Free Online Math Courses
These math courses, available for free online from top universities across the country, can benefit students and professionals both. There are courses here that will help you brush up on your basic algebra and geometry skills, and classes on matrix theory and complex numbers for more advanced students.
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Advanced Algorithms - MIT
This course is a first-year graduate course in algorithms. Emphasis is placed on fundamental algorithms and advanced methods of algorithmic design, analysis, and implementation. Techniques to be covered include amortization, randomization, fingerprinting, word-level parallelism, bit scaling, dynamic programming, network flow, linear programming, fixed-parameter algorithms, and approximation algorithms. Domains include string algorithms, network optimization, parallel algorithms, computational geometry, online algorithms, external memory, cache, and streaming algorithms, and data structures.
Advanced Analytic Methods in Science and Engineering is a comprehensive treatment of the advanced methods of applied mathematics. It was designed to strengthen the mathematical abilities of graduate students and train them to think on their own.
This course analyzes the functions of a complex variable and the calculus of residues. It also covers subjects such as ordinary differential equations, partial differential equations, Bessel and Legendre functions, and the Sturm-Liouville theory.
The topics for this course cover various aspects of complexity theory, such as the basic time and space classes, the polynomial-time hierarchy and the randomized classes . This is a pure theory class, so no applications were involved.
The topics covered in this course include interactive proofs, zero-knowledge proofs, zero-knowledge proofs of knowledge, non-interactive zero-knowledge proofs, secure protocols, two-party secure computation, multiparty secure computation, and chosen-ciphertext security.
Algebra I - MIT
This undergraduate level Algebra I course covers groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups.
Algebraic Geometry - MIT
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.
Algebraic Topology - MIT
This course is a first course in algebraic topology. The emphasis is on homology and cohomology theory, including cup products, Kunneth formulas, intersection pairings, and the Lefschetz fixed point theorem.
Algebraic Topology - MIT
In this second term of Algebraic Topology, the topics covered include fibrations, homotopy groups, the Hurewicz theorem, vector bundles, characteristic classes, cobordism, and possible further topics at the discretion of the instructor.
An Introduction to Complex Numbers - The Open University
This unit looks at complex numbers. You will learn how they are defined, examine their geometric representation and then move on to looking at the methods for finding the nth roots of complex numbers and the solutions to simple polynominal equations.
Analysing Skid Marks - The Open University
This unit is the second in the MSXR209 series of five units on mathematical modelling. In this unit you are asked to relate the stages of the mathematical modelling process to a previously formulated mathematical model. This example, that of skid mark produced by vehicle tyres, is typical of accounts of modelling that you may see in books, or produced in the workplace. The aim of this unit is to help you to draw out and to clarify mathematical modelling ideas by considering the example. It assumes that you have studied Modelling pollution in the Great Lakes (MSXR209_1).
Analysis II - MIT
This course continues from Analysis I (18.100B), in the direction of manifolds and global analysis. The first half of the course covers multivariable calculus. The rest of the course covers the theory of differential forms in n-dimensional vector spaces and manifolds.
Analytic Number Theory - MIT
This course is an introduction to analytic number theory, including the use of zeta functions and L-functions to prove distribution results concerning prime numbers (e.g., the prime number theorem in arithmetic progressions).
Applied Parallel Computing is an advanced interdisciplinary introduction to applied parallel computing on modern supercomputers.
Behavior of Algorithms - MIT
This course is a study of Behavior of Algorithms and covers an area of current interest in theoretical computer science. The topics vary from term to term. During this term, we discuss rigorous approaches to explaining the typical performance of algorithms with a focus on the following approaches: smoothed analysis, condition numbers/parametric analysis, and subclassing inputs.
Calculus - MIT
This is an undergraduate course on calculus of several variables. It covers all of the topics covered in Calculus II (18.02), but presents them in greater depth. These topics are vector algebra in 3-space, determinants, matrices, vector-valued functions of one variable, space motion, scalar functions of several variables, partial differentiation, gradient, optimization techniques, double integrals, line integrals in the plane, exact differentials, conservative fields, Green's theorem, triple integrals, line and surface integrals in space, the divergence theorem, and Stokes' theorem. Additional topics covered in 18.022 are geometry, vector fields, and linear algebra.
Calculus II - UMass Boston
Continuation of MATH 140. Topics include transcendental functions, techniques of integration, applications of the integral, improper integrals, l'Hospital's rule, sequences, and series.
