Mathematics

Whether they had any interest in mathematics in high school, students often discover a new appreciation for the field at Sarah Lawrence College. In our courses—which reveal the inherent elegance of mathematics as a reflection of the world and how it works—abstract concepts literally come to life. That vitality further emerges as faculty members adapt course content to fit student needs, emphasizing the historical context and philosophical underpinnings behind ideas and theories.

By practicing rigorous logic, creative problem solving, and abstract thought in small seminar discussions, students cultivate habits of mind that they can apply to every interest. With well-developed, rational thinking and problem-solving skills, many students continue their studies in mathematics, computer science, philosophy, medicine, law, or business; others go into a range of careers in fields such as insurance, technology, defense, and industry.

Mathematics 2022-2023 Courses

First-Year Studies: Pattern

FYS—Year | 10 credits

This seminar will study patterns in nature and design from the mathematical point of view. Examples will be primarily visual, including beadwork, braids, tilings, trees, waves, and crystals, among others. The workshop format of the class will give students the opportunity to discover, collaboratively, the structures that govern patterns. Students can expect to use both visual and logical reasoning to answer open-ended problems that involve hands-on experimentation and creative problem solving. By the end of the year, students will know how to reproduce a given pattern in one, two, or three dimensions; how to identify its symmetries; and how to compare it to related structures. For conference, there is a possibility of service-learning placements in community-based organizations, depending on availability. During the fall semester, students will meet with the instructor weekly for individual conferences. In the spring, we will meet weekly or every other week, depending on students’ needs and the progress of their conference projects. This course is recommended for any student interested in mathematics as the science of patterns, as well as those intending to study visual art or education. No particular math background is required.

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An Introduction to Statistical Methods and Analysis

Open, Lecture—Fall | 5 credits | Remote

Prerequisite: basic high-school algebra and plane coordinate geometry

Variance, correlation coefficient, regression analysis, statistical significance, and margin of error—you’ve heard these terms and other statistical phrases bantered about before, and you’ve seen them interspersed in news reports and research articles. But what do they mean? And why are they so important? Serving as an introduction to the concepts, techniques, and reasoning central to the understanding of data, this lecture course focuses on the fundamental methods of statistical analysis used to gain insight into diverse areas of human interest. The use, misuse, and abuse of statistics will be the central focus of the course; specific topics of exploration will be drawn from experimental design theory, sampling theory, data analysis, and statistical inference. Applications will be considered in current events, business, psychology, politics, medicine, and other areas of the natural and social sciences. Statistical (spreadsheet) software will be introduced and used extensively in this course, but no prior experience with the technology is assumed. Group conferences, conducted in workshop mode, will serve to reinforce student understanding of the course material. This lecture is recommended for anybody wishing to be a better-informed consumer of data and strongly recommended for those planning to pursue advanced undergraduate or graduate research in the natural sciences or social sciences.

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Symmetry of Ornament

Open, Small Lecture—Spring | 5 credits

This lecture will present a formal analysis of ornament, using the mathematical principles of symmetry. Symmetric designs appear in material cultures from around the world and throughout history, from Bronze Age ceramics, 15th-century Islamic tiling, Latin American textiles, and Fijian bark cloth to the Arts and Crafts movement. Symmetry is a correspondence among the parts of a figure or object. Such a correspondence is often described in terms of an operation (“isometry,” in mathematical terms); for example, we will show that the symmetries of designs that repeat in one or two directions are comprised of just four types of operations: translation, rotation, reflection, and glide reflection. The collection of all possible symmetries of a figure comprises its “symmetry group,” and we will use this to classify finite and infinite ornamental designs. Many of the cultural artifacts that we study predate the mathematical theory of groups; in this sense, the lecture introduces the prehistory of modern mathematics. Museum visits and group conferences will offer students direct experience analyzing examples of visual structures in decorative art and design.

