesthetic eye with the creation of the pastoral genre and the idea of agreeable and tamed landscapes or set a model for a utilitarian relationship to nature with Hesiod and Vergil’s agricultural treaties. We will also analyze specific places, such as the forest in medieval chivalric romances and American “wilderness” fictions, or chaotic landscapes admired and imagined by the Romantics, or the sea as depicted in Melville’s Moby Dick. The 17th-century scientific revolution and its mathematical and mechanistic approach to nature will lead us to discuss with Descartes the concept of animality in parallel with contemporary philosophers such as Deleuze and Guattari, who make use of models like the burrow or territoriality imported from the animal realm. Going into a completely different direction, we will question the characteristics of a Judeo-Christian conception of the world, organized around a remote and immaterial god, in direct opposition to a more organic understanding of nature as a “motherly” and immanent figure with all of the reservations that such a figure implies. These are some of the questions that we will explore, and the focus of our discussions will be to bring new voices in order to deconstruct the Eurocentric concept of “nature.”

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Chaos

Open, Seminar—Fall

This course introduces the beautiful world of nonlinear and chaotic dynamics and also provides the mathematical and numerical tools to explore the astounding patterns that can arise from these inherently unpredictable systems. We shall see how chaos emerges from fairly simple nonlinear dynamical systems; utilize numerical methods to simulate the dynamics of chaotic systems; and explore characteristics of chaos using iterated maps, bifurcation diagrams, phase space, Poincaré sections, Lyapunov exponents, and fractal dimensions. Class time will oscillate between the presentation of new material and workshops for hands-on exploration. Students are encouraged to build and/or analyze their own chaotic system as potential conference projects. No previous programming experience is required, and all relevant mathematical concepts will be introduced.

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Electromagnetism and Light (Calculus-Based General Physics)

Intermediate, Seminar—Spring

Students are encouraged to have completed Classical Mechanics, or equivalent, along with Calculus II, or equivalent.

This is the follow-on course to Classical Mechanics, where we will be covering 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. Seminars and weekly laboratory meetings will incorporate technology-based, exploratory, and problem-solving activities.

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