Calculus II - Dixie State
This course is the continuation of MATH 1210. Topics covered includes arc length, area of a surface of revolution, moments and centers of mass, integration techniques, sequences and series, parametrization of curves and polar coordinates, vectors in 3-space, quadric surfaces and cylindrical and spherical coordinates.
Calculus III - UMass Boston
This course is an introduction to the calculus of functions of several variables. It begins with studying the basic objects of multidimensional geometry: vectors and vector operations, lines, planes, cylinders, quadric surfaces, and various coordinate systems. It continues with the elementary differential geometry of vector functions and space curves. After this, it extends the basic tools of differential calculus - limits, continuity, derivatives, linearization, and optimization - to multidimensional problems. The course will conclude with a study of integration in higher dimensions, culminating in a multidimensional version of the substitution rule.
This is an undergraduate course on differential calculus in one and several dimensions. It is intended as a one and a half term course in calculus for students who have studied calculus in high school. The format allows it to be entirely self contained, so that it is possible to follow it without any background in calculus.
Combinatorial Analysis - MIT
This course analyzes combinatorial problems and methods for their solution. Prior experience with abstraction and proofs is helpful. Topics include: Enumeration, generating functions, recurrence relations, construction of bijections, introduction to graph theory, network algorithms and, extremal combinatorics.
Combinatorial Optimization provides a thorough treatment of linear programming and combinatorial optimization. Topics include network flow, matching theory, matroid optimization, and approximation algorithms for NP-hard problems.
This is a graduate-level course in combinatorial theory. The content varies year to year, according to the interests of the instructor and the students. The topic of this course is hyperplane arrangements, including background material from the theory of posets and matroids.
This course serves as an introduction to major topics of modern enumerative and algebraic combinatorics with emphasis on partition identities, young tableaux bijections, spanning trees in graphs, and random generation of combinatorial objects. There is some discussion of various applications and connections to other fields.
Commutative Algebra - MIT
In this course students will learn about Noetherian rings and modules, Hilbert basis theorem, Cayley-Hamilton theorem, integral dependence, Noether normalization, the Nullstellensatz, localization, primary decomposition, DVRs, filtrations, length, Artin rings, Hilbert polynomials, tensor products, and dimension theory.
Complex Numbers - The Open University
You may have met complex numbers before, but not had experience in manipulating them. This unit gives an accessible introduction to complex numbers, which are very important in science and technology, as well as mathematics. The unit includes definitions, concepts and techniques which will be very helpful and interesting to a wide variety of people with a reasonable background in algebra and trigonometry.
The following topics are covered in the course: complex algebra and functions; analyticity; contour integration, Cauchy's theorem; singularities, Taylor and Laurent series; residues, evaluation of integrals; multivalued functions, potential theory in two dimensions; Fourier analysis and Laplace transforms.
Computational Discrete Mathematics - Open Learning
This course presents material in discrete mathematics and computation theory with a strong emphasis on practical algorithms and experiential learning.
This course provides a review of linear algebra, including applications to networks, structures, and estimation, Lagrange multipliers. Also covered are: differential equations of equilibrium; Laplace's equation and potential flow; boundary-value problems; minimum principles and calculus of variations; Fourier series; discrete Fourier transform; convolution; and applications.
Developing Modeling Skills - The Open University
This unit is the third in the MSXR209 series of five units on mathematical modellng. It provides an overview of the processes involved in developing models, starting by explaining how to specify the purpose of the model. It then moves on to look at aspects involved in creating models, such as simplifying problems, choosing variables and parameters, formulating relationships and finding solutions. You will also look at interpreting results and evaluating models. This unit assumes that you have previously studied Modelling pollution in the Great Lakes (MSXR209_1) and Analysing skid marks (MSXR209_2).
Diagrams, Charts and Graphs - The Open University
Diagrams, charts and graphs are used by all sorts of people to express information in a visual way, whether it's in a report by a colleague or a plan from your interior designer. This unit will teach you how to interpret these tools and how to use them yourself to convey information more effectively.
Differential Analysis - MIT
This is the first semester of a two-semester sequence on Differential Analysis. Topics include fundamental solutions for elliptic; hyperbolic and parabolic differential operators; method of characteristics; review of Lebesgue integration; distributions; fourier transform; homogeneous distributions; asymptotic methods.
Differential Equations - MIT
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.
