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.