Universität zu Köln MI

Algebra at Köln

Summer school on Geometry of Representations

University of Cologne, July 26 to August 1, 2009

Arrival: Sunday, July 26.
Departure: Saturday, August 1.
Lectures will start on Monday morning. The summer school will close on Friday late afternoon.
A poster of the summer school (pdf format) can be downloaded here.
schedule of talks (pdf)
NEW: pictures from the summer school

Speakers and programme

There will be three series of lectures, to be given by William Crawley-Boevey, Tamás Hausel and Markus Reineke. In addition, there will be problem classes and discussion sessions. Each lecture series will consist of four lectures, each lecture will be 75 minutes. There will be two lectures each morning from Monday to Friday, one lecture on Monday afternoon and one on Friday afternoon. Problem sessions and discussions will run in the afternoons, apart from Wednesday afternoon that is reserved for an excursion.

Lecture series will be given by

William Crawley-Boevey, Leeds
Tamás Hausel, Oxford
Markus Reineke, Wuppertal

Contents of the lecture series and reading material

William Crawley-Boevey: Quiver-theoretic techniques for the Deligne-Simpson Problem
Abstract. Given a finite sequence of conjugacy classes in the general linear group, the Deligne-Simpson Problem asks whether or not there exist matrices in these conjugacy classes, whose product is the identity, and which have no common invariant subspace. I shall survey methods from the representation theory of quivers, and related ideas, which lead to a solution of the DSP in terms of root systems.
The lectures will mainly cover material from the following articles:
W. Crawley-Boevey, Geometry of the moment map for representations of quivers, Compositio Math., 126 (2001), 257-293
W. Crawley-Boevey, On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero, Duke Math. J. 118 (2003), 339-352
W. Crawley-Boevey, Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity, Publ. Math. Inst. Hautes Etudes Sci. 100 (2004), 171-207.
W. Crawley-Boevey and P. Shaw, Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem, Adv. Math. 201 (2006), 180-208.
W. Crawley-Boevey, Connections for weighted projective lines, arxiv.org/abs/0904.3430

Lec 1. Deformed and multiplicative preprojective algebras and middle convolution.
Lec 2. Monodromy, logarithmic connections and parabolic bundles
Lec 3. Varieties of representations and a sufficient condition for the DSP
Lec 4. Perpendicular categories and necessity of the condition for the DSP

Notes (taken by Felix Dietlein) are here

Background reading.
Some representation theory of quivers, such as: M. Brion, Representations of quivers, Notes de l'école d'été "Geometric Methods in Representation Theory" (Grenoble, 2008), www-fourier.ujf-grenoble.fr/~mbrion/notes_quivers_rev.pdf
Some homological algebra, including the notion of an abelian category and Ext groups.
Some algebraic geometry, including the notion of a vector bundle and a coherent sheaf.

Related reading.
V. P. Kostov, The Deligne-Simpson problem - a survey, J. Algebra 281 (2004), 83-108, arxiv.org/abs/math.RA/0206298
Books discussing Hilbert's 21st problem / the Riemann-Hilbert problem, such as:
D. V. Anosov and A. A. Bolibruch, The Riemann-Hilbert problem, Vieweg 1994
C. Sabbah, Isomonodromic deformations and Frobenius manifolds, Springer-Verlag 2007.
Tamás Hausel
In my talks I plan to explain the arithmetic study of quiver varieties and the computation of Betti numbers in http://arxiv.org/abs/0811.1569 and how it implies Kac's conjecture on representations of quivers. As time permits I will then show how the same arithmetic technique together with the character theory of finite groups and algebras Lie type, yield results on the cohomology of character and quiver varieties, following http://arxiv.org/abs/math/0612668 and http://arxiv.org/abs/0810.2076.
The mathematics I will need to use include:
basics of GIT quotients
arithmetic vs. cohomology of varieties
representations of quivers and Kac-Moody algebras
character theory of finite groups e.g. GL_n(F_q)

Slides of the lectures are here
Problem sheets are here: first problem sheet, second problem sheet, third problem sheet.

Markus Reineke
Roughly my plan for the Summer School is to talk about applications of Hall algebras to quiver moduli. If time allows, I will also discuss applications to Donaldson-Thomas invariants.
Most of the material which I will cover is contained in the ICRA-overview paper "Moduli of representations of quiver", http://arxiv.org/abs/0802.2147
An older overview paper is http://arxiv.org/abs/math/0304193.
I'd propose Bill's lecture notes "Lectures on representations of quiver" and "Geometry of representations of algebras" on http://www.amsta.leeds.ac.uk/~pmtwc/ as a good starting point for those not familiar with quiver representations.


We have reserved a number of rooms in Hotel Flandrischer Hof, which is in walking distance to both the university and the city centre. You can book your room when you register for the summer school. If you are not applying for financial support and want to make your own reservation, then here you can find links to some other hotels and websites offering accomodation at Köln .
If you need assistance with the hotel reservation, please contact Mrs Koersgen (Monday to Friday 11am-3pm, phone +49 221 470 5709, email nkoersge@math.uni-koeln.de).


Limited financial support is expected to be available, in particular for PhD students and postdocs. It will usually cover the cost of accommodation, while travel expenses should be paid by the participants' home institution whenever possible. More detail on support and how to apply for it will be given on the registration web page. The summer school is supported by Graduiertenkolleg 1269 (doctoral training programme).


Please register here.

Registered Participants

Travel information

Maps and information on public transport can be found here.
If you have problems finding your way, or if your travel is seriously delayed, please contact the conference secretary Mrs Wehmeyer (phone +49 (0) 221 470 2893, from Cologne simply dial 470 2893).


Robert Hartmann, Steffen Koenig, Peter Littelmann and Qunhua Liu

Contact: Robert Hartmann, Steffen Koenig, Peter Littelmann and Qunhua Liu .

Universität zu Köln Mathematisches Institut, AG Algebra
Weyertal 86-90
D-50931 Köln

rhartman@math.uni-koeln.de, skoenig@math.uni-koeln.de, qliu@math.uni-koeln.de
Letzte Änderung: 30.11.2008