Erlangen program in geometry and analysis: SL(2,R) case study (MAGIC071) |
GeneralDescription
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. SemesterAutumn 2016 (Monday, October 3 to Friday, December 9) Timetable
PrerequisitesThere 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
Lecturer
Students
Bibliography
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.) AssessmentThe assessment will check your understanding of the basic SL(2,R) structures. It is not required to know the advanced results presented in the course. It is enough to complete 60 percents of the examp paper to pass the assessment.
Final assessment
FilesFiles marked L are intended to be displayed on the main screen during lectures.
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