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Lecture Notes
Algebra and Computation
These are the lecture notes by Madhu Sudan on the course Algebra and Computation, offered in MIT in 1999.
Real-time TrafficAlgebra | Algorithms | Computation |
| Added | 26 Jan 2009 |
| Updated | 26 Jan 2009 |
| Year | 1999 |
| Authors | Madhu Sudan |
The Table of Contents
Lecture 1: Overview; Preliminaries.
Lecture 2: Polynomials; The factorization and GCD problems.
Lecture 3: Polynomials and error-correcting codes. The Euclidean algorithm for GCD. Applications. Resultant.
Lecture 4: Properties of the resultant. Applications. Towards factorization of polynomials: Finding square roots modulo a prime.
Lecture 5: Factorization of univariate polynomials over finite fields.
Lecture 6: Berlekamp's deterministic algorithm for factoring univariate polynomials. Existence, density and properties of irreducible polynomials.
Lecture 7: Factoring bivariate polynomials. Hensel's lifting.
Lecture 8: Factoring bivariate polynomials (contd.). Digression: Applications to error-correction algorithms.
Lecture 9: Irreducibility testing; Black-box factoring of multivariate polynomials.
Lecture 10: Factoring polynomials over the rationals; Reduction to basis reduction.
Lecture 11: LLL's Basis reduction algorithm.
Lecture 12: (2nd Phase of course) Ideals and Varieties. Division in Ideals; Groebner bases.
Lecture 13: Construction and uniqueness of Groebner bases. Solution to the Ideal Membership problem.
Lecture 14: Upper bound on the degrees for ideal generation. Complexity lower bound.
Lecture 15: Varieties. Emptiness of a variety. Elimination. Hilbert's Nullstellensatz.
Lecture 16: Strong form of Hilbert's Nullstellensatz. Quantifier Elimination.
Lecture 17: Quantifier elimination (contd.); Bezout's Thm.
Lecture 18: Bezout's Thm. and some applications.
Lecture 19: Algebraic models of computation; Ben-Or Cleve result.
Lecture 20: Blum-Shub-Smale model of computation (contd.) Undecidability of Mandelbrot Set.
Lecture 21: Algebaic settings for the P=NP question
Lecture 22: An Arthur-Merlin proof for the Hilbert's Nullstellensatz.
Lecture 23: Non-uniform lower bounds; Linear independence; algebraic independence; Strassen's degree bound.
Lecture 24: Ben-Or's lower bounds based on no. of connected components; Survey of other methods: Volume; Euler characteristic and Betti numbers.
Lecture 25: Mulmuley's Algebraic PRAM without bit operations. Lower bounds for LP and Max Flow.
Lecture 26: Summary. What we didn't cover ..
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