Mathematics

Open, Small Lecture—Fall

This lecture will present 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 will 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, 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. Weekly hands-on laboratory sessions will reinforce the programming concepts covered in class.

Faculty

• Michael Siff

Intermediate, Seminar—Fall

What is truth? One idea is that if that something is true, if using a small collection of assumptions and a few basic rules, then we can prove our claim from those assumptions. In this course, we will focus on mathematical proof and how it represents a formal backbone on which both theoretical computer science and most of the rest of mathematics can be constructed. We will work from the ground up, starting with Boolean logic and deductive arguments and moving onto elementary number theory where we will focus on the use of induction (e.g., to establish a formula for triangular numbers) and contradiction (e.g., to prove the infinitude of primes). We will introduce a formal notion of algorithm and demonstrate how number theory can be used to explain the principles behind modern cryptography. We will study sets, both finite and infinite, and how they can be used to formalize relations and functions. We will introduce basic notions of combinatorics (a fancy word for counting) with emphasis on the import of exponential growth. (Example: If you walked into a pizza shop that offered ten different toppings, would you want to sample all of the combinations?) We will then use our improved methods of counting to develop a basic theory of probability. Near the end of the semester, we will apply most, if not all, of the earlier concepts to introduce the theory of graphs. Along the way, we will explore the interplay between discrete mathematics and computation including recursion and programming; big-oh notation and categorizing the efficiency of algorithms; and graph theory's role in circuit design. Time permitting, we will circle back to the original question and investigate the limits of proof by trying to show that some true things can never be proven! For conference, students will do a deep dive on a related mathematical or computational topic. Examples include, but are not limited to: studying the proof and applications of a famous theorem (e.g., Fermat's "little" theorem); explaining how computers can be used (without so-called "artificial intelligence (AI)") to verify the validity of a formal proof; learning to program using logic; measuring the performance of related algorithms; or designing and simulating nontrivial digital circuits.

Faculty

• Michael Siff

Intermediate, Seminar—Spring

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 was not until the 1980s—over half a 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, the Deutsch-Jozsa algorithm, Grover's search algorithm, Shor's factoring algorithm, quantum teleportation, and applications to cryptography. 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.

Faculty

• James Marshall

Intermediate, Seminar—Spring

In this course, we will study a wide range of data structures and algorithms that are important for the design of sophisticated computer programs along with techniques for managing program complexity. Throughout the course, we will use Java, a strongly typed, object-oriented programming language. Topics covered will include types and polymorphism, arrays, linked lists, stacks, queues, priority queues, heaps, dictionaries, balanced trees, and graphs, as well as several important algorithms for manipulating those structures. We will also study techniques for analyzing the efficiency of algorithms. The central theme tying all of these topics together is the idea of abstraction and the related notions of information hiding and encapsulation, which we will emphasize throughout the course. Weekly hands-on lab sessions will reinforce the concepts covered in class.

Faculty

• Michael Siff

Open, Lecture—Fall

A while ago, I went to a visual arts museum, and they had their collection displayed in an unusual fashion. Instead of grouping art in rooms according to genre, chronology, nationality or particular artists, the art was arranged by intriguing concepts. A room might contain an O’Keeffe painting, a centuries-old indigenous piece from Australia, a Rodin sculpture, and a poem that were, in some way, connected by a fascinating idea. I want to recapitulate something like this experience. Every class session will begin with some concept from mathematics, poetry, philosophy, astronomy, and more; then we will gradually explore different music that engages with that concept in some way. The musical examples every week will span centuries and cultures—one week might have an avant-garde piano sonata by Boulez, a 1980's art-rock song by Laurie Anderson, and a Kendrick Lamar album; the next week might have an ancient Sumerian song, a piece by Debussy, and a work from the Indian Carnatic tradition. Gradually, more and more connections between the seemingly disparate topics will be revealed. So, okay, it isn’t everything exactly—and it’s more like “across the course of a semester” rather than “all at once”—but, by the end, you will know a whole lot more across a wide range of disciplines. And, most importantly, we will listen to a metric ton of fantastic music.

Faculty

• Patrick Muchmore

Open, Lecture—Spring

We will begin in Babel itself, the ancient site of Babylon, where archaeologists have discovered many tablets about music. Nearby sites have the earliest examples of musical notation, some dating as far back as 1400 BCE. We will learn some aspects of how their music worked and begin building a vocabulary for talking about and notating music in general. Across the course of the semester, we will learn many different musical languages, such as the music of Ancient Greece, the old court music of Japan, drum ensembles of central Africa, and the world of European classical music. We will also delve into many different modern musics, including the rise of sampling and turntablism in hip-hop, the theory of so-called “atonal” music, and the development of electronic sound. In short, the course will be devoted to learning a sampling of crucial aspects of the multitudinous vocabularies and grammars that pervade music across the world and across time. No prior study of nor the ability to read music is required. By the end of the semester, students will be able to read basic musical ideas in a few different notation systems and will have some understanding of important aspects of not only standard European music theories but also many others that are too-often learned only by specialists.

Faculty

• Patrick Muchmore

First-Year Studies—Spring

This course will provide an overview of the pivotal developments in 20th-century physics that dramatically overturned the centuries-old scientific understanding of the fundamental laws of our universe. In this seminar, we will discuss readings, walk through thought experiments, and unravel paradoxes to understand the concepts behind Einstein’s theories of special and general relativity, debate various interpretations of quantum mechanics, and explore the open questions that are motivating theoretical physics research in the 21st century. Biweekly in spring, students will meet with the instructor for individual conferences.

Faculty

• Merideth Frey

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 courses. Laboratory 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 experiment or an American Journal of Physics article to replicate, analyze, and present their findings at the semiannual Sarah Lawrence College Science & Mathematics Poster Session.

Faculty

• Sarah Racz

Open, Large seminar—Spring

This course will provide an overview of the pivotal developments in 20th-century physics that dramatically overturned the centuries-old scientific understanding of the fundamental laws of our universe. In this seminar, we will discuss readings, walk through thought experiments, and unravel paradoxes to understand the concepts behind Einstein’s theories of special and general relativity, debate various interpretations of quantum mechanics, and explore the open questions that are motivating theoretical physics research in the 21st century.

Faculty

• Merideth Frey

Advanced, Seminar—Spring

This course will introduce classical field theory, which provides the description for many physical phenomena. In this course, students will develop the foundational tools that can be used to understand electromagnetism, general relativity, and fluid mechanics. We will study classical scalar and vector fields, which serve as a precursor for tensor and quantum field theories. Along the way, we will cover the fundamentals of classical mechanics for particle systems through Lagrangians and Hamiltonians before moving to continuous systems; study Noether's theorem and the relationship between conservation laws and symmetries; dive into Maxwell's theory of electrodynamics; and more. For conference work, students may complete coursework in an advanced physics topic of their choosing.

Faculty

• Sarah Racz