groups, rings and modules and algebras and representation … · groups, rings and modules and...
Post on 28-Jul-2020
25 Views
Preview:
TRANSCRIPT
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 Gordonigordon@ed.ac.uk
School of Mathematics, University of Edinburgh
Perth 5 Oct 2017
Groups, Rings andModules
andAlgebras andRepresentation
Theory
Iain Gordon
The Algebra Team
Subject Matter
Content of theCourse
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:igordon@ed.ac.uk
Groups, Rings andModules
andAlgebras andRepresentation
Theory
Iain Gordon
The Algebra Team
Subject Matter
Content of theCourse
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:
colva@mcs.st-and.ac.uk
Groups, Rings andModules
andAlgebras andRepresentation
Theory
Iain Gordon
The Algebra Team
Subject Matter
Content of theCourse
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:
mq3@st-andrews.ac.uk
Groups, Rings andModules
andAlgebras andRepresentation
Theory
Iain Gordon
The Algebra Team
Subject Matter
Content of theCourse
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
I E-mail:ellen.henke@abdn.ac.uk
Groups, Rings andModules
andAlgebras andRepresentation
Theory
Iain Gordon
The Algebra Team
Subject Matter
Content of theCourse
Greg Stevenson
I University: Glasgow
I Research interests: tensorand triangulatedcategories
I Web page:www.maths.gla.ac.uk
/∼gstevenson/I E-mail:
Gregory.Stevenson@glasgow.ac.uk
Groups, Rings andModules
andAlgebras andRepresentation
Theory
Iain Gordon
The Algebra Team
Subject Matter
Content of theCourse
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
I E-mail:charles.strickland-constable@ed.ac.uk
Groups, Rings andModules
andAlgebras andRepresentation
Theory
Iain Gordon
The Algebra Team
Subject Matter
Content of theCourse
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/
I E-mail:l.ciobanu@hw.ac.uk
Groups, Rings andModules
andAlgebras andRepresentation
Theory
Iain Gordon
The Algebra Team
Subject Matter
Content of theCourse
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.
Groups, Rings andModules
andAlgebras andRepresentation
Theory
Iain Gordon
The Algebra Team
Subject Matter
Content of theCourse
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,. . . .
Groups, Rings andModules
andAlgebras andRepresentation
Theory
Iain Gordon
The Algebra Team
Subject Matter
Content of theCourse
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)
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).
top related