Groups, Rings and Modules and Algebras and Representation … · Groups, Rings and Modules and Algebras and Representation Theory Iain Gordon The Algebra Team Subject Matter Content

Groups, Rings andModules

andAlgebras andRepresentation

Theory

Iain Gordon

The Algebra Team

Subject Matter

Content of theCourse

Groups, Rings and Modulesand

Algebras and Representation Theory

Iain [email protected]

School of Mathematics, University of Edinburgh

Perth 5 Oct 2017



Theory

Iain Gordon

The Algebra Team

Subject Matter


Iain Gordon (Stream leader)

I University: Edinburgh

I Research interests:Geometric RepresentationTheory and applicationsto Algebraic Geometryand to Combinatorics

I Web page:http://www.maths.ed.ac.uk/∼igordon/

I E-mail:[email protected]



Theory

Iain Gordon

The Algebra Team

Subject Matter


Colva Roney-Dougal

I University: St Andrews

I Research interests:symmetry and inference,permutation groups,matrix groups, constraintsatisfaction

I Web page:www-groups.mcs.st-and.ac.uk

/∼colva/I E-mail:

[email protected]



Theory

Iain Gordon

The Algebra Team

Subject Matter


Martyn Quick

I University: St Andrews

I Research interests: grouptheory, finite and infinite

I Web page:www-groups.mcs.st-and.ac.uk

/∼martyn/index.htmlI E-mail:

[email protected]



Theory

Iain Gordon

The Algebra Team

Subject Matter


Ellen Henke

I University: Aberdeen

I Research interests: Finitegroup theory, particularlyfusion systems

I Web page:www.maths.abdn.ac.uk/ncs/

people/profiles/ellen.henke




Theory

Iain Gordon

The Algebra Team

Subject Matter


Greg Stevenson

I University: Glasgow

I Research interests: tensorand triangulatedcategories

I Web page:www.maths.gla.ac.uk

/∼gstevenson/I E-mail:

[email protected]



Theory

Iain Gordon

The Algebra Team

Subject Matter


Charlie Strickland-Constable

I University: Edinburgh

I Research interests:generalised geometriesand mathematical physics

I Web page:http://www.maths.ed.ac.uk/

school-of-mathematics/people?person=590




Theory

Iain Gordon

The Algebra Team

Subject Matter


Laura Ciobanu

I University: Heriot-Watt

I Research interests:Combinatorial, geometricand algorithmic grouptheory. Combinatoricsand theoretical computerscience.

I Web page:http://www.macs.hw.ac.uk/∼lc45/




Theory

Iain Gordon

The Algebra Team

Subject Matter


What is Algebra?

I Traditionally: Theory of polynomials and solvingequations.

I 19th, 20th Centuries: Theory of various abstractalgebraic structures.

I Algebraic structure: A set with some operations definedon it.



Theory

Iain Gordon

The Algebra Team

Subject Matter


Areas of Algebra

Division according to the number of operations and theirproperties.

I Classical structures:I Groups, rings, fieldsI Linear spaces, modulesI Algebras, Lie algebras

I ‘Modern’ structures:I Lattices, semigroups, general/universal algebras,

boolean algebras, quasigroups, semirings, Hopf algebras,vertex operator algebras, differential gradedalgebras,. . . .



Theory

Iain Gordon

The Algebra Team

Subject Matter


Content

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)



Theory

Iain Gordon

The Algebra Team

Subject Matter


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:



Theory

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The Algebra Team

Subject Matter


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:



Theory

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The Algebra Team

Subject Matter


Noncommutative rings (5 lectures)

I Topics:I Finitely generated modules over principal ideal domains

and applicationsI The Artin–Wedderburn theorem

I Lecturers:



Theory

Iain Gordon

The Algebra Team

Subject Matter


Representation theory (5 lectures)

I Topics:I Representations and charactersI Orthogonality relationsI Induced representationsI Computing character tablesI Applications

I Lecturer:



Theory

Iain Gordon

The Algebra Team

Subject Matter


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.



Theory

Iain Gordon

The Algebra Team

Subject Matter


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).

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Groups, Rings and Modules and Algebras and Representation … · Groups, Rings and Modules and Algebras and Representation Theory Iain Gordon The Algebra Team Subject Matter Content