Groups, Rings and Modules and Algebras and Representation Theory Iain Gordon The Algebra Team Subject Matter Content of the Course Groups, Rings and Modules and Algebras and Representation Theory Iain Gordon [email protected]School of Mathematics, University of Edinburgh Perth 5 Oct 2017
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In this course we will concentrate on the classical structures:
Groups, Rings and Modules
I Part 1: Groups (5 lectures)
I Part 2: Commutative Rings (5 lectures)
Algebras and Representation Theory
I Part 1: Noncommutative Algebra (5 lectures)
I Part 2: Representation theory (5 lectures)
Groups, Rings andModules
andAlgebras andRepresentation
Theory
Iain Gordon
The Algebra Team
Subject Matter
Content of theCourse
Groups (5 lectures)
I Topics:I Simple groups, Jordan–Holder theorem, direct and
semidirect productsI Permutation representations and group actionsI Sylow Thorems and applicationsI Abelian, soluble and nilpotent groupsI Free groups and presentations
I Lecturers:
Groups, Rings andModules
andAlgebras andRepresentation
Theory
Iain Gordon
The Algebra Team
Subject Matter
Content of theCourse
Commutative rings (5 lectures)
I Topics:I Modules: introductionI Chain conditions and Hilbert’s basis theoremI Fields and numbersI Affine algebraic geometry and Hilbert’s Nullstellensatz
I Lecturers:
Groups, Rings andModules
andAlgebras andRepresentation
Theory
Iain Gordon
The Algebra Team
Subject Matter
Content of theCourse
Noncommutative rings (5 lectures)
I Topics:I Finitely generated modules over principal ideal domains
and applicationsI The Artin–Wedderburn theorem
I Lecturers:
Groups, Rings andModules
andAlgebras andRepresentation
Theory
Iain Gordon
The Algebra Team
Subject Matter
Content of theCourse
Representation theory (5 lectures)
I Topics:I Representations and charactersI Orthogonality relationsI Induced representationsI Computing character tablesI Applications
I Lecturer:
Groups, Rings andModules
andAlgebras andRepresentation
Theory
Iain Gordon
The Algebra Team
Subject Matter
Content of theCourse
Prerequisites
You should be familiar and comfortable with:
I Basic linear algebra
I Definitions and examples of groups, rings, fields
I Basic algebra concepts such as homomorphisms
I Basic notions of group theory: permutations, symmetricgroups, Lagrange’s theorem, normal subgroups andfactor groups
If you want to join in 2nd term you should know:
I The notion of a module and related concepts.
I Basics on Noetherian and Artinian modules.
I Some commutative algebra, in particular the notion of aprincipal ideal domain.
Groups, Rings andModules
andAlgebras andRepresentation
Theory
Iain Gordon
The Algebra Team
Subject Matter
Content of theCourse
Other Details
I Lecture time: Mondays 1pm–3pm
I First lecture: next Monday, 9 Oct, from St Andrews
I Tutorial and IT support: this is arranged locally
I Assessment: continuous; four take-home sets ofproblems (two in each term).