Css 307 1 Algebra And

Agenda: [Introduction] Administrivia and course structure, introducing groups and actions via permutation puzzles. Agenda: [Polynomial multiplication] Adjoining artificial roots of unity, the Schönhage-Strassen algorithm for polynomial ... Agenda: [Fast Fourier Transform] The Fast Fourier Transform algorithm, and application to polynomial multiplication over "nice" ... Agenda: [Group theory basics] Crash course on basic group theory and a SageMath demo. Agenda: [Applications of fast PolyMult] Finding quotients and remainders efficiently, general multipoint evaluations, and ... Agenda: [Bivariate factorisation - I] Proof of CRT, Gauss Lemma, Resultants and bivariate GCD.

Agenda: [Factorising integer polynomials - II] Gram-Schmidt orthogonalisation, and the Lenstra-Lenstra-Lovasz algorithm to find ... Agenda: [Commutators and conjugates] Understanding commutators and conjugates; solving a Rubik's cube (and other perm. Agenda: [Towers of recognisable subgroups] Subnormality, solvability, nilpotence tests, Schreier's lemma and application to GI for ... Agenda: [Set-stabilisers and blocks] Finishing up GI for bounded colour multiplicity. Set-stabilizers and blocks of imprimitivity. Agenda: [Algorithms for perm. groups] Computing orbits and stabilisers efficiently, towards membership testing. Agenda: [Univariate factorisation] Repeated factors and derivatives, distinct degree factorisation and the Cantor-Zassenhaus ...

Agenda: [Membership test + more] Efficient membership test for permutation groups, and other algos. Revisiting Graph ... Agenda: [Factorising integer polynomials - I] Bounds on coefficient sizes of factors of integer polynomials, adapting bivariate ... Agenda: [Divide and conquer methods] Finding blocks, and using blocks for divide and conquer methods, overview of trivalent ...