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MTH 577: Computational Algebra

Term: Fall 2012

This is a graduate course in computational algebra.  Computational Algebra is not necessary computer algebra as preceived  by some people. Instead, it is more the study of computational methods in the study of algebra and related areas.

Goal:  The purpose of this course is to provide the necessary background on computational techniques of algebra for the working algebraist.  Graduate students working in algebra or related areas will find this course very useful. Moreover, research mathematicians, engineers, physicists, might find parts of this course helpful.

Our goal will be to focus on computational techniques for some of the classical problems that have determined the development of algebra and computationally are still relevant today.

Prerequisites:  A solid background of abstract algebra is required. MTH 571-572 should be enough.  Please talk to me before registering for the course if you have any doubts about the requirements.

TextBook:
• Computational Algebra, T. Shaska,  Lecture Notes
Use of these lecture notes:  These lecture notes will be available sometime in the Summer 2013.  The topics here should be used selectively from the instructor having in mind her/his research interests or the background of the students.  They are initially intended for students who have had at least two years of graduate algebra.

Contents

Preface

Part I: Computational Group Theory

1. Preliminaries
• An introduction to GAP
• Group actions
• Orbits and stabilizers
• Random elements
2. Permutation groups
• Stabilizer chains
• Natural actions and decompositions
• Primitive groups
• The O'Nan-Scott theorem
• Composition series
3. Finitely presented groups
• Free groups and presentations
• Tietze transformations
• Quotient subgroups
• Coset enumeration
4. Extension of groups
• Group extensions
• Central extensions
• Cohomology
• An application: Determining the automorphism group of a cyclic curve.
5. Representation Theory
• Characters of finite groups
• Character tables
6. Matrix Groups
• Composition series
• Aschbacher's theorem

Part II:  Algebraic Equations

• Polynomials
• Factorization of polynomials
• Discriminant
• Resultants
• Decomposition of polynomials
• Univariate polynomial equations, Galois theory
• Solving the quintic
• Computing the Galois groups of polynomials
• Constructive Galois Theory
• Systems of polynomial equations, Grobner bases
• Introduction to Grobner bases
• Ordering on the monomials of k[x]
• Monomial ideals and Dickson's lemma
• Hilbert's basis theorem and Grobner bases
• Buchberger's algorithm
• Elimination theory
• Extension theorem
• Implicitization
• Singular points
• Resultants and extension theorem
• An application: Computing the equation for moduli spaces of coverings.

Part III:  Invariant Theory

• Introduction to the theory of invariants
• Classical invariant theory
• Invariants of binary forms
• Invariant of ternary forms
• Computational invariant theory
• An application: Invariants of binary sextics
• Invariants of finite groups

• Ideals

Part V: Computational Algebraic Geometry

• Projective algebraic geometry
• The projective space and projective varieties
• Projective closure
• Hypersurfaces
• Bezout's theorem
• Dimension of varieties
• The Hilbert function and the dimension of the variety
• Properties of dimension
• Dimension and nonsingularity
• The tangent cone
• Curves and their Jacobians
• Singularities
• Plucker formulas
• Divisors and the Picard group
• Abel's theorem
• Theta functions of algebraic curves
• Classical theory of theta functions
• Picard's formula and other hyperelliptic curves
• Thomae's formula for hyperelliptic curves
• Determining the fundamental theta functions
• Computing in the Jacobian
• Theta functions and integrable systems
• An application to differential equations
• Moduli spaces of coverings
• Coverings of the Riemann sphere and their ramification
• Braid action and Nielsen classes
• Symmetrized Hurwitz spaces
• Solving the system of equations: Grobner bases versus resultants
• An example
• Algebraic surfaces
• xx
• Toric varieties