Geometric and analytic problems related to Cartan connections

Research program at the International Erwin Schrödinger Institute (ESI) in Vienna.


Time: First half of 2004, main activity between the beginning of January and the end of April

The traditional "Winter School on Geometry and Physics" in Srni (Czech Republic) will take place January 17 - 24, 2004. While this school is formally independent of the program, some of the topics will be related, and there will be only little activity in Vienna during this week.
In the early phase of the program, the 27th meeting of the Seminar Sophus Lie will take place at the ESI on January 9 and 10. Participants of the program are welcome to take part in this meeting.

 Thomas P. Branson (University of Iowa)

 Andreas Cap (University of Vienna)

 Jan Slovak ( Masaryk University Brno)

Background: The concept of a Cartan connection was introduced and sucessfully applied by E. Cartan  during the first few decades of the 20th century. In the process of the clarification and formalization of many of Cartan's ideas during the second half of the 20th century, main emphasis was put on principal connections, while Cartan connections were rather forgotten.

The renewed interest in conformal and quaternionic structures as well as the relations to complex analysis via CR structures brought Cartan connections back to general interest during the last decades. These structures have the common property that their homogeneous model is a generalized flag manifold, i.e. the quotient of a semisimple Lie group by a parabolic subgroup. Starting from the pioneering work of N. Tanaka on Cartan connections associated to certain types of differential systems, the theory of the geometric structures defined by Cartan connections of that type, which have been named parabolic geometries, was developed extensively during the last years, both in the general picture and for specific structures.   In particular, several new and efficient methods for constructing and studying invaraint differential operators have been developed. Much of the recent work on parabolic geometries has close connections to representation theory and twistor theory.

Independently of the geometric developments there has been a lot of interest in analytic problems related to conformal and CR structures during the last decades. Early examples of this include the solution of the Yamabe problem and its CR analogue, as well as work on the relations of conformally invariant operators to sharp inequalities, for example of Sobolev and Moser-Trudinger type. More recent work on curvature prescription problems involves newly dicovered curvature quantities (like Q-curvature) which is closely related to higher order conformally invariant operators.

Aims and topics: Apart from advancing the above mentioned subjects, the main aim of the program is to intensify the  communication and collaboration of scientists approaching the field from different directions. Specific topics include:

Visitors: We have preliminarily agreed for dates with the following participants (partially supported from other funds):

 P. Albin (Stanford)  March 22 - 29
 D.V. Alekseevsky (Hull)  January 8 - 11
 T.N. Bailey (Edinburgh)  March 6 - 16
 L. Barberis  February 23 - March 4
 H. Baum (Berlin)  March 28 - April 8
 O. Biquard (Strasbourg)  February 23 - March 1
 T. Branson (Iowa)  January 5 - 17, then Srni, then in Vienna until January 27;
and March 10 - April 16
 J. Bures (Prague)  end of January - end of April
 D.M.J. Calderbank (Edinburgh)  March 28 - April 8
 S.Y.A. Chang (Princeton)  February 25 - March 7
 M. Cowling (Sydney)   January 5 - 16
 B. Doubrov  February 2 - 20
 D. Duchemin (Strasbourg)  March 8 - 21
 M.G. Eastwood (Adelaide)  Srni, then in Vienna until February 9
 A. Fino (Torino)  February 22 - March 4
 D. Fox (Atlanta)  January 25 - February 7
 K. Galicky (Albuquerque)  March 31 - April 14
 A.R. Gover (Auckland)   January 5 - 17, then in Srni;  
 C.R. Graham (Seattle)  March 16 - April 3
 M. Gursky (Notre Dame)  February 29 - March 6
 O. Hijazi (Nancy)  March 29 - April 3
 K. Hirachi (Tokyo)  March 8 - April 5
 D. Hong (Iowa)  March 14 - 27
 P. Julg (Orleans)  February 5 - 19
 J. Konderak (Bari)  January 24 - 31
 L. Krump (Prague)  four weeks, starting around January 25
 M. Kuranishi (Columbia)  one week in January or in April
 F. Leitner (Leipzig)  March 28 - April 8
 L. Mason (Oxford)  March 28 - April 3
 T. Morimoto (Nara)  January 4 - 12
 P.-A. Nagy (Berlin)  April 1 - 7
 P. Nurowsky (Warsaw)  January 5 - 17
  B. Ørsted (Odense)  January 9 - 18
 S. Salamon (Torino)  one week in April
  G. Schmalz (Bonn)  February 8 - 14
  L. Schwachhöfer (Dortmund)  March 14 - 27
 U. Semmelmann (München)  February 25 - March 5
 J. Silhan (Auckland)  January 5 - 15
 P. Somberg (Prague)  four weeks, starting around January 25
 V. Soucek (Prague)  end of January - end of April
 R. Stanton (Columbus)  March 22 - April 7
 J. Tafel (Warsaw)  April 13 - 26
 W. Ugalde (Purdue University)  March 12 - 25
 A. Villanueva (Iowa)  March 14 - 22
 G. Weingart (Philadelphia)  February 25 - March 5
 K. Yamaguchi (Sapporo)  February 8 - 21
 P. Yang (Princeton)  February 25 - March 7
 D. Zaitsev (Dublin)  March 19 - 28

Some touristic links: