Online courses directory (118)

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

6.844 is a graduate introduction to programming theory, logic of programming, and computability, with the programming language Scheme used to crystallize computability constructions and as an object of study itself. Topics covered include: programming and computability theory based on a term-rewriting, "substitution" model of computation by Scheme programs with side-effects; computation as algebraic manipulation: Scheme evaluation as algebraic manipulation and term rewriting theory; paradoxes from self-application and introduction to formal programming semantics; undecidability of the Halting Problem for Scheme; properties of recursively enumerable sets, leading to Incompleteness Theorems for Scheme equivalences; logic for program specification and verification; and Hilbert's Tenth Problem.

Geometry of Manifolds analyzes topics such as the differentiable manifolds and vector fields and forms. It also makes an introduction to Lie groups, the de Rham theorem, and Riemannian manifolds.

Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time.

This course is an introduction to linear optimization and its extensions emphasizing the underlying mathematical structures, geometrical ideas, algorithms and solutions of practical problems. The topics covered include: formulations, the geometry of linear optimization, duality theory, the simplex method, sensitivity analysis, robust optimization, large scale optimization network flows, solving problems with an exponential number of constraints and the ellipsoid method, interior point methods, semidefinite optimization, solving real world problems problems with computer software, discrete optimization formulations and algorithms.

Double affine Hecke algebras (DAHA), also called Cherednik algebras, and their representations appear in many contexts: integrable systems (Calogero-Moser and Ruijsenaars models), algebraic geometry (Hilbert schemes), orthogonal polynomials, Lie theory, quantum groups, etc. In this course we will review the basic theory of DAHA and their representations, emphasizing their connections with other subjects and open problems.

Numerical methods for solving problems arising in heat and mass transfer, fluid mechanics, chemical reaction engineering, and molecular simulation. Topics: numerical linear algebra, solution of nonlinear algebraic equations and ordinary differential equations, solution of partial differential equations (e.g. Navier-Stokes), numerical methods in molecular simulation (dynamics, geometry optimization). All methods are presented within the context of chemical engineering problems. Familiarity with structured programming is assumed. The examples will use MATLAB®.

Acknowledgements

The instructor would like to thank Robert Ashcraft, Sandeep Sharma, David Weingeist, and Nikolay Zaborenko for their work in preparing materials for this course site.

This course is divided into two sections, Part I and Part II.  Part I, found here, provides an introduction to statistical theory. A brief review of probability will be given mainly as background material, however, it is assumed to be known. Topics include normal distribution, limit theorems, Bayesian concepts, and testing, among others. 

Part II prepares students for the remainder of the econometrics sequence and and can be found by visiting 14.381 Fall 2006. 

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 course is an introduction to differential geometry. The course itself is mathematically rigorous, but still emphasizes concrete aspects of geometry, centered on the notion of curvature.

Geometry and Quantum Field Theory, designed for mathematicians, is a rigorous introduction to perturbative quantum field theory, using the language of functional integrals. It covers the basics of classical field theory, free quantum theories and Feynman diagrams. The goal is to discuss, using mathematical language, a number of basic notions and results of QFT that are necessary to understand talks and papers in QFT and String Theory.

This course teaches the art of guessing results and solving problems without doing a proof or an exact calculation. Techniques include extreme-cases reasoning, dimensional analysis, successive approximation, discretization, generalization, and pictorial analysis. Applications include mental calculation, solid geometry, musical intervals, logarithms, integration, infinite series, solitaire, and differential equations. (No epsilons or deltas are harmed by taking this course.) This course is offered during the Independent Activities Period (IAP), which is a special 4-week term at MIT that runs from the first week of January until the end of the month.

This course covers differential, integral and vector calculus for functions of more than one variable. These mathematical tools and methods are used extensively in the physical sciences, engineering, economics and computer graphics.

Course Formats

The materials have been organized to support independent study. The website includes all of the materials you will need to understand the concepts covered in this subject. The materials in this course include:

• Lecture Videos recorded on the MIT campus
• Recitation Videos with problem-solving tips
• Examples of solutions to sample problems
• Problem for you to solve, with solutions
• Exams with solutions
• Interactive Java Applets ("Mathlets") to reinforce key concepts

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