Differential Equations - The Open University
This unit extends the ideas introduced in the unit on first-order differential equations to a particular type of second-order differential equations which has a variety of applications. The unit assumes that you have previously had a basic grounding in calculus, know something about first-order differential equations and some familiarity with complex numbers.
Differential Geometry - MIT
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.
Distrubted Algorithms - MIT
Distributed algorithms are algorithms designed to run on multiple processors, without tight centralized control. In general, they are harder to design and harder to understand than single-processor sequential algorithms. Distributed algorithms are used in many practical systems, ranging from large computer networks to multiprocessor shared-memory systems. They also have a rich theory, which forms the subject matter for this course.
Engineering Statistics - Open Learning
Statics is a sophomore level engineering course, offered in all mechanical and civil engineering programs. We study methods of quantifying the forces between bodies, for example parts of mechanical, structural, and biological systems.
Error-Correcting Codes - MIT
This course introduces students to iterative decoding algorithms and the codes to which they are applied, including Turbo Codes, Low-Density Parity-Check Codes, and Serially-Concatenated Codes. The course will begin with an introduction to the fundamental problems of Coding Theory and their mathematical formulations. This will be followed by a study of Belief Propagation--the probabilistic heuristic which underlies iterative decoding algorithms. Belief Propagation will then be applied to the decoding of Turbo, LDPC, and Serially-Concatenated codes. The technical portion of the course will conclude with a study of tools for explaining and predicting the behavior of iterative decoding algorithms, including EXIT charts and Density Evolution.
Experiences of Learning Mathematics - The Open University
This unit is aimed at teachers who wish to review how they go about the practice of teaching maths, those who are considering becoming maths teachers, or those who are studying maths courses and would like to understand more about the teaching process.
Exploring Data: Graphs and Numerical Summaries - The Open University
This Unit will introduce you to a number of ways of representing data graphically and of summarising data numerically. You will learn the uses for pie charts, bar charts, histograms and scatterplots. You will also be introduced to various ways of summarising data and methods for assessing location and dispersion.
Exploring Distance Time Graphs - The Open University
Graphs are a common way of presenting information. However, like any other type of representation, graphs rely on shared understandings of symbols and styles to convey meaning. Also, graphs are normally drawn specifically with the intention of presenting information in a particularly favourable or unfavourable light, to convince you of an argument or to influence your decisions.
Finding Information in Mathematics and Statistics - The Open University
This unit will help you to identify and use information in maths and statistics, whether for your work, study or personal purposes. Experiment with some of the key resources in this subject area, and learn about the skills which will enable you to plan searches for information, so you can find what you are looking for more easily. Discover the meaning of information quality, and learn how to evaluate the information you come across. You will also be introduced to the many different ways of organising your own information, and learn how to reference it properly in your work. Finally, discover how to keep up to date with the latest developments in your area of interest by using tools such as RSS and mailing lists.
First-Order Differential Equations - The Open University
This unit introduces the topic of differential equations. The subject is developed without assuming that you have come across it before, but it is taken for granted that you have a basic grounding in calculus. In particular, you will need to have a good grasp of the basic rules for differentiation and integration.
18.103 picks up where 18.100B (Analysis I) left off. Topics covered include the theory of the Lebesgue integral with applications to probability, Fourier series, and Fourier integrals.
This is an advanced undergraduate course dealing with calculus in one complex variable with geometric emphasis. Since the course Analysis I (18.100B) is a prerequisite, topological notions like compactness, connectedness, and related properties of continuous functions are taken for granted.
Geometric Combinatorics - MIT
This course offers an introduction to discrete and computational geometry. Emphasis is placed on teaching methods in combinatorial geometry. Many results presented are recent, and include open (as yet unsolved) problems.
Geometry - The Open University
Geometry is concerned with the various aspects of size, shape and space. In this unit, you will explore the concepts of angles, shapes, symmetry, area and volume through interactive activities.
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.
Geometry of Manifolds - MIT
This is a second-semester graduate course on the geometry of manifolds. The main emphasis is on the geometry of symplectic manifolds, but the material also includes long digressions into complex geometry and the geometry of 4-manifolds, with special emphasis on topological considerations.
This course covers the same material as Differential Equations (18.03) with more emphasis on theory. In addition, it treats mathematical aspects of ordinary differential equations such as existence theorems.
In this course on the mathematics of infinite random matrices, students will learn about the tools such as the Stieltjes transform and Free Probability used to characterize infinite random matrices.
