This undergraduate level Algebra I course covers groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups.

This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

Algebra II is the second semester of a year-long introduction to modern algebra. The course focuses on group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.
These notes, which were created by students in a recent on-campus 18.702 Algebra II class, are offered here to supplement the materials included in OCW’s version of 18.702. They have not been checked for accuracy by the instructors of that class or by other MIT faculty members.

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.

Bycatch, the unintended capture of animals in commercial fishing gear, is a hot topic in marine conservation today. The surprisingly high level of bycatch about 25% of the entire global catch is responsible for the decline of hundreds of thousands of dolphins, whales, porpoises, seabirds and sea turtles each year. Through this curricular unit, students analyze the significance of bycatch in the global ecosystem and propose solutions to help reduce bycatch. They become familiar with current attempts to reduce the fishing mortality of these animals. Through the associated activities, the challenges faced today are reinforced and students are stimulated to brainstorm about possible engineering designs or policy changes that could reduce the magnitude of bycatch.

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.

Is climate change real? Yes, it is! And technologies to reduce Greenhouse Gas (GHG) emissions are being developed. One type of technology that is imperative in the short run is biofuels; however, biofuels must meet specifications for gasoline, diesel, and jet fuel, or catastrophic damage could occur. This course will examine the chemistry of technologies of bio-based sources for power generation and transportation fuels. We'll consider various biomasses that can be utilized for fuel generation, understand the processes necessary for biomass processing, explore biorefining, and analyze how biofuels can be used in current fuel infrastructure.