Spring School on A-infinity Structures
Rothenberge, 27 March – 31 March 2006
Organizers: Henning Krause (Paderborn), Markus
Reineke (Münster)
Supported by: Graduiertenkolleg
„Analytische Topologie und Metageometrie“
The Spring School:
The
topic of the Spring School is the role
of A-infinity structures in various contexts of Representation Theory,
Homological Algebra and Algebraic Topology.
The
Spring School is intended for Ph.D. students and (early) postdocs working on
and interested in these fields.
The
Spring School will follow the tradition of previous spring/summer schools in
representation theory. Every participant chooses a subject of the program (see
below) and gives a talk.The typical participant will not be an expert in the
subject, but working in one of the fields indicated above and interested in
jointly learning a new subject. There will be no special prerequisities except
for standard knowledge in homological algebra, but participants are expected to
prepare themselves and their talks in advance. It is recommended that this is
done in small groups; in particular, the talks could/should be subdivided into
smaller parts.
At
the end of the week, there will be invited lectures by Bernhard Keller on more
recent developments in the field.
Program:
1.
Karsten
Schmidt: Motivating Examples I (45 min)
Two problems motivating the study of
A-infinity structures will be presented: the problem of reconstruction of a
complex of modules from its homology, and the problem of reconstruction of a
module category from an Ext-algebra. The solution to these problems will be
discussed in talks 8 and 9. The reference is [Ke01, 2.1].
2.
Michael
Ehrig: Topological Origin of A-infinity Structures
(45 min)
As the (historically) first example of
A-infinity structures, the A-infinity algebra structure of the loop space of a
topological space will be explained. These relations to topology will be
deepened in talk 10. The reference is [Ke01, 2.2].
3.
Klaus Löhrke, Roland Olbricht:
Differential Graded Algebras and Algebraic Triangulated Categories (90 min)
The concept of Algebraic Triangulated
Categories will be introduced. It will be shown that such categories appear as
derived categories of differential graded algebras. This result will be
generalized to the A-infinity context in talk 13. The primary reference is
[Ke94, 4.3] (where algebraic triangulated categories are called stable
categories). This should also be compared with [Ke06, Thm.
3.8].
4.
David Pauksztello, Marcel Wiedemann:
Definition of A-infinity Algebras, Modules and Categories (2x60 min)
The definitions of A-infinity algebras,
modules, and categories, their morphisms and functors will be given and discussed in detail. The
reference is [Ke01, 3.1, 3.4, 4.2, 7.2-7.4].
5.
Hideto Asashiba: Model Category Structure (60 min)
The concept of model categories will be
introduced. As a first example, the model category structure on the category of
chain complexes will be explained. Moreover, the model category structure on
the category of DG algebras will be explained. References are [Ke05, 4.1, 4.2].
6.
Pedro
Nicolas: The Bar Construction (60 min)
The characterization of A-infinity
algebras via the bar construction will be explained. The primary reference is
[Ke01, 3.6], see also [Ke05, 4.3, 4.4].
7.
Birgit
Huber: The Homology of DG algebras has an A-infinity Structure (60 min)
It will be shown that the homology of a DG algebra admits an A-infinity algebra structure. Minimal
models for A-infinity algebras will be introduced. The reference is [Ke01, 3.3]
8.
Guodong Zhou:
Motivating Examples II (45 min)
The reconstruction of a complex from its
homology via A-infinity structures will be explained. The reference is [Ke01,
4.3].
9.
Thomas Bliem,
Steffen Oppermann: Motivation Examples III (90 min)
The reconstruction of a module category
from an Ext-algebra will be explained. The primary reference is [Ke01, 5., 6.]. This should be complemented by [Ke02].
10.
Julia
Singer: The
The relations of A-infinity structures to
topology will be discussed.
11.
Xuan Yang:
Operads (90 min)
Operads will
be introduced and related to A-infinity structures. References are [MS], [M,
4.8].
12.
Kristian Brüning: Hochschild Cohomology and A-infinity Structures (60 min)
Applications of Hochschild
(co)homology to A-infinity algebras will be discussed. The reference is [K].
13.
