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.

2019-2020 Courses

Mathematics

An Introduction to Statistical Methods and Analysis

Open , Lecture—Fall

Prerequisite: basic high-school algebra and geometry.

Correlation, regression, 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, 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 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. 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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Calculus I: The Study of Motion and Change

Open , Seminar—Fall

Prerequisites: successful completion of trigonometry and precalculus courses. Students concerned about meeting the prerequisites should contact the instructor. This course is also offered in the spring semester.

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 and study the motion and change happening around us. 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, 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. For conference work, students may choose to undertake a deeper investigation of a single topic or application of calculus or conduct a study of some other mathematically-related topic. This seminar 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.

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Calculus II: Further Study of Motion and Change

Open , Seminar—Fall

Prerequisites: one year of high-school calculus or one semester of college-level calculus. Students concerned about meeting the course prerequisites are encouraged to contact the instructor as soon as possible. This course is also being offered in the spring semester of this academic year.

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

Faculty

Strange Universes: An Introduction to Non-Euclidean Geometry

Open , Seminar—Fall

If you draw two straight lines on a piece of paper, it’s not difficult to keep them from crossing. Imagine, however, that the lines extend in both directions off the page and without end. Do these hypothetical lines cross? Surprisingly, this mundane question goes to the heart of our modern conception of space. Your experience might suggest that the lines will cross unless they head off the edge of the page at exactly the same angle. In that case we call the lines parallel; and this is the answer Euclid asserts with his fifth (or “parallel”) postulate of the “Elements.” Roughly 2,000 years later, mathematicians came to the shocking realization that lines need not obey the parallel postulate. The resulting non-Euclidean geometries were so unexpected to the mathematicians who first conceived of them that one, János Bolyai, remarked, “Out of nothing I have created a strange new universe.” This course will explore the alternatives to Euclidean geometry that first appeared in the 19th century. These include hyperbolic, spherical, and projective geometry, as well as more idiosyncratic geometries that we will devise together. Our exploration of these strange universes will be aided by visualizations that include drawing, computer-graphics animation, and video-game technology. Throughout, we will discuss the impact of the non-Euclidean revolution on astronomy, philosophy, and culture.

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Calculus I: The Study of Motion and Change

Open , Seminar—Spring

Prerequisites: successful completion of courses in trigonometry and precalculus. Students concerned about meeting the prerequisites should contact the instructor. This course is also offered in the fall semester.

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 and study the motion and change happening around us. 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, 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. For conference work, students may choose to undertake a deeper investigation of a single topic or application of calculus or conduct a study of some other mathematically-related topic. This seminar 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.

Faculty

TBA

Calculus II: Further Study of Motion and Change

Open , Seminar—Spring

Prerequisite: one year of high-school calculus or one semester of college-level calculus. Students concerned about meeting the prerequisite should contact the instructor. This course is also offered in the fall semester.

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 calculus or conduct a study of some other mathematically-related topic. This seminar 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. The theory of limits, differentiation, and integration will be briefly reviewed at the beginning of the term.

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

Intermediate , Seminar—Year

Prerequisite: successful completion of Calculus II or a score of 4 or 5 on the AP Calculus BC exam.

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 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.

Faculty

Discrete Mathematics: Gateway to Higher Mathematics

Intermediate , Seminar—Fall

Prior study of calculus is highly recommended.

There is an enormous, vivid world of mathematics beyond what students encounter in high-school algebra, geometry, and calculus courses. This seminar provides an introduction to this realm of elegant and powerful mathematical ideas. With an explicit goal of improving students’ mathematical reasoning and problem-solving skills, this seminar provides the ultimate intellectual workout. Five important themes are interwoven in the course: logic, proof, combinatorial analysis, discrete structures, and philosophy. For conference work, students may design and execute any appropriate project involving mathematics. A must for students interested in pursuing advanced mathematical study, this course is also highly recommended for students with a passion for computer science, engineering, law, logic, and/or philosophy.

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Abstract Algebra: Theory and Applications

Advanced , Seminar—Spring

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. Abstract algebra examines sets of objects (numbers, matrices, polynomials, functions, ideas) and operations on these 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 these. 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.

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Computational Number Theory

Open , Seminar—Spring

Number theory is one of the oldest and most beautiful fields of mathematics, and many of the ideas it has generated over the millennia are just now becoming crucially important in the information age. This course will serve as an introduction to number theory, computer programming, and the interplay between the two. Topics will include divisibility, prime factorization, modular arithmetic, cryptography, and algorithms, with other topics selected based on class interest. We will spend some time formulating conjectures, generating evidence to support or disprove them, and attempting to prove the ones that seem true. We will also address algorithmic questions such as run-time efficiency and compare and contrast different mathematical algorithms that theoretically achieve the same goal but differ practically in consequential ways.

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Computer Organization

Intermediate , Seminar—Fall

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

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. 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

Prerequisite: Familiarity with linear algebra or equivalent mathematical preparation.

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; it 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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Compilers

Intermediate , Seminar—Spring

Permission of the instructor is required. Students should have at least one semester of programming experience and, preferably, some familiarity with computer organization.

