MAGIC071: Erlangen program in geometry and analysis: SL(2,R) case study

Course details

Semester

Autumn 2020
Monday, October 5th to Friday, December 11th

Hours

Live lecture hours
10
Recorded lecture hours
0
Total advised study hours
40

Timetable

Tuesdays
10:05 - 10:55

Course forum

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Description

The Erlangen program of F.Klein (influenced by S.Lie) defines geometry as a study of invariants under a certain group action. This approach proved to be fruitful much beyond the traditional geometry. For example, special relativity is the study of invariants of Minkowski space-time under the Lorentz group action. Another example is complex analysis as study of objects invariant under the conformal maps. 

In this course we consider in details SL(2,R) group and corresponding geometrical and analytical invariants with their interrelations. The course has a multi-subject nature touching algebra, geometry and analysis. 

There are no prerequisites beyond a standard undergraduate curriculum: elements of group theory, linear algebra, real and complex analysis. Some knowledge of Lie groups and Hilbert spaces would be helpful but is not obligatory. 

Lectures will demonstrate numerous connections between various areas of mathematics. Therefore the course will benefit students wishing to see their research field in a broader context. 

The best approximation to the Lecture Notes at the moment is my paper 'Erlangen Program at Large: Outline', see "Course Materials" section and the Bibliography. This will be significantly edited and expanded during the semester. 

Prerequisites

There are no prerequisites beyond a standard undergraduate curriculum: elements of group theory, linear algebra, real and complex analysis.

Some knowledge of Lie groups and Hilbert spaces would be helpful but is not obligatory. 

Syllabus

  • SL(2,R) group and Moebius transformations of the real line.
  • Complex, dual and double numbers and Clifford algebras with two generators.
  • Iwasawa decomposition of SL(2,R).
  • Moebius transformations in the upper half-plane.
  • Cycles (quadrics) as geometric SL(2,R)-invariants.
  • Filmore-Springer-Cnops construction and algebraic invariants of cycles.
  • Linearised Moebius action in the in the space of function: Hardy and Bergman spaces.
  • Cauchy integral formula as a wavelet transform.
  • Cauchy-Riemann and Laplace equations from the invariant vector fields.
  • Laurent and Taylor expansions over the eigenvectors of rotations.
  • Functional calculus as an intertwining operator.
  • Prolongation of representations and functional calculus of non-selfadjoint operators.
  • Spectrum of an operator as a support of the functional calculus.

Lecturer

  • VK

    Dr Vladimir V. Kisil

    University
    University of Leeds

Bibliography

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