
Hecke Algebras and Orthogonal
Polynomials

Research Focus Group





Summer/Fall 2004
Review Session in algebra for the
Prelim Exam (2004)
for graduate students.
Sept 810, Sept 13 (WednesdayFriday, Monday)
We will have 2 hours of lecture/review in the mornings
from 10 AM  noon, in 693 Kerr,
and problem sessions in the afternoon, from 3 PM  5 PM.
Check back here for details.
Click here for a .ps version of some
practice problems,
or in
pdf
which have just been updated, with a new section on category theory
(and a typo fixed).
Please do as many as you can by the afternoon sessions.
There YOU will present solutions and have a chance to ask questions.
Solutions will be posted here, but only AFTER you've had some time
to solve these on your own.
Thanks everyone for participating! Here are some solutions to
section 1 (groups)
,
section 5 (groups)
,
section 4 (modules)
,
section 3 (fields and galois theory).

More info on the exam:
The algebra portion
of the exam will be based on the syllabus for Math 250A,B,C. A
short course description and more detailed outline are below, along w/
suggested texts.
Course
Description for 250ABC: Group and rings. Sylow theorems, abelian
groups, JordanHolder
theorem. Rings, unique factorization. Algebras, and modules. Fields and vector spaces over fields.
Field extensions. Commutative rings. Representation theory and its
applications.
TOPICAL OUTLINE  sept 8, 9
(probably):
Group Theory: groups as symmetries,
homomorphisms, subgroups and quotient groups, group actions, abelian groups.
Basic concepts including group actions, Sylow's theorems, classifications of
finite abelian groups and groups of small order, the JordanHolder theorem.
Introduction to rings, ring
homomorphisms and ideals, field of fractions of a commutative domain; polynomial
rings; factorial rings; principal ideal domains and Euclidean domains;
polynomial extensions of factorial domains. Unique factorization. Free
groups, group representations.
TOPICAL OUTLINE  sept 10, 13
(probably):
Modules and vector spaces. Dual
vector spaces, multilinear functions, determinants, bilinear forms, tensor
products and tensor algebras, the
classification of finitely generated abelian groups and endomorphisms of finitedimensional
vector spaces, field theory and Galois theory. Time permitting also
category theory.
texts:
250ABC Textbook: Abstract algebra,
Dummit and Foote.
Additional reference books:
Artin, Algebra.
Rose, A course on
group theory.
Dixon, Problems
in group theory.
Hall, The theory
of groups.