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

Intermediate, Seminar—Year | 10 credits

Prerequisite: successful completion of 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 I: The Study of Motion and Change

Open, Seminar—Fall | 5 credits

Prerequisite: successful completion of study in trigonometry and pre-calculus topics

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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Proof and Paradox

Intermediate, Seminar—Fall | 5 credits

Prerequisite: one year of college-level mathematics or the equivalent

One of the remarkable ironies of modern mathematics is that the success of its methodology has exposed its own limitations. In particular, the advances in mathematical foundations and logic of the early 20th century precipitated Kurt Gödel's incompleteness theorems—which establish that, for any effective axiomatic system of mathematics, there are mathematical truths that mathematics cannot prove. Gödel's proof is remarkable for both its philosophical implications and its very ingenuity. To prepare our study of the proof, the seminar will review basic logic, set theory, elementary number theory and the standard techniques of mathematical proof. Having completed a close reading of Gödel's proof, we will then explore the relationship between proof and understanding in more recent mathematical practice. Students will have an opportunity to strengthen their mathematical reading and writing abilities while engaging contemporary mathematical issues concerning the progress of the discipline, the role of computers in proof, and best practices in mathematical exposition. This seminar is recommended not only for the mathematically inclined but also for students interested in computer science, law, or philosophy.

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Calculus II: Further Study of Motion and Change

Open, Seminar—Spring | 5 credits

Prerequisite: The minimum required preparation for the study of Calculus II is successful completion of study in Calculus I

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.

Abstract Algebra

Intermediate/Advanced, Seminar—Spring | 5 credits

Prerequisite: completion of Discrete Mathematics or another proof-based course

In pre-college mathematics courses, we learned the basic methodology and notions of algebra. We appointed letters of the alphabet to abstractly represent unknown or unspecified quantities. We discovered how to translate real-world (and often complicated) problems into simple equations whose solutions, if they could be found, held the key to greater understanding. But algebra does not end there. Advanced algebra examines sets of objects (numbers, matrices, polynomials, functions, ideas) and operations on those sets. The approach is typically axiomatic: One assumes a small number of basic properties, or axioms, and attempts to deduce all other properties of the mathematical system from those. Such abstraction allows us to study, simultaneously, all structures satisfying a given set of axioms and to recognize both their commonalties and their differences. Specific topics to be covered include groups, actions, isomorphism, symmetry, permutations, rings, fields, and applications of these algebraic structures to questions outside of mathematics. The pace and level of discussion is aimed at students who have experience reading and writing proofs.

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First-Year Studies: Achilles, the Tortoise, and the Mystery of the Undecidable

FYS—Year

In this course, we will take an extended journey through Douglas Hofstadter’s Pulitzer Prize-winning book, Gödel, Escher, Bach, which has been called “an entire humanistic education between the covers of a single book.” The key question at the heart of the book is: How can minds possibly arise from mere matter? Few people would claim that individual neurons in a brain are “conscious” in anything like the normal sense in which we experience consciousness. Yet, consciousness and self-awareness emerge, somehow, out of a myriad of neuronal firings and molecular interactions. How can individually meaningless physical events in a brain, even vast numbers of them, give rise to meaningful awareness, to a sense of self? And could we duplicate such a process in a machine? Considering these questions will lead us to explore a wide range of ideas—from the foundations of mathematics and computer science to molecular biology, art, and music and to the research frontiers of modern-day cognitive science and neuroscience. Along the way, we will closely examine Gödel's incompleteness theorem, the drawings of M. C. Escher, the music of J. S. Bach, mathematical logic and formal systems, the limits of computation, and the past history and future prospects of artificial intelligence. During the fall semester, students will meet with the instructor weekly for individual conferences. In the spring, we will meet weekly or every other week, depending on students’ needs.

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Introduction to Computer Programming

Open, Small Lecture—Fall

This lecture presents 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, the theory of algorithms, and the benefits of clearly written, well-structured programs. Fundamental topics include: how computers represent and manipulate numbers, text, and other data (such as images and sound); variables and symbolic abstraction; Boolean logic; conditional, iterative, and recursive computation; functional abstraction (“black boxes”); and standard data structures, such as arrays, lists, and dictionaries. We will learn introductory computer graphics and how to process simple user interactions via mouse and keyboard. We will also consider the role of randomness in otherwise deterministic computation, basic sorting and searching algorithms, how programs can communicate across networks, and some principles of game design. Toward the end of the semester, we will investigate somewhat larger programming projects and, so, will discuss file processing; modules and data abstraction; and object-oriented concepts such as classes, methods, and inheritance. As we proceed, we will debate the relative merits of writing programs from scratch versus leveraging existing libraries of code. Discussion topics will also include the distinction between decidable and tractable problems, the relationship between programming and artificial intelligence, the importance of algorithmic efficiency to computer security, and Moore’s Law and its impact on the evolution on programming languages and programming style. Weekly hands-on laboratory sessions will reinforce the programming concepts covered in class.

