Mathematics

When used well, mathematics is a powerful set of tools for understanding the world. When used in other ways, mathematics can serve to uphold and perpetuate inequality and injustice. In this course, we will investigate how mathematical tools can be used to understand, document, and work against inequity and injustice, including topics such as voting rights, health disparities, access to education, “big data” algorithms that control aspects of our lives, the carceral system, and environmental justice.

Our existence lies in a perpetual state of change. An apple falls from a tree, clouds move across expansive farmland, blocking out the sun for days; meanwhile, satellites zip around the Earth, transmitting and receiving signals to our cell phones. Calculus was invented to develop a language to accurately describe the motion and change happening all around us. The ancient Greeks began a detailed study of change, but they were scared to wrestle with the infinite; so it was not until the 17th century that Isaac Newton and Gottfried Leibniz, among others, tamed the infinite and gave birth to this extremely successful branch of mathematics. Though just a few hundred years old, calculus has become an indispensable research tool in both the natural and social sciences. Our study begins with the central concept of the limit and proceeds to explore the dual processes of differentiation and integration. Numerous applications of the theory will be examined. Weekly group conferences will be run in hands-on workshop mode. This course is intended for students interested in advanced study in mathematics or sciences, students preparing for careers in the health sciences or engineering, and any student wishing to broaden and enrich the life of the mind. 

Variance, correlation coefficient, regression analysis, statistical significance, and margin of error—these terms and other statistical phrases have been bantered about before and seen 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 course will focus 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 many other areas of the natural and social sciences. Statistical software will be introduced and used extensively in this course, but no prior experience with spreadsheet technology is assumed. Group conferences, conducted in workshop mode, will serve to reinforce student understanding of the course material. This course is recommended for any student 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. 

Calculus is the mathematical gift that keeps on giving—thank you, Newton and company! In this course, students will expand their knowledge of limits, derivatives, and integrals with concepts and techniques that will enable them to solve many important problems in mathematics and the sciences. By the end of the course, students will be able to judge whether answers provided by engine services such as WolframAlpha or ChatGPT are correct. Topics will include differentiation review, integration review, integration with non-polynomial functions, applications of integration (finding area, volume, length, center of mass, moment of inertia, probability), advanced techniques for integration (substitution, integration-by-parts, partial fractions), infinite sequences, infinite series, convergent and divergent sums, power series, differential equations and modeling dynamical systems, and, time permitting, parametric equations of a curve and polar coordinates. Students will work on a conference project related to the mathematical topics covered in class and are free to choose technical, historical, crafty, computational, or creative projects.

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 will introduce 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. Specific topics to be covered will 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.

This course will begin with an exploration of advanced mathematical foundations, including logic, set theory, methods of proof, and properties of real numbers and functions. Each of these topics will bridge both theoretical mathematical structures and applications to a broad range of real-world problems. We will then build on the methods and concepts of precollege algebra to analyze abstract systems that consist of mathematical objects (for example, numbers, functions, matrices, or permutations) and operations on them. By assuming a small number of basic properties—called axioms—of these systems, we will deduce other, more complex properties that can help us analyze a diverse number of abstract systems that, perhaps surprisingly, have common properties. Specific topics in abstract algebra will include groups, isomorphisms, symmetries, permutations, rings, and fields. Conference work may focus on any advanced topic relating to mathematics, including theoretical mathematical ideas or their applications to problems outside of mathematics.

This course will revitalize students' relationship with math, leading them to develop practical mathematical skills in contexts that are rewarding and meaningful both in and out of school. Students will strengthen their mathematical reasoning and problem-solving skills through important, real-world applications, including measurement, finances, critical consumption of statistics in the media, scientific thinking, and epidemiology. This course will give students the tools and the confidence to engage with mathematical concepts in other academic areas, leading students to discover the joy of engaging with the beautiful ideas of mathematics. Each group conference will address a special topic in mathematics based on students' interests. Topics might include math and democracy, mathematics in the arts, children’s understanding of math, and precalculus studies for students wishing to better prepare themselves for the study of calculus. No prior mathematics knowledge is required, as everyone can learn mathematics with understanding.