Integral Equations - MIT
This course emphasizes concepts and techniques for solving integral equations from an applied mathematics perspective. Material is selected from the following topics: Volterra and Fredholm equations, Fredholm theory, the Hilbert-Schmidt theorem; Wiener-Hopf Method; Wiener-Hopf Method and partial differential equations; the Hilbert Problem and singular integral equations of Cauchy type; inverse scattering transform; and group theory. Examples are taken from fluid and solid mechanics, acoustics, quantum mechanics, and other applications.
Interpreting Data: Boxplots and Tables - The Open University
This unit is concerned with two main topics. In Section 1, you will learn about another kind of graphical display, the boxplot. A boxplot is a fairly simple graphic, which displays certain summary statistics of a set of data. Boxplots are particularly useful for assessing quickly the location, dispersion, and symmetry or skewness of a set of data, and for making comparisons of these features in two or more data sets.
This course teaches techniques for the design and analysis of efficient algorithms, emphasizing methods useful in practice. Topics covered include: sorting; search trees, heaps, and hashing; divide-and-conquer; dynamic programming; amortized analysis; graph algorithms; shortest paths; network flow; computational geometry; number-theoretic algorithms; polynomial and matrix calculations; caching; and parallel computing.
This course introduces the basic computational methods used to understand the cell on a molecular level. It covers subjects such as the sequence alignment algorithms: dynamic programming, hashing, suffix trees, and Gibbs sampling. Furthermore, it focuses on computational approaches to: genetic and physical mapping; genome sequencing, assembly, and annotation; RNA expression and secondary structure; protein structure and folding; and molecular interactions and dynamics.
This course is devoted to the theory of Lie Groups with emphasis on its connections with Differential Geometry.
This course is offered to undergraduates and introduces students to the formulation, methodology, and techniques for numerical solution of engineering problems. Topics covered include: fundamental principles of digital computing and the implications for algorithm accuracy and stability, error propagation and stability, the solution of systems of linear equations, including direct and iterative techniques, roots of equations and systems of equations, numerical interpolation, differentiation and integration, fundamentals of finite-difference solutions to ordinary differential equations, and error and convergence analysis. The subject is taught the first half of the term.
This course provides a solid introduction to Partial Differential Equations for advanced undergraduate students. The focus is on linear second order uniformly elliptic and parabolic equations.
This course provides an elementary introduction to probability and statistics with applications. Topics include: basic probability models; combinatorics; random variables; discrete and continuous probability distributions; statistical estimation and testing; confidence intervals; and an introduction to linear regression.
Introduction to Topology - MIT
This course introduces topology, covering topics fundamental to modern analysis and geometry. It also deals with subjects like topological spaces and continuous functions, connectedness, compactness, separation axioms, and selected further topics such as function spaces, metrization theorems, embedding theorems and the fundamental group.
Linear Algebra - MIT
This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices.
This is a communication intensive supplement to Linear Algebra (18.06). The main emphasis is on the methods of creating rigorous and elegant proofs and presenting them clearly in writing. The course starts with the standard linear algebra syllabus and eventually develops the techniques to approach a more advanced topic: abstract root systems in a Euclidean space.
This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods and tools for solving these PDEs, such as separation of variables, Fourier series and transforms, eigenvalue problems, and Green's functions.
Math 105-Precalculus Mathematics - Capilano University
This is a functions course. Topics include: graphs, properties of functions, mathematical models, trigonometry, logarithms and exponential functions. Primarily for students who anticipate taking science calculus courses or who require a Principles of Math 12 equivalent course.
Math 1210-Calculus I - Dixie State
Designed for students intending to earn an Associate of Science degree and then transfer to a mathematics, engineering program, or other calculus-based major at a four-year institution. Students will gain a basic understanding of calculus, the mathematics of motion and change. Topics include limits and continuity, differentiation, applications of differentiation, integration, applications of integration, derivatives of exponential functions, logarithmic functions, inverse trigonometric functions, hyperbolic functions and related integrals. Students must have a working knowledge of college algebra and trigonometry.
Math 1220 Calculus II - Dixie State
This course is the continuation of MATH 1210. Topics covered includes arc length, area of a surface of revolution, moments and centers of mass, integration techniques, sequences and series, parametrization of curves and polar coordinates, vectors in 3-space, quadric surfaces and cylindrical and spherical coordinates.
