Module Theory (IISER, Pune)
MTH 390 January-April 2009
Reference :
Further Linear Algebra : Blyth and Robertson
Algebra : Dummit and Foote
Advanced Linear Algebra : Ronan
On the Conjugacy Classes in the Unitary, Symplectic and Orthogonal groups : G. E. Wall, J. Austr. Math. Soc. 3 (1963) 1-62
Numbers of Conjugacy Classes in Some Finite Classical Groups : I. G. Macdonal Bull. Austr. Math. Soc. vol 23 (1981), 23-48
This is an elective course offered this semester.
Course Content :
Modules, Structure Theory of Modules over PID, Application to Conjugacy classes in GL_n(F), Rational and Jordan Canonical Forms, Conjugacy Classes in Orthogonal Groups etc.
Time : It's a 3 credit course. We will meet 2 hours a week.
Prerequisite : Group Theory, Vector Spaces.
Grading Method : One end-sem exam + regular presentations + a project.
Topics Covered :
Revision of Vector Spaces and Linear Transformation, Conjugacy in GL_n, Similarity of Linear Transformation, Modules over a ring, Vector space as k[X]-module for a given L.T., Jordon Canonical Forms, Nilpotent Transformation, Semisimple transformation, Jordan Decomposition of a Linear Transformation, Canonical Forms, Structure Theory of Modules over PID, Revisit to Jordan Canonical and rational Canonical Forms.
Reports on the content of the course by students:
LakshmiPriya - Jordon Canonical Forms via Module Theory
Kapil Gupta - Cayley-Hamilton Theory using Structure Theorem of Modules over PID
Avi Prasanna - Triangular Forms and Jordon Canonical Forms
Arvind - Free Modules over Z, Free Modules over PID