### Math 8300: Topics in Algebra (2015--2016)

#### Instructor:

Dr. Dihua Jiang
Office: VinH 224, Telephone: 625--7532, E-mail: dhjiang@math.umn.edu
#### Lectures:

Lecture: 2:30--3:20pm, MWF VH 209 (Office Hours: by appointment)
#### Course Description:

This is a two-semester course for Transformation Groups, Representation Theory, and Harmonic Analysis

Let X be an affine space or a vector space with an action of a group G.

There are many fundamental problems connected to such a pair (X,G).

In Algebraic Aspect:

1) We may consider the action of G on the space of polynomials, P(X),

over the affine space X. How to decompose the space P(X) as a

G-module? Can one tell the multiplicity in terms of the geometry of

the pair (X,G)?

2) If the multiplicity is one, what can one say about the geometry of

the pair (X,G)? How to classify all the pairs (X,G) with the multiplicity

one property? Good candidates of such examples are from the so called

affine symmetric spaces or more generally spherical varieties.

In Arithmetic Aspect:

What happens if one consider the base fields to be number fields,

or local fields? In principle, the Galois cohomology plays important roles

for classification of the rational structures. We will discuss many examples,

instead the general theory.

In Harmonic Analysis Aspect:

One may consider to extend harmonic analysis and representation theory

for groups to the setting of the pairs (X,G). One may also

consider to extend the theory of automophic forms and the Langlands

program from the reductive algebraic groups to homogeneous spaces.

In the Fall, 2015, we discuss the algebraic aspect and arithmetic

aspect of the theory, with many examples.

In the Spring, 2016, we continue with the arithmetic aspect of the

theory and discuss various approaches to extend the harmonic

analysis and representation theory from groups to homogeneous spaces.

References are for the Algebraic Aspect:

(1) Linear Algebraic Groups. By T. A. Springer

ISBN: 978-0-8176-4839-8 (Print) 978-0-8176-4840-4 (Online)

(2)Linear Algebraic Monoids. By Lex E. Renner

ISBN: 978-3-540-24241-3 (Print) 978-3-540-27556-5 (Online)

(3) Homogeneous Spaces and Equivariant Embeddings. By D.A. Timashev

ISBN: 978-3-642-18398-0 (Print) 978-3-642-18399-7 (Online)
#### Homework and Exams:

Homework Problems will be assigned, but no exams are required. Students may give reports to the class.