Algebra I is the first semester of a year-long introduction to modern algebra. Algebra is a fundamental subject, used in many advanced math courses and with applications in computer science, chemistry, etc. The focus of this class is studying groups, linear algebra, and geometry in different forms.
These notes, which were created by students in a recent on-campus 18.701 Algebra I class, are offered here to supplement the materials included in OCW’s version of 18.701. They have not been checked for accuracy by the instructors of that class or by other MIT faculty members.

This course covers the applications of algebra to combinatorics. Topics include enumeration methods, permutations, partitions, partially ordered sets and lattices, Young tableaux, graph theory, matrix tree theorem, electrical networks, convex polytopes, and more.

This is the first semester of a two-semester sequence on Algebraic Geometry. The goal of the course is to introduce the basic notions and techniques of modern algebraic geometry. It covers fundamental notions and results about algebraic varieties over an algebraically closed field; relations between complex algebraic varieties and complex analytic varieties; and examples with emphasis on algebraic curves and surfaces. This course is an introduction to the language of schemes and properties of morphisms.

This course covers the fundamental notions and results about algebraic varieties over an algebraically closed field. It also analyzes the relations between complex algebraic varieties and complex analytic varieties.

This course provides an introduction to the language of schemes, properties of morphisms, and sheaf cohomology. Together with 18.725 Algebraic Geometry, students gain an understanding of the basic notions and techniques of modern algebraic geometry.

This research-oriented course will focus on algebraic and computational techniques for optimization problems involving polynomial equations and inequalities with particular emphasis on the connections with semidefinite optimization. The course will develop in a parallel fashion several algebraic and numerical approaches to polynomial systems, with a view towards methods that simultaneously incorporate both elements. We will study both the complex and real cases, developing techniques of general applicability, and stressing convexity-based ideas, complexity results, and efficient implementations. Although we will use examples from several engineering areas, particular emphasis will be given to those arising from systems and control applications.

This is a course on the singular homology of topological spaces. Topics include: Singular homology, CW complexes, Homological algebra, Cohomology, and Poincare duality.

This is the second part of the two-course series on algebraic topology. Topics include basic homotopy theory, obstruction theory, classifying spaces, spectral sequences, characteristic classes, and Steenrod operations.

This is a research-oriented course on algorithm engineering, which will cover both the theory and practice of algorithms and data structures. Students will learn about models of computation, algorithm design and analysis, and performance engineering of algorithm implementations. We will study the design and implementation of sequential, parallel, cache-efficient, external-memory, and write-efficient algorithms for fundamental problems in computing. Many of the principles of algorithm engineering will be illustrated in the context of parallel algorithms and graph problems.

This course is organized around algorithmic issues that arise in machine learning. Modern machine learning systems are often built on top of algorithms that do not have provable guarantees, and it is the subject of debate when and why they work. In this class, we focus on designing algorithms whose performance we can rigorously analyze for fundamental machine learning problems.

6.890 Algorithmic Lower Bounds: Fun with Hardness Proofs is a class taking a practical approach to proving problems can’t be solved efficiently (in polynomial time and assuming standard complexity-theoretic assumptions like P ≠ NP). The class focuses on reductions and techniques for proving problems are computationally hard for a variety of complexity classes. Along the way, the class will create many interesting gadgets, learn many hardness proof styles, explore the connection between games and computation, survey several important problems and complexity classes, and crush hopes and dreams (for fast optimal solutions).

This course is offered to undergraduates and addresses several algorithmic challenges in computational biology. The principles of algorithmic design for biological datasets are studied and existing algorithms analyzed for application to real datasets. Topics covered include: biological sequence analysis, gene identification, regulatory motif discovery, genome assembly, genome duplication and rearrangements, evolutionary theory, clustering algorithms, and scale-free networks.