What do TikTok, Bitcoin, ChatGPT, self-driving vehicles, and Zoom have in common? The answer lies in this course, which focuses on how digital technologies have rapidly altered (and continue to alter) daily life. In this course, we will develop a series of core principles that attempt to explain the rapid change and forge a reasoned path to the future. We will 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 can 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 TikTok 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. A famous musician infuriates their fanbase by releasing a song supporting an authoritarian politician—but it turns out to be a deepfake. A core tension in the course is whether and how the internet should be regulated and how to strike a balance among privacy, security, and free speech. We will consider major US Supreme Court cases that chart slow-motion government reaction to the high-speed change of today's wired world. In fall, students will meet weekly with the instructor for individual conferences; in spring, individual conferences will be biweekly. 

This lecture will be 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.

This course will explore the concepts of emergence and complexity within natural and artificial systems. Simple computational rules interacting in complex, nonlinear ways can produce rich and unexpected patterns of behavior and may account for much of what we think of as beautiful or interesting in the world. Taking this as our theme, we will investigate a multitude of topics, including: fractals and the Mandelbrot set; chaos theory and strange attractors; cellular automata, such as the Wolfram rules and Conway's Game of Life; self-organizing and emergent systems; formal models of computation such as Turing machines; artificial neural networks; genetic algorithms; and artificial life. The central questions motivating our study will be: How does complexity arise in nature? Can complexity be quantified and objectively measured? Can we capture the patterns of nature as computational rules in a computer program? What is the essence of computation, and what are its limits? Throughout the course, we will emphasize mathematical concepts and computer experimentation rather than programming, using the computer as a laboratory in which to design and run simulations of complex systems and observe their behaviors.

This course will be an introduction to computer programming through the lens of old-school, arcade-style video games such as Pong, Adventure, Breakout, Pac-Man, Space Invaders, and Tetris. We will learn programming from the ground up and demonstrate how it can be used as a general-purpose, problem-solving tool. The course will emphasize the power of abstraction and the benefits of clearly written, well-structured code, covering topics such as variables, conditionals, iteration, functions, lists, and objects. We will focus on event-driven programming and interactive game loops. We will consider when it makes sense to build software from scratch and when it might be more prudent to make use of existing libraries and frameworks rather than reinventing the wheel. Some of the early history of video games and their lasting cultural importance will also be discussed. Students will design and implement their own low-res, but fun-to-play, games. No prior experience with programming or web design is necessary (nor expected, nor even desirable).

The field of artificial intelligence (AI) is concerned with reproducing in computers the abilities of human intelligence. In recent years, exciting new approaches to AI have been developed, inspired by a wide range of biological processes and structures that are capable of self-organization, adaptation, and learning. These sources of inspiration include biological evolution, neurophysiology, and animal behavior. This course is an in-depth introduction to the algorithms and methodologies of biologically-inspired AI and is intended for students with prior programming experience. We will focus primarily on machine-learning techniques—including genetic algorithms, reinforcement learning, artificial neural networks, and deep learning—from both a theoretical and a practical perspective. Throughout the course, we will use the Python programming language to implement and experiment with these algorithms in detail. Students will have many opportunities for extended exploration through open-ended, hands-on lab exercises and conference work.

Data are everywhere. And data contain plenty of valuable hidden information that is waiting to be uncovered. How can we use data properly to help inform policy decisions? In this research workshop, we will learn the essential skills and contemporary methods for conducting applied studies of economic, political, social, and policy issues using data. We will discuss how to properly formulate a research hypothesis, how to select and organize quantitative data, how to construct relevant variables, how to select empirical research methods, and how to present and communicate your research findings. The course will cover a range of contemporary applied research methods that emphasize causal inference, including panel data, fixed effects, difference-in-difference, matching, Regression Discontinuity Design, instrumental variables, and so on. We will start with finding correlations among variables of interest (e.g., How do X and Y relate to each other?), but will focus more on making causal inferences (e.g., Does X cause Y?). We will learn Stata, a relatively advanced statistical package used widely by the social science and science research communities. The ultimate goal of the course will be to help students write a successful applied conference project. But first, do no harm!

