MT6264 Algebra - I

(January 2022) Ph.D. course at IISER Pune

Instructor: Dr Anupam Singh

Schedule : TBA

Evaluation : mid-semester exam 30 % end-semester exam 30 % tests 20+20 %

Prerequisite : enthusiasm towards mathematics

Goal of the course : This course is a compulsory part of coursework to get a Ph.D. degree at IISER Pune. The course aims to revise basic algebra learned during the undergraduate degree programme and introduces topics that will be useful towards research in mathematics.

IISER Pune Maths PhD webpage: http://sites.iiserpune.ac.in/~mathphd/

Proposed course content :

Group Theory

• Examples, category of groups, Action of a group on a set. Subgroups, isomorphism theorems. Group actions: Permutation representations, action on itself by left multiplication, action on itself by conjugation.

• Automorphisms: Inner automorphisms, automorphism groups of some finite groups: dihedral, quaternions, cyclic. Statement of Sylow’s theorem, Direct and Semidirect products. Simple groups, composition series, Jordan-Hölder Series, A_n is simple.

• Category Theory, Objects, morphisms, functors.

• Free groups: words, construction, and uniqueness. Universal property, adjointness with forgetful functor. Finitely generated and finitely presented groups.

Rings and Modules

• Definitions (review): integral domains, euclidean domains, pid, ufd, fields. Examples: Polynomials rings, Matrix rings, group rings. Ideals and Quotient rings, prime and maximal ideals. Chinese Reminder Theorem. Nilradical and Jacobson radical.

• Modules, Definition, Z-modules, F[x]-modules. Direct sums and free modules - construction and universal property.

Linear Algebra :

• Bilinear Forms: Symmetric forms. Orthogonal bases, ordered fields, Gram Schmidt, Sylvester’s theorem. Eigen vectors of linear maps, Spectral theorem (Hermitian, Unitary, Symmetric case). Structure theorem for alternating forms.

• Tensors: Tensor products of modules. Examples. Universal property, Adjointness with Hom. Tensor product of homomorphisms, associativity, symmetry, tensor product of algebras.

• Symmetric and Exterior algebras: Linear functions on tensor products of vector spaces, determinants. Symmetric algebras, universal properties, alternating algebras, universal properties, symmetric and alternating tensors.

• Modules over a PID and Canonical forms: Structure of finitely generated modules over a PID. Canonical forms. Rational Canonical Form. Jordan Canoncial Form.

Textbooks and references :

• • Dummit & Foote: Abstract Algebra.

  • Hungerford: Algebra.

  • Lang: Algebra.

  • Bourbaki: Algebra.

  • Alperin & Bell: Groups and Representations.

Class begins from 17 January

18/01/22 Groups, homomorphism, subgroups, quotient group, cosets, Lagrange's theorem

19/01/22 Group action, Class Equation, and applications, Cayley's theorem

Assignment 1

25/01/22 Sylow's Theorem, applications, classifying groups

26/01/22 Tutorial

01/02/22 Free basis and groups

02/02/22 Presentation, direct product, and free product

08/02/22 Category, universal objects, products, co-products

09/02/22 representations, functors, Cayley graph

Test I on 15/02/22

15/02/22 Rings and Modules, rings of fraction.

16/02/22 Tutorial

22/02/22 Modules of fractions

23/02/22 Tutorial

01/03/22 Structure of modules over PID - free, torsion free

02/03/22 Structure of modules over PID

07 - 22 March 2022 Mid-sem exam + mid sem break

Mid sem exam 15/03/22

22/03/22 Vector space, linear transformation

23/03/22 Tutorial

29/03/22 Primary decomposition, diagonalisation, upper triangulation

30/03/22 Jordan-Chevalley decomposition, Jordan canonical forms

05/04/22 Rational Canonical forms

06/04/22 Tutorial

12/04/22 Bilinear forms, classification of alternating forms

13/04/22 Quadratic forms, Classification of symmetric forms

19/04/22 Spectral theorem

20/04/22 Tutorial

26/04/22 Tensor algebra, symmetric tensor algebra

27/04/22 Alternating tensor algebra, symmetric and alternating tensors

03/05/22 Holiday

04/05/22 Tutorial

End sem exam