Animation is a compelling and effective form of expression; it engages viewers and makes difficult concepts easier to grasp. Today’s animation industry creates films, special effects, and games with stunning visual detail and quality. This graduate class will investigate the algorithms that make these animations possible: keyframing, inverse kinematics, physical simulation, optimization, optimal control, motion capture, and data-driven methods. Our study will also reveal the shortcomings of these sophisticated tools. The students will propose improvements and explore new methods for computer animation in semester-long research projects. The course should appeal to both students with general interest in computer graphics and students interested in new applications of machine learning, robotics, biomechanics, physics, applied mathematics and scientific computing.

This is a graduate-level introduction to the principles of statistical inference with probabilistic models defined using graphical representations. The material in this course constitutes a common foundation for work in machine learning, signal processing, artificial intelligence, computer vision, control, and communication. Ultimately, the subject is about teaching you contemporary approaches to, and perspectives on, problems of statistical inference.

This course introduces the unique characteristics of militaries and explores the roles they play in the societies they are constructed to defend, with a special focus on the relationships between the military and their civilian leaders and popular publics. Topics include a modern history of relations between US presidents and the military, coups and military governments, public trust in the military, racial integration of the military, and the military-industrial (and tech!) complex.

This course will provide an overview of a new vision for Human-Computer Interaction (HCI) in which people are surrounded by intelligent and intuitive interfaces embedded in the everyday objects around them. It will focus on understanding enabling technologies and studying applications and experiments, and, to a lesser extent, it will address the socio-cultural impact. Students will read and discuss the most relevant articles in related areas: smart environments, smart networked objects, augmented and mixed realities, ubiquitous computing, pervasive computing, tangible computing, intelligent interfaces and wearable computing. Finally, they will be asked to come up with new ideas and start innovative projects in this area.

This course focuses on the Great Depression and World War II and how they led to a major reordering of American politics and society. We will examine how ordinary people experienced these crises and how those experiences changed their outlook on politics and the world around them.

This subject, cross-listed in Literature and Women’s Studies, examines a range of American women authors from the seventeenth century to the present. It aims to introduce a number of literary genres and styles- the captivity narrative, slave novel, sensational, sentimental, realistic, and postmodern fiction- and also to address significant historical events in American women’s history: Puritanism, the American Revolution, industrialization and urbanization in the nineteenth century, the Harlem Renaissance, World War II, the 60s civil rights movements. A primary focus will be themes studied and understood through the lens of gender: war, violence, and sexual exploitation (Keller, Rowlandson, Rowson); the relationship between women and religion (Rowlandson, Rowson, Stowe); labor, poverty, and working conditions for women (Fern, Davis, Wharton); captivity and slavery (Rowlandson, Jacobs); class struggle (Fern, Davis, Wharton, Larsen); race and identity (Keller, Jacobs, Larsen, Morrison); feminist revisions of history (Stowe, Morrison, Keller); and the myth of the fallen woman (take your pick). Essays and in-class reports will focus more particularly on specific writers and themes and will stress the skills of close reading, annotation, research, and uses of multimedia where appropriate.

What is a “life” when it’s written down? How does memory inform the present? Why are autobiographies and memoirs so popular? This course will address these questions among others, considering the relationship between biography, autobiography, and memoir and between personal and social themes. We will examine classic authors such as Mary Rowlandson, Benjamin Franklin, Frederick Douglass, Harriet Jacobs, and Mark Twain; then more recent examples like Tobias Wolff, Art Spiegelman, Sherman Alexie, Shirley Geok-lin Lim, Edwidge Danticat, and Alison Bechdel.

“What then is the American, this new man?” asked J. Hector St-John de Crèvecoeur in his Letters from an American Farmer in 1782. This subject takes Crèvecoeur’s question as the starting point for an examination of the changing meanings of national identity in the American past. We will consider a diverse collection of classic texts in American history to see how Americans have defined themselves and their nation in politics, literature, art, and popular culture. As a communications-intensive subject, students will be expected to engage intensively with the material through frequent oral and written exercises.