Compilers are often known as translators—and for good reason: Their job is to take programs written in one language and translate them to another language (usually assembly or machine language) that a computer can execute. It is perhaps the ideal meeting between the theoretical and practical sides of computer science. Modern compiler implementation offers a synthesis of: (1) language theory, how languages (both natural languages and programming languages) can be represented on and recognized by a computer; (2) software design and development, how practical software can be developed in a modular way—e.g., how components of one compiler can be connected to components of another compiler to form a new compiler; and (3) computer architecture, understanding how modern computers work. During the semester, we will write a program implementing a nontrivial compiler for a novel programming language (partly of our own design). Topics we will cover along the way include the difference between interpreters and compilers, regular expressions and finite automata, context-free grammars and the Chomsky hierarchy, type checking and type inference, contrasts between syntax and semantics, and graph coloring as applied to register allocation. Conference work will allow students to pursue different aspects of compilers, such as compilation of object-oriented languages, automatic garbage collection, compiler optimizations, just-in-time compilation, WebAssembly, and applications of compiler technology to natural-language translation.

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Exploring the Universe: Astronomy and Cosmology

Open , Lecture—Year

This yearlong course will provide a broad introduction to our current knowledge of the universe without requiring previous background in college-level science and math. Topics covered will include the history of our understanding of the universe; our current knowledge of the solar system, including the Sun, planets, moons, asteroids, and comets; the nature, life cycle, and properties of stars, as well as neutron stars and black holes; the possibility of extraterrestrial life; our knowledge of distant galaxies; and the description of the universe as a whole, its development from the Big Bang, and the unresolved questions concerning its origin and ultimate fate. Classes will incorporate discussions and some problem-solving activities. The course will also include occasional evening meetings for telescope observations.

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Classical Mechanics (Calculus-Based General Physics)

Open , Seminar—Fall

Permission of the instructor is required. Students are encouraged to have completed one semester of calculus as a prerequisite. It is strongly recommended that students who have not completed a second semester of calculus enroll in Calculus II, as well. Calculus II, or equivalent, is highly recommended in order to take Electromagnetism and Light (Calculus-Based General Physics) in the spring.

Calculus-based general physics is a standard course at most institutions; as such, this course will prepare you for more advanced work in physical science, engineering, or the health fields. (Alternatively, the algebra-based Introduction to Mechanics will also suffice for pre-medical students.) The course will cover introductory classical mechanics, including kinematics, dynamics, momentum, energy, and gravity. Emphasis will be placed on scientific skills, including: problem solving, development of physical intuition, scientific communication, use of technology, and development and execution of experiments. The best way to develop scientific skills is to practice the scientific process. We will focus on learning physics through discovering, testing, analyzing, and applying fundamental physics concepts in an interactive classroom, as well as in weekly laboratory meetings.

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Introduction to Mechanics (General Physics Without Calculus)

Open , Seminar—Fall

This course, or equivalent, is required to take Introduction to Electromagnetism, Light, and Modern Physics (General Physics Without Calculus) in the spring.

This course covers introductory classical mechanics, including dynamics, kinematics, momentum, energy, and gravity. Students considering careers in architecture or the health sciences, as well as those interested in physics for physics’ sake, should take either this course or Classical Mechanics. Emphasis will be placed on scientific skills, including problem solving, development of physical intuition, scientific communication, use of technology, and development and execution of experiments. Seminars will incorporate discussion, exploratory activities, and problem-solving activities. In addition, the class will meet weekly to conduct laboratory work. A background in calculus is not required.

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20th-Century Physics Through Three Pivotal Papers

Intermediate , Seminar—Fall

Prerequisites: one year of general physics and one year of calculus.

This course takes an in-depth look at three pivotal papers in 20th-century physics pertaining to special relativity and fundamental interpretations of quantum mechanics that transformed and defined our way of thinking in modern science. In this seminar-style class, we will deeply read, dissect, and discuss these three primary sources. In the process, we will together derive the predictions of special relativity; debate the various interpretations of quantum mechanics revolving around the famous Einstein, Podolsky, Rosen (EPR) paradox; and explore experiments meant to test our fundamental understanding of quantum mechanics.

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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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Resonance and Its Applications

Intermediate , Seminar—Spring

This is a lab-based course designed to teach students critical advanced laboratory skills while exploring the fascinating phenomenon of resonance and its many applications. The course will be broken into three main units: mechanical resonators, electronic resonators, and quantum mechanical resonators. Resonators are physical systems that undergo periodic motion and react quite dramatically to being driven at particular frequencies (like the opera singer hitting just the right note to break a wine glass). These systems are very common in everyday life, as well as inside many important technological devices. Each unit will explore a particular application of resonance (e.g., building an AM radio receiver for electronic resonance and using our benchtop NMR system to explore quantum mechanical resonance). Although some class time will be spent going over the relevant theory, the majority of the class time will be spent designing and doing experiments using advanced lab equipment, analyzing data using Jupyter (iPython) notebooks, and reporting the results using LaTeX. For conference work, students are encouraged to develop their own experimental question, design their own experiment to answer that question, do the experiment, analyze the data, and present their findings at the Science and Mathematics Poster Session.

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