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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.

Semester

Autumn 2018 (Monday, October 8 to Friday, December 14)

Hours

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

Timetable

  • Wed 10:05 - 10:55

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


Vladimir V. Kisil
Email kisilv@maths.leeds.ac.uk
Phone (0113) 343 5173
Interests Symmetries in analysis and applications
Photo of Vladimir V. Kisil


Bibliography


GEOMETRY OF MÖBIUS TRANSFORMATIONS. Elliptic, Parabolic and Hyperbolic Actions of SL(2,R)Vladimir V Kisil
Advances in Applied AnalysisRogosin, Sergei V.; Koroleva, Anna A. (Eds.)
Erlangen program for geometry and analysis: SL(2,R) case studyVladimir V. Kisil
Erlangen Programme at Large: An Overview.Vladimir V. Kisil
How to use the dedicated software (CAS)Vladimir V. Kisil


Note:

Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)

Assessment



The course is assessed by the take home exam paper in January exam period. To pass the exam you will need to provide correct answers for 60% of questions.
Compared to previous years, this course will be mainly on the geometric part. Thus, this year exam paper will have mainly geometry questions.
You may practise on exercises uploaded to File section of the course.

No assignments have been set for this course.

Recorded Lectures


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