Bernhard
Keller: Algebraic Triangulated Categories and A-infinity Structures (3x60 min)
The characterization of certain compactly
generated algebraic triangulated categories as compact objects in the derived
category of an A-infinity algebra will be explained. The reference is [Ke05].
Talks will start on Monday at
References:
(All
papers of Bernhard Keller can also be retrieved from his homepage http://www.math.jussieu.fr/~keller/publ/index.html
For
further references, see also this
dvi-file.)
[Ke94]
Bernhard Keller: Deriving DG categories. Ann. Sci. Ecole
Norm. Sup. (4), 27(1):63-102, 1994.
[Ke01]
Bernhard Keller: Introduction to A-infinity algebras and modules. Homology Homotopy Appl. 3(1):1-35, 2001. math.RA/9910179
[Ke02]
Bernhard Keller: A-infinity algebras in representation theory.
http://www.math.jussieu.fr/~keller/publ/art.dvi
[Ke05]
Bernhard Keller: A-infinity algebras, modules and functor
categories. math.RT/0510508
[Ke06]
Bernhard Keller: On differential graded categories. math.KT/0601185
[K]
Tornike Kadeishvili: On the
homology of fibre spaces. Uspekhi Mat. Nauk 35:3 (1980), 183-188. math.AT/0504437
[MS]
James E. McClure, Jeffrey H. Smith: Operads and cosimplicial objects: an introduction. In: Axiomatic,
enriched and motivic homotopy
theory, volume 131 of NATO Sci. Ser. II Math. Phys.
Chem., 133-171. Kluwe Acad. Publ.,
[M]
Martin Markl: Models for operads.
Comm. Algebra, 24(4):1471-1500, 1996.
Schedule:
Monday:
Tuesday:
10:45-11:45 Wiedemann
12:00 Lunch
14:00-15:00 Asashiba
15:00 Coffee
15:30-16:30 Nicolas
16:45-17:45 Huber
18:00 Dinner
Wednesday:
Thursday:
Friday:
11:00-12:00 Brüning
12:00 Lunch
14:00-15:00 Keller, part 3
Participation:
Due
to the restrictions of the location, the number of participants is strictly
limited to 24.
Accomodation
and full boarding will be covered by the Graduiertenkolleg
“Analytische Topologie und Metageometrie”. However, travel
expenses can not be covered.
If
you want to participate in the Spring School, please send an email to H.
Krause or M.
Reineke, indicating which topic you are willing to give a talk about.
The
deadline for registration was
List of participants:
Name |
Email address |
|
Hideto Asashiba |
asashiba at sci.osaka-cu.ac.jp |
|
Thomas Bliem |
tbliem at math.uni-koeln.de |
|
Kristian Brüning |
bruening at math.uni-paderborn.de |
|
Nicolay Dichev |
dichev at math.upb.de |
|
Michael Ehrig |
mehrig at math.uni-koeln.de |
|
Birgit Huber |
bhuber at math.uni-paderborn.de |
|
Bernhard Keller |
keller at math.jussieu.fr |
|
Hennig Krause |
hkrause at math.uni-paderborn.de |
|
Klaus Löhrke |
klaus.loerke at muenster.de |
|
Vanessa Miemietz |
miemietz at maths.ox.ac.uk |
|
Pedro Nicolas |
nicolas at math.jussieu.fr |
|
Roland Olbricht |
olbricht at math.uni-muenster.de |
|
Steffen Oppermann |
sopperma at math.uni-koeln.de |
|
David Pauksztello |
davidp at maths.leeds.ac.uk |
|
Markus Reineke |
reinekem at math.uni-muenster.de |
|
Karsten Schmidt |
kschmidt at math.uni-paderborn.de |
|
Julia Singer |
singer at math.uni-bonn.de |
|
Catharina Stroppel |
cs at maths.gla.ac.uk |
|
Marcel Wiedemann |
mat4mw at leeds.ac.uk |
|
Stefan Wolf |
swolf at math.uni-paderborn.de |
|
Xuan Yang |
xuyang at uni-osnabrueck.de |
|
Guodong Zhou |
zhouguodong at math.jussieu.fr |
Rothenberge:
Some
description of the location „Landhaus Rothenberge“ and some travel information (in German) can be found
here.
M. Reineke, 07.04.2006