### 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,

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.