Summer School on Local Systems

This is a summer school of the SFB/TR 45 Bonn-Essen-Mainz financed by Deutsche Forschungsgemeinschaft. It takes place August 31 - September 3, 2009 at the University of Mainz. The workshop intends to improve the training of PhD students and postdocs in the area, in particular of the members of the SFB/TR 45.

Speakers

• Frits Beukers (University of Utrecht)
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• Michael Dettweiler (University of Heidelberg)
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• Gert Heckman (University of Nijmegen)
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Preliminary Schedule

Lectures

Frits Beukers: Hypergeometric functions in one variable 

• Lecture 1: We give a brief explanation of the relevant terminology, local exponents, Fuchsian equations and monodromy for linear differential equations in the complex plane. Then we introduce Gauss’ hypergeometric function with examples.
  
• Lecture 2: We deal with rigidity of the Gauss’ hypergeometric differential equation and some consequences. Then we discuss the monodromy of Gauss’ hypergeometric equation by use of the Schwarz map.
  
• Lecture 3: We introduce hypergeometric differential equations in one variable of higher order. This is another example of a rigid local system. On the basis of that one easily reconstructs the abstract monodromy group, which turns out to be a complex reflection group.
  
• Lecture 4: In this lecture we show how to construct the monodromy of a one variable hypergeometric equation with respect to an explicitly given basis of solutions. There are two approaches, namely Mellin-Barnes integrals on the one hand and the recent Golyshev-Mellit approach via Fourier transforms of products of gamma-factors on the other hand.
   
  

Michael Dettweiler: Monodromy and arithmetic of rigid local systems.

• Lecture 1: We outline N. Katz' theory of the additive middle convolution of sheaves on the affine line. Using this, we explain a theorem of Katz by which every rigid local system can be obtained by  iteratively applying
  a certain sequence of middle convolutions and tensor products to a rank-one-object.
  
• Lecture 2: We interprete the middle convolution as variation of parabolic cohomology and show how to compute the monodromy of rigid local systems explicitly. We give several examples of rigid and non-rigid local systems with G_2-monodromy.
  
• Lecture 3: Every rigid local system is closely related to a certain GKZ-differential equation, where GKZ stands for Gelfand, Kapranov and Zelevinsky. We relate the underlying motives and Galois representations, as well as variations of Hodge-Structures.
  
• Lecture 4: We show how it is possible in some cases to apply the automorphic lifting results of Clozel, Harris and Taylor to specializations
  of the Galois representation accociated to a rigid local system.

Gert Heckman: Complex hyperbolic structures for tori of type ADE

• For each root lattice L of type A,D or E consider the complex torus H with rational character L, and let W be the associated Weyl group. On the complement of the discriminant D in W\H there exists a one parameter family of complex hyperbolic functions, coming from a suitable hypergeometric system on W\H. Our main goal is to understand for which parameter this hyperbolic structure leads to a ball quotient structure. It is believed that these hypergeometric systems are motivic, in the sense that they come from algebraic geometry via suitable period maps. The example of this period map for the moduli space of quartic curves will be discussed in some details.
  

     Birational Hyperbolic Geometry

     Complex Geometric Structures

Applications and Financial Support

Full financial support is available for members of the SFB/TR 45. There is limited support for other participants, too. Please indicate in the application form if you would like us to reserve a hotel room for you.

Travel information

All lectures will take place in the Mathematics Department of the University of Mainz, Staudinger Weg 9, in room 05-426. Here is a map of the campus. The closest airport is located at Frankfurt/Main. Find your way from the airport to the institute here.

Further information

If you have additional questions, please contact the organizers.

• Stefan Müller-Stach (sfb45@mathematik.uni-mainz.de)
• Duco van Straten (straten@mathematik.uni-mainz.de)