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Computer Organization

Intermediate, Seminar—Year

This course investigates how computers are designed “underneath the hood” and how basic building blocks can be combined to make powerful machines that execute intricate algorithms. There are two essential categories of components in modern computers: the hardware (the physical medium of computation) and the software (the instructions executed by the computer). As technology becomes more complex, the distinction between hardware and software blurs. We will study why this happens, as well as why hardware designers need to be concerned with the way software designers write programs and vice versa. Along the way, we will learn how computers work from higher-level programming languages such as Python and JavaScript, to system-level languages C and Java, down to the basic zeroes and ones of machine code. Topics include Boolean logic, digital-circuit design, computer arithmetic, assembly and machine languages, memory hierarchies, and parallel processing. Special attention will be given to the RISC architectures—now the world’s most common, general-purpose microprocessors. In particular, we will focus on the ARM architecture and Apple’s new M1 processors. Time permitting, we will investigate the relationship between energy consumption and the rise of multicore and mobile architectures.

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Quantum Computing

Intermediate, Seminar—Fall

Physicists and philosophers have been trying to understand the strangeness of the subatomic world as revealed by quantum theory since its inception back in the 1920s, but it wasn’t until the 1980s—more than a half-century after the development of the theory—that computer scientists first began to suspect that quantum physics might hold profound implications for computing, as well, and that its inherent weirdness might possibly be transformed into a source of immense computational power. This dawning realization was followed soon afterward by key theoretical and practical advances, including the discovery of several important algorithms for quantum computers that could potentially revolutionize (and disrupt) the cryptographic systems protecting practically all of our society’s electronic banking, commerce, telecommunications, and national security systems. Around the same time, researchers succeeded in building the first working quantum computers, albeit on a very small scale. Today, the multidisciplinary field of quantum computing lies at the intersection of computer science, mathematics, physics, and engineering and is one of the most active and fascinating areas in science, with potentially far-reaching consequences for the future. This course will introduce students to the theory and applications of quantum computing from the perspective of computer science. Topics to be covered will include bits and qubits, quantum logic gates and reversible computing, Deutsch’s algorithm, Grover’s search algorithm, Shor’s factoring algorithm, quantum teleportation, and applications to cryptography. No advanced background in physics, mathematics, or computer programming is necessary beyond a basic familiarity with linear algebra. We will study the quantitative, mathematical theory of quantum computing in detail but will also consider broader philosophical questions about the nature of physical reality, as well as the future of computing technologies.

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Privacy, Technology, and the Law

Open, Seminar—Spring

What do Bitcoin, NFTs, Zoom, 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 will 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 required legal thinking to evolve: 1) whether a pilot (and passengers) of a plane is trespassing when the plane flies over someone’s backyard; and 2) whether the police can listen to a phone call from a phone booth (remember those?) without a warrant. Quickly, we will arrive at the age of information and will be able to update these 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. 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, and 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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Bio-Inspired Artificial Intelligence

Intermediate, Seminar—Spring

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 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—such as evolutionary computation and 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 techniques in detail. Students will have many opportunities for extended exploration through open-ended, hands-on lab exercises and conference work.

