Mathematics

First-Year Studies—Year

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. 

Faculty

• Michael Siff

Open, Small Lecture—Fall

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.

Faculty

• James Marshall

Open, Small Lecture—Spring

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.

Faculty

• James Marshall

Intermediate, Seminar—Spring

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.

Faculty

• James Marshall

Intermediate/Advanced, Seminar—Fall

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!

Faculty

• An Li

Open, Small Lecture—Year

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?

Faculty

• Jamee Moudud

Open, Seminar—Fall and Spring

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.

Faculty

• Melvin Irizarry-Gelpi

Open, Small Lecture—Spring

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.

Faculty

• Roy Ben-Shai

First-Year Studies—Year

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.

Faculty

• Sarah Racz

Open, Small Lecture—Fall

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.

Faculty

• Merideth Frey
• Sarah Racz

Intermediate, Seminar—Fall

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 classtime 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, then present their findings at the semiannual Sarah Lawrence College Science & Mathematics Poster Session.

Faculty

• Merideth Frey
• Sarah Racz

Open, Seminar—Year

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.

Faculty

• Sarah Racz

Open, Small Lecture—Spring

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.

Faculty

• Merideth Frey

Intermediate, Small Lecture—Spring

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 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.

Faculty

• Merideth Frey
• Sarah Racz

Advanced, Seminar—Spring

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.

Faculty

• Merideth Frey