This course is designed for students interested in the social sciences who wish to understand the methodology and techniques involved in the estimation of structural relationships between variables (i.e., regression analysis). The course is intended for students who wish to be able to carry out empirical work in their particular field, both at Sarah Lawrence College and beyond, and critically engage with empirical work done by academic or professional social scientists. In fall, the course will cover the theoretical and applied statistical principles that underlie Ordinary Least Squares (OLS) regression techniques. The course will begin with a review of basic statistical and probability theory, as well as relevant mathematical techniques. We will then study the assumptions needed to obtain the Best Linear Unbiased Estimates (BLUE) conditions of a regression equation. Particular emphasis will be placed on the assumptions regarding the distribution of a model’s error term and other BLUE conditions. The course will cover hypothesis testing, sample selection, and the critical role of the t- and F-statistic in determining the statistical significance of an econometric model and its associated slope or “β” parameters. Further, we will address three main problems associated with the violation of a particular BLUE assumption: multicollinearity, serial correlation, and heteroscedasticity. We will learn how to identify, address, and remedy each of these problems. In addition, the course will take a similar approach to understanding and correcting model specification errors. In spring, the course will build on fall learning by introducing advanced econometrics topics. We will study difference-in-difference estimators, autoregressive dependent lag (ARDL) models, co-integration, and error correction models involving nonstationary time series. We will investigate simultaneous equations systems, vector error correction (VEC), and vector autoregressive (VAR) models. The final part of the course will involve the study of panel data, as well as logit and probit models. Students will receive ample exposure to concrete issues while also being encouraged to consider basic methodological questions, including the debates between John Maynard Keynes and Jan Tinbergen regarding the power and limitations of econometric analysis. Spring is particularly relevant to students who wish to pursue graduate studies in a social-science discipline but equally relevant for other types of graduate degrees that involve knowledge of intermediate-level quantitative analysis. The practical "hands-on" approach taken in this course will be useful to those students who wish to do future conference projects, internships, or enter the job market in the social (or natural) sciences with significant empirical content. The goal is for students to be able to analyze questions such as: What is the relationship between slavery and industrialization in the United States? What effects do race, gender, and educational attainment have in the determination of wages? How does the female literacy rate affect the child mortality rate? How can one model the effect of economic growth on carbon-dioxide emissions? What is the relationship among sociopolitical instability, inequality, and economic growth? How do geographic location and state spending affect average public-school teacher salaries? How does one study global inequalities in terms of access to COVID-19 vaccines?

Calculus is the mathematical gift that keeps on giving—thank you, Newton and company! In this course, students will expand their knowledge of limits, derivatives, and integrals with concepts and techniques that will enable them to solve many important problems in mathematics and the sciences. By the end of the course, students will be able to judge whether answers provided by engine services such as WolframAlpha or ChatGPT are correct. Topics will include differentiation review, integration review, integration with non-polynomial functions, applications of integration (finding area, volume, length, center of mass, moment of inertia, probability), advanced techniques for integration (substitution, integration-by-parts, partial fractions), infinite sequences, infinite series, convergent and divergent sums, power series, differential equations and modeling dynamical systems, and, time permitting, parametric equations of a curve and polar coordinates. Students will work on a conference project related to the mathematical topics covered in class and are free to choose technical, historical, crafty, computational, or creative projects.