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Econometric Analysis: Structural Explorations in the Social Sciences

Open, Lecture—Year

This course is designed for all students interested in the social sciences who wish to understand the methodology and techniques involved in the estimation of structural relationships between variables. 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 empirical work done by academic or professional social scientists. The practical, hands-on approach taken in this course will be useful to those students who wish to do future conference projects in the social (or natural) sciences with significant empirical content. The course will also be invaluable for students who are seeking internships, planning to enter the job market, or desiring to pursue graduate education in the social sciences and public policy. After taking this course, students will be able to analyze questions such as the following: What is the relationship between slavery and the development of capitalist 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 do socioeconomic factors determine the crime rate in the United States? During the course of the year, we will study all of these questions. In the first semester, we will cover the theoretical and applied statistical principles that underlie Ordinary Least Squares (OLS) regression techniques. We will begin with the assumptions needed to obtain the Best Linear Unbiased Estimates of a regression equation, also known as the “BLUE” conditions. Particular emphasis will be placed on the assumptions regarding the distribution of a model’s error term and other BLUE conditions. We will also 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 the 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, we will take a similar approach to understanding and correcting model specification errors. The spring semester class will build on the fall class by introducing students to advanced topics in econometrics. 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 seminar will involve the study of panel data, as well as logit/probit models. As with the fall class, the spring class will also be very “hands-on,” in that students will get ample exposure to concrete issues while also being encouraged to consider basic methodological questions (e.g., the debates between John Maynard Keynes and Jan Tinbergen) regarding the power and limitations of econometric analysis. The spring semester is particularly relevant to students who wish to pursue graduate studies in a social-science discipline, although it will be equally relevant for those seeking other types of graduate degrees that involve knowledge of intermediate-level quantitative analysis.

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Intermediate Microeconomics: Conflicts, Coordination, and Institutions

Intermediate, Seminar—Fall

Economics was born in the 18th century, around the same time that capitalism emerged in Europe. Since then, economists have sought to understand the ways in which people allocate, produce, exchange, and distribute things in capitalist societies and how such activities impact people’s welfare. For the most part of the 20th century, microeconomics centered on the “efficiency” of the free market. Since the late 20th century, contending and critical paradigms have successfully challenged the narrow definition of “efficiency” and broadened the scope of analysis from the free market to a variety of institutions. In this course, we will examine the fundamental questions, such as: What are the incentives of individual decision making under different circumstances? How do individuals make decisions? What are the social consequences of individual decision making? We will not only learn about traditional issues such as how individual consumers and firms make decisions and the welfare properties of the market but also examine how individuals interact with each other, the power relationship between individuals, the power relationship on the labor market and the credit market and inside the firms, the situations where individuals care about other than their self-interests, the successful and unsuccessful coordination of individuals, and the institutional solutions for improving social welfare.

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Intermediate Macroeconomics: Theory and Policies

Intermediate, Seminar—Spring

Keynes not only revolutionized economic theory in 1937 but also led generations of economists to believe that the government should play an active role in managing a country’s aggregate demand. Yet, since the 1980s, the theoretical and policy world of mainstream economics took a great U-turn and, once again, embraced the fundamental role of the free market. In macroeconomics, this is reflected by the pursuit of goals such as fiscal austerity, a balanced budget, financial deregulation, and liberalization of international finance. In this course, we will examine the fundamental debates in macroeconomic theory and policymaking. The standard analytical framework of aggregate demand, aggregate supply, labor market, inflation, exchange rate, and economic growth will be used as our entry point of analysis. On top of that, we will examine multiple theoretical and empirical perspectives on money, credit and financial markets, consumption, investment, governmental spending, unemployment, international finance, growth and distribution, economic crisis, technological change, and long waves of capitalist societies. More recent progressive theories and policies will be discussed, such as universal basic income and job guarantee, modern monetary theory, etc.

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Classical Mechanics (Calculus-Based General Physics)

Open, Small Lecture—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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Electromagnetism & Light (Calculus-Based General Physics)

Open, Small Lecture—Spring

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

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Quantum Mechanics

Intermediate/Advanced, Seminar—Spring

There are three kinds of people: those who understand quantum mechanics; those who do not understand quantum mechanics; and those who both simultaneously understand and do not understand quantum mechanics. This course will provide an introduction to the theoretical foundations of quantum mechanics. Topics will include: the classical physics paradigm, quantum state vectors, quantum operators and observables, commutator relations, the Schrödinger equation and time-evolution, the quantum harmonic potential, the quantum Coulomb potential and the hydrogen atom, angular momentum and spin, and the Feynman path integral formalism. No cats will be harmed. Familiarity with introductory physics, complex numbers, vectors, dot and cross products, and matrices is useful but not required.

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