Math 230-Calculus III - Capilano University
To extend the language and concepts of the calculus of functions of one independent variable to functions of two or more independent variables. Specifically, the limit concept, continuity, the derivative and the integral will be generalized. The appropriate use of graphing calculators and CAS is explored.
Mathematical Exposition - MIT
This course provides techniques of effective presentation of mathematical material. Each section of this course is associated with a regular mathematics subject, and uses the material of that subject as a basis for written and oral presentations. The section presented here is on chaotic dynamical systems.
Mathematical Language - The Open University
In our everyday lives we use we use language to develop ideas and to communicate them to other people. In this unit we examine ways in which language is adapted to express mathematical ideas.
This graduate-level course is a continuation of Mathematical Methods for Engineers I (18.085). Topics include numerical methods; initial-value problems; network flows; and optimization.
Find out what solid-state physics has brought to Electromagnetism in the last 20 years. This course surveys the physics and mathematics of nanophotonics—electromagnetic waves in media structured on the scale of the wavelength.
Mathematical Statistics - MIT
This graduate level mathematics course covers decision theory, estimation, confidence intervals, and hypothesis testing. The course also introduces students to large sample theory. Other topics covered include asymptotic efficiency of estimates, exponential families, and sequential analysis.
This is an introductory course in Discrete Mathematics oriented toward Computer Science and Engineering. The course divides roughly into thirds: 1. Fundamental Concepts of Mathematics: Definitions, Proofs, Sets, Functions, Relations. 2. Discrete Structures: Modular Arithmetic, Graphs, State Machines, Counting. 3. Discrete Probability Theory.
Maths Everywhere - The Open University
This unit explores reasons for studying mathematics, practical applications of mathematical ideas and aims to help you to recognise mathematics when you come across it. It introduces the you to the graphics calculator, and takes you through a series of exercises from the Calculator Book, Tapping into Mathematics With the TI-83 Graphics Calculator. The unit ends by asking you to reflect on the process of studying mathematics. In order to complete this unit you will need to have obtained a Texas Instruments TI-83 calculator and the book Tapping into Mathematics With the TI-83 Graphics Calculator (ISBN 0201175479).
Measure and Integration - MIT
This graduate-level course covers Lebesgue's integration theory with applications to analysis, including an introduction to convolution and the Fourier transform.
Modeling Displacements and Velocities - The Open University
In this unit you will see first how to convert vectors from geometric form, in terms of a magnitude and direction, to component form, and then how conversion in the opposite sense is accomplished. The ability to convert between these different forms of a vector is useful in certain problems involving displacement and velocity, as shown in Section 2, in which you will also work with bearings.
Modelling Heat Transfer - The Open University
This unit is the fourth in the MSXR209 series of five units on mathematical modelling. In this unit you will be taken through the whole modelling process in detail, from creating a first simple model, through evaluating it, to the subsequent revision of the model by changing one of the assumptions. The problem that will be examined is one based on heat transfer. This unit assumes you have studied Modelling pollution in the Great Lakes (MSXR209_1), Analysing skid marks (MSXR209_2) and Developing modelling skills (MSXR209_3).
Modelling Pollution in the Great Lakes - The Open University
This unit is the first in the MSXR209 series of five units that introduce the idea of modelling with mathematics. This unit centres on a mathematical model of how pollution levels in the Great Lakes of North America vary over a period of time. It demonstrates that, by keeping the model as simple as possible extremely complex systems can be understood and predicted.
Modelling Pollution in the Great Lakes: A Review - The Open University
This is the fifth and final unit in the MSXR209 series on mathematical modelling. In this unit we revisit the model developed in the first unit of this series on pollution in the Great Lakes of North America. Here we evaluate and revise the original model by comparing its predictions against data from the lakes before finally reflecting on the techniques used. This unit assumes you have studied Modelling pollution in the Great Lakes (MSXR209_1), Analysing skid marks (MSXR209_2), Developing modelling skills (MSXR209_3) and Modelling heat transfer (MSXR209_4).
Modelling Static Problems - The Open University
This unit lays the foundation of the subject of mechanics. Mechanics is concerned with how and why objects stay put, and how and why they move. In particular, this unit - Modelling Static Problems - considers why objects stay put. And it assumes that you have a good working knowledge of vectors.
Modelling with First Order Differential Equations - The Open University
This unit lays the foundation of Newtonian mechanics and in particular the procedure for solving dynamics problems. The preresquisite skills needed for this unit are the ability to solve first and second-order differential equations, a knowledge of vectors, and an understanding of the concept of a force.