The magnum opus, Ethics, of great early modern Jewish philosopher Baruch Spinoza (1633-1672) will serve as the focus of this course. German philosopher Friedrich Heinrich Jacobi once wrote that “Spinoza is the only philosopher who had the courage to take philosophy seriously; if we want to be philosophers, we can only be Spinozists.” Even if Jacobi’s statement is exaggerated, it is certainly true that studying Spinoza will make us better philosophers. But Spinoza promises much more. He claims that those who follow the guide of his Ethics become freer, wiser, and, above all, happier. Ethics is a notoriously difficult and enigmatic text, written in the form of geometrical proofs, even concerning psychological, moral, and theological matters. Yet, many philosophers and poets considered it exceptionally beautiful. Among the questions the book tackles are: What determines our desires, and in what ways can we, or should we, control them? In what ways can we be free, and in what ways are our behaviors and desires predetermined? In what ways can we be unique, and in what ways are we an inherent part of a greater whole? As we will learn, Spinoza argued that God and Nature are synonyms and that, to achieve an eternal and blissful life, we do not need to die and go to heaven. We do not even need to change the world or ourselves. All we need is to understand the way things are.

Our everyday experiences with the world around us give us an intuitive knowledge of some of the principles of physics; however, many areas of contemporary physics study the unseen—literally! This course will guide students through the core principles needed to understand modern physics and to think like a physicist. As we develop our knowledge of physics, we will study puzzles, thought experiments, and toy models of the real world to uncover the nature of our universe. Unlike traditional introductory physics courses, we will start with the modern formulations of classical mechanics, which lay the groundwork for how physical theories, including quantum mechanics, have been developed over approximately the last 100 years. We will also see how forces, such as the electromagnetic force and gravity, can be understood as field theories acting everywhere in space. As we develop our physics toolbox, we will focus on building a deep and intuitive understanding of the material, including the fundamental mathematics needed to study physics. This course will be mathematically rigorous; and while prior exposure to calculus will be helpful, a deep interest in mathematical reasoning will be essential. This seminar will focus on understanding the real-world physics at play. Work in this course will largely consist of problem sets designed to develop thinking and showcase progress over the course of the year. Biweekly in fall, students will have individual conferences with the instructor to explore a physics topic while developing skills to read and analyze research articles. In alternate weeks, students will meet for group conferences as problem-solving sessions. Occasionally, we will conduct a lab during group conference so students can experience the physics that they are studying. Biweekly in spring, students will meet with the instructor for individual conferences.

General physics is a standard course at most institutions; as such, this course will prepare students for more advanced work in physical science, engineering, or the health fields. Lectures will be accessible at all levels; and through group conference, students will have the option of either taking an algebra-based or calculus-based course. 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, through problem-solving, as well as in weekly lab meetings.

Explore the beautiful mathematics and physics of waves through both theory and experiment. This course will teach students valuable mathematical methods and basic computational skills that are necessary for more advanced physical-science classes. Lab class time will include using advanced lab equipment, analyzing data using Jupyter (IPython) notebooks, learning numerical techniques, and reporting the results using LaTeX. For conference work, students are encouraged to choose an American Journal of Physics article to replicate, analyze, and then present their findings at the semi-annual Sarah Lawrence College Science & Mathematics Poster Session.

This course 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, break down the role of time in fundamental physics, and discuss popular science books and articles, along with science-inspired works of fiction, to make sense of this fascinating topic. Time stops for no one, but let us take some time to appreciate its uniqueness.

General physics is a standard course at most institutions; as such, this course will prepare students for more advanced work in physical science, engineering, or the health fields. Lectures will be accessible at all levels; and through group conference, students will have the option of either taking an algebra-based or calculus-based course. 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 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, through problem-solving, as well as in weekly lab meetings.

We encounter temperature on a daily basis when we check our weather apps and have undoubtedly heard discussions about the greenhouse effect and Earth’s warming climate. But what do scientists mean by warming? How can they model it? And what even is temperature? In this course, we will dig into the fascinating world of thermal physics, which is important for delving into many more advanced topics in physics, geosciences, or chemistry. Topics will include: thermodynamics, including energy, temperature, work, heat, and ideal gases; statistical mechanics, including entropy, partition functions, distributions, chemical potential, nonideal gases, bosonic gas, and fermionic gas; and applications from physics, chemistry, and engineering, such as engines, refrigerators, Bose-Einstein condensates, black holes, and climate models. For conference work, students will be encouraged to model a simple thermal system of their choice, using the mathematical and numerical methods developed throughout the course.