Math Video Lectures
Abstract Algebra (Harvard)
Algebra is the language of modern mathematics. This course introduces students to that language through a study of groups, group actions, vector spaces, linear algebra, and the theory of fields.
Review of linear algebra. Permutations. Quotient groups, first isomorphism theorem. Abstract linear operators and how to calculate with them. Orthogonal groups. Isometrics of plane figures. Group actions. A5 and the symmetries of an icosahedron. Rings. Extensions of rings. Euclidean domains, PIDs, UFDs. Structure of ring of integers in a quadratic field.
Sets, Counting, and Probability (Harvard)
This online math course develops the mathematics needed to formulate and analyze probability models for idealized situations drawn from everyday life. Topics include elementary set theory, techniques for systematic counting, axioms for probability, conditional probability, discrete random variables, infinite geometric series, and random walks. Applications to card games like bridge and poker, to gambling, to sports, to election results, and to inference in fields like history and genealogy, national security, and theology. The emphasis is on careful application of basic principles rather than on memorizing and using formulas.
Probability, Intuition, and Axioms. Probability by Counting and Inclusion-Exclusion. Principles of Counting. Conditional Probability. Conditional Craps. Lying Witnesses and Simpson's Paradox. Random Variables. Expectation. Tartan Dice; Terminated Geometric; World Series Pitchers; Negative Hypergeometric; Coin Tossing. Gambling. Expected Lead Time; Bijections between Paths. Variables. Inequality.
A Beautiful Mind: Genius, Madness, Reawakening: John Nash
Dr. Sylvia Nasar, the author of "A Beautiful Mind" tells the extraordinary story of mathematician John Nash a drama about the mystery of the human mind and shares some of her experiences in writing her prize-winning biography. "A Beautiful Mind" won the 1998 National Book Critics Circle Award for Biography. It was also a finalist for the Pulitzer Prize, the Helen Bernstein Journalism Award, and the Rhone Poulenc Prize for Science Writing and, most recently, was honored by the three leading American mathematics societies. Translated into more than a half dozen foreign languages, "A Beautiful Mind" was adapted for the screen and released as feature film in 2001, directed by Ron Howard and starring Russell Crowe.
Kurt Gödel and the Limits of Mathematics
Kurt Gödel was one of the foremost mathematicians and logicians of the 20th century, best known for his famous incompleteness theorem, which tells us that there are mathematical 'blind spots': parts of mathematics that traditional methods of proof cannot access. The theorem has far-reaching consequences for computing and even for our understanding of the nature of the human mind. Mark Colyvan from the University of Sydney introduces us to this strange and paradoxical result.
Quaternion Julia Sets. Diamond of Quaternionic Julia Sets. Air on the Dirac Strings. A Volume of Julia Sets. Dynamics of the Quaternion. Interactive Visualization of Quaternion Julia Sets. Quaternion Handshake.
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