Algebra at Köln
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Summer school on Geometry of Representations
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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
Contents
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.
Accommodation
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).
Support
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).
Registration
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).
Organisers
Robert Hartmann, Steffen Koenig, Peter Littelmann and Qunhua Liu
Contact:
Robert Hartmann,
Steffen Koenig,
Peter Littelmann
and Qunhua Liu
.
rhartman@math.uni-koeln.de, skoenig@math.uni-koeln.de, qliu@math.uni-koeln.de
Letzte Änderung: 30.11.2008
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