Modelling with Fourier Series - The Open University
This unit shows how partial differential equations can be used to model phenomena such as waves and heat transfer. The prerequisite requirements to gain full advantage from this unit are an understanding of ordinary differential equations and basic familiarity with partial differential equations.
Multivariable Calculus - MIT
This course covers vector and multi-variable calculus. It is the second semester in the freshman calculus sequence. Topics include vectors and matrices, partial derivatives, double and triple integrals, and vector calculus in 2 and 3-space.
This graduate-level course provides a unified treatment of nonlinear oscillations and wave phenomena with applications to mechanical, optical, geophysical, fluid, electrical and flow-structure interaction problems.
This course provides an introduction to the theory and phenomenology of nonlinear dynamics and chaos in dissipative systems. The content is structured to be of general interest to undergraduates in science and engineering.
Number Systems (M208_6) - The Open University
Number systems and the rules for combining numbers can be daunting. This unit will help you to understand the detail of rational and real numbers, complex numbers and integers. You will also be introduced to modular arithmetic and the concept of a relation between elements of a set.
Numbers (M208_6) - The Open University
This unit will help you understand more about real numbers and their properties. It will explain the relationship between real numbers and recurring decimals, explain irrational numbers and discuss inequalities. The unit will help you to use the Triangle Inequality, the Binomial Theorem and the Least Upper Bound Property.
Numbers, Units and Arithmetic (MU120_4M1) - The Open University
Do fractions and decimals make you apprehensive about maths? Do you lack confidence in dealing with numbers? If so, then this unit is for you. The unit will explain the basics of working with positive and negative numbers and how to multiply and divide with fractions and decimals.
Prices (MU120_2) - The Open University
This unit looks at a wide variety of ways of comparing prices and the construction of a price index. You will also look at the Retail Price Index (RPI) and the Consumer Price Index (CPI), indices used by the UK Government to calculate the percentage by which prices in general have risen over any given period. You will also look at the important statistical and mathematical ideas that contribute to the construction of a price index.
Principles of Applied Mathematics is a study of illustrative topics in discrete applied mathematics including sorting algorithms, information theory, coding theory, secret codes, generating functions, linear programming, game theory. There is an emphasis on topics that have direct application in the real world.
Problem Solving Seminar - MIT
This course, which is geared toward Freshmen, is an undergraduate seminar on mathematical problem solving. It is intended for students who enjoy solving challenging mathematical problems and who are interested in learning various techniques and background information useful for problem solving. Students in this course are expected to compete in a nationwide mathematics contest for undergraduates.
Quantum Computation - MIT
This course provides an introduction to the theory and practice of quantum computation. Topics covered include: physics of information processing, quantum logic, quantum algorithms including Shor's factoring algorithm and Grover's search algorithm, quantum error correction, quantum communication, and cryptography.
This course is an introduction to the basics of random matrix theory, motivated by engineering and scientific applications.
This graduate-level subject explores various mathematical aspects of (discrete) random walks and (continuum) diffusion. Applications include polymers, disordered media, turbulence, diffusion-limited aggregation, granular flow, and derivative securities.
Randomized Algorithms - MIT
This course examines how randomization can be used to make algorithms simpler and more efficient via random sampling, random selection of witnesses, symmetry breaking, and Markov chains. Topics covered include: randomized computation; data structures (hash tables, skip lists); graph algorithms (minimum spanning trees, shortest paths, minimum cuts); geometric algorithms (convex hulls, linear programming in fixed or arbitrary dimension); approximate counting; parallel algorithms; online algorithms; derandomization techniques; and tools for probabilistic analysis of algorithms.
Ratio, Proportion and Percentages - The Open University
From politics to cookery, ratios, proportions and percentages are part of everyday life. This unit is designed to help you become more familiar with how figures can be manipulated, then you can check whether that discount really is as big as they claim!
Real Functions and Graphs - The Open University
Sometimes the best way to understand a set of data is to sketch a simple graph. This exercise can reveal hidden trends and meanings not clear from just looking at the numbers. In this unit you will review the various approaches to sketching graphs and learn some more advanced techniques.
Rounding and Estimation - The Open University
Scientific calculators are a wonderful invention, but they're only as good as the people who use them. If you often get an unexpected – or ridiculous – result when you press the ‘enter’ button, this unit is for you. Learn how to do a calculation correctly and get the right answer every time.
