groups, rings and modules and algebras and representation … · groups, rings and modules and...
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![Page 1: 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](https://reader035.vdocuments.us/reader035/viewer/2022081504/5f7a90e56cb8fd2fbe2278b4/html5/thumbnails/1.jpg)
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
![Page 2: 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](https://reader035.vdocuments.us/reader035/viewer/2022081504/5f7a90e56cb8fd2fbe2278b4/html5/thumbnails/2.jpg)
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:[email protected]
![Page 3: 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](https://reader035.vdocuments.us/reader035/viewer/2022081504/5f7a90e56cb8fd2fbe2278b4/html5/thumbnails/3.jpg)
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:
![Page 4: 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](https://reader035.vdocuments.us/reader035/viewer/2022081504/5f7a90e56cb8fd2fbe2278b4/html5/thumbnails/4.jpg)
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:
![Page 5: 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](https://reader035.vdocuments.us/reader035/viewer/2022081504/5f7a90e56cb8fd2fbe2278b4/html5/thumbnails/5.jpg)
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:[email protected]
![Page 6: 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](https://reader035.vdocuments.us/reader035/viewer/2022081504/5f7a90e56cb8fd2fbe2278b4/html5/thumbnails/6.jpg)
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:
![Page 7: 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](https://reader035.vdocuments.us/reader035/viewer/2022081504/5f7a90e56cb8fd2fbe2278b4/html5/thumbnails/7.jpg)
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:[email protected]
![Page 8: 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](https://reader035.vdocuments.us/reader035/viewer/2022081504/5f7a90e56cb8fd2fbe2278b4/html5/thumbnails/8.jpg)
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:[email protected]
![Page 9: 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](https://reader035.vdocuments.us/reader035/viewer/2022081504/5f7a90e56cb8fd2fbe2278b4/html5/thumbnails/9.jpg)
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.
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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,. . . .
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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)
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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:
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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:
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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:
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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:
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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.
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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).