Seminar in Algebra and Number Theory: Computational Commutative Algebra and Algebraic Geometry - MIT
In this undergraduate level seminar series, topics vary from year to year. Students present and discuss the subject matter, and are provided with instruction and practice in written and oral communication. Some experience with proofs required. The topic for fall 2008: Computational algebra and algebraic geometry.
This is a seminar for mathematics majors, where the students present the lectures. No prior experience giving lectures is necessary.
An undergraduate level seminar for mathematics majors. Students present and discuss subject matter taken from current journals or books. Instruction and practice in written and oral communication is provided. The topics vary from year to year. The topic for this term is Applications to Number Theory.
Seminar in Geometry - MIT
In this course, students take turns in giving lectures. For the most part, the lectures are based on Robert Osserman's classic book A Survey of Minimal Surfaces, Dover Phoenix Editions. New York: Dover Publications, May 1, 2002. ISBN: 0486495140.
Seminar in Topology - MIT
In this course, students present and discuss the subject matter with faculty guidance. Topics presented by the students include the fundamental group and covering spaces. Instruction and practice in written and oral communication are provided to the students.
Simplicity Theory - MIT
This is an advanced topics course in model theory whose main theme is simple theories. We treat simple theories in the framework of compact abstract theories, which is more general than that of first order theories. We cover the basic properties of independence (i.e., non-dividing) in simple theories, the characterization of simple theories by the existence of a notion of independence, and hyperimaginary canonical bases.
Single Variable Calculus - MIT
This introductory calculus course covers differentiation and integration of functions of one variable, with applications.
Sqaures, Roots and Powers - The Open University
From paving your patio to measuring the ingredients for your latest recipe, squares, roots and powers really are part of everyday life. This unit reviews the basics of all three and also describes scientific notation, which is a convenient way of writing or displaying large numbers.
This course is a broad treatment of statistics, concentrating on specific statistical techniques used in science and industry. Topics include: hypothesis testing and estimation, confidence intervals, chi-square tests, nonparametric statistics, analysis of variance, regression, correlation, decision theory, and Bayesian statistics.
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
Surfaces - The Open University
Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.
Symmetry - The Open University
We all encounter symmetry in our everyday lives, in both natural and man-made structures. The mathematical concepts surrounding symmetry can be a bit more difficult to grasp. This unit explains such concepts as direct and indirect symmetries, Cayley tables and groups through exercises, audio and video.
Tapping into Mathematics - The Open University
Do you have a graphics or scientific calculator? If so, this unit will help you to understand the different functions and facilities available. With a focus on arithmetic, you will learn what a powerful tool this type of calculator can be.
This seminar focuses on the knowledge and skills necessary for teaching science and engineering in higher education. Topics include: using current research in student learning to improve teaching; developing courses; lecturing; promoting students' ability to think critically and solve problems; communicating with a diverse student body; using educational technology; creating effective assignments and tests; and utilizing feedback to improve instruction. Students research and teach a topic of particular interest. This subject is appropriate for both novices and those with teaching experience.
The Art of Counting - MIT
The subject of enumerative combinatorics deals with counting the number of elements of a finite set. For instance, the number of ways to write a positive integer n as a sum of positive integers, taking order into account, is 2n-1. We will be concerned primarily with bijective proofs, i.e., showing that two sets have the same number of elements by exhibiting a bijection (one-to-one correspondence) between them. This is a subject which requires little mathematical background to reach the frontiers of current research. Students will therefore have the opportunity to do original research. It might be necessary to limit enrollment.
Theory of Computation - MIT
This graduate level course is more extensive and theoretical treatment of the material in Computability, and Complexity (6.045J / 18.400J). Topics include Automata and Language Theory, Computability Theory, and Complexity Theory.
Theory of Numbers - MIT
This course provides an elementary introduction to number theory with no algebraic prerequisites. Topics include primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions and elliptic curves.
Theory of Probability - MIT
This course covers the laws of large numbers and central limit theorems for sums of independent random variables. It also analyzes topics such as the conditioning and martingales, the Brownian motion and the elements of diffusion theory.
The course consists of a sampling of topics from algebraic combinatorics. The topics include the matrix-tree theorem and other applications of linear algebra, applications of commutative and exterior algebra to counting faces of simplicial complexes, and applications of algebra to tilings.
The main aims of this seminar will be to go over the classification of surfaces (Enriques-Castelnuovo for characteristic zero, Bombieri-Mumford for characteristic p), while working out plenty of examples, and treating their geometry and arithmetic as far as possible.
The topics for this course vary each semester. This semester, the course aims to introduce techniques for studying intersection theory on moduli spaces. In particular, it covers the geometry of homogeneous varieties, the Deligne-Mumford moduli spaces of stable curves and the Kontsevich moduli spaces of stable maps using intersection theory.
This course is a first course in algebraic number theory. Topics to be covered include number fields, class numbers, Dirichlet's units theorem, cyclotomic fields, local fields, valuations, decomposition and inertia groups, ramification, basic analytic methods, and basic class field theory. An additional theme running throughout the course will be the use of computer algebra to investigate number-theoretic questions; this theme will appear primarily in the problem sets.
The goal of this course is to describe some of the tools which enter into the proof of Sullivan's conjecture.
In this graduate-level course, we will be covering advanced topics in combinatorial optimization. We will start with non-bipartite matchings and cover many results extending the fundamental results of matchings, flows and matroids. The emphasis is on the derivation of purely combinatorial results, including min-max relations, and not so much on the corresponding algorithmic questions of how to find such objects. The intended audience consists of Ph.D. students interested in optimization, combinatorics, or combinatorial algorithms.
Topics in Geometry - MIT
This is an introductory (i.e. first year graduate students are welcome and expected) course in generalized geometry, with a special emphasis on Dirac geometry, as developed by Courant, Weinstein, and Severa, as well as generalized complex geometry, as introduced by Hitchin. Dirac geometry is based on the idea of unifying the geometry of a Poisson structure with that of a closed 2-form, whereas generalized complex geometry unifies complex and symplectic geometry. For this reason, the latter is intimately related to the ideas of mirror symmetry.
This course covers harmonic theory on complex manifolds, the Hodge decomposition theorem, the Hard Lefschetz theorem, and Vanishing theorems. Some results and tools on deformation and uniformization of complex manifolds are also discussed.
This graduate-level course focuses on one-dimensional nonparametric statistics developed mainly from around 1945 and deals with order statistics and ranks, allowing very general distributions.
The main goal of this course is to study the generalization ability of a number of popular machine learning algorithms such as boosting, support vector machines and neural networks. Topics include Vapnik-Chervonenkis theory, concentration inequalities in product spaces, and other elements of empirical process theory.
We will discuss numerous research problems that are related to the internet. Sample topics include: routing algorithms such as BGP, communication protocols such as TCP, algorithms for intelligently selecting a resource in the face of uncertainty, bandwidth sensing tools, load balancing algorithms, streaming protocols, determining the structure of the internet, cost optimization, DNS-related problems, visualization, and large-scale data processing. The seminar is intended for students who are ready to work on challenging research problems.
This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic.
Using Vectors to Model - The Open University
This unit introduces the topic of vectors. The subject is developed without assuming you have come across it before, but the unit assumes that you have previously had a basic grounding in algebra and trigonometry, and how to use Cartesian coordinates for specifying a point in a plane.
Vectors and Conics - The Open University
Attempts to answer problems in areas as diverse as science, technology and economics involve solving simultaneous linear equations. In this unit we look at some of the equations that represent points, lines and planes in mathematics. We explore concepts such as Euclidean space, vectors, dot products and conics.
Wave Propagation - MIT
This course discusses the Linearized theory of wave phenomena in applied mechanics. Examples are chosen from elasticity, acoustics, geophysics, hydrodynamics and other subjects. The topics include: basic concepts, one dimensional examples, characteristics, dispersion and group velocity, scattering, transmission and reflection, two dimensional reflection and refraction across an interface, mode conversion in elastic waves, diffraction and parabolic approximation, radiation from a line source, surface Rayleigh waves and Love waves in elastic media, waves on the sea surface and internal waves in a stratified fluid, waves in moving media, ship wave pattern, atmospheric lee waves behind an obstacle, and waves through a laminated media.
Wavelets are localized basis functions, good for representing short-time events. The coefficients at each scale are filtered and subsampled to give coefficients at the next scale. This is Mallat's pyramid algorithm for multiresolution, connecting wavelets to filter banks. Wavelets and multiscale algorithms for compression and signal/image processing are developed. Subject is project-based for engineering and scientific applications.
Working on Your Own Mathematics - The Open University
This unit focuses on your initial encounters with research. It invites you to think about how perceptions of mathematics have influenced you in your prior learning, your teaching and the attitudes of learners.