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Conferencier: Sergei Yuzvinsky
Titre du cours: Cohomology of Orlik-Salomon
Algebras
If is a complex hyperplane arrangement and is its complement then a rank 1 local system on is defined by an element where is the (graded) Orlik-Solomon algebra of . On the other hand, the multiplication by in defines a cochain complex whose cohomology is denoted by . The cohomology and are closely related to each other and studying these relations is one of the main theme of the arrangement theory.
We propose a short course of lectures exploring that can be given by two or three lecturers. Alternatively it can be given as a survey talk. The syllabus may include the following.
1. Definition of the Orlik-Solomon algebra and main properties.
2. Vanishing results for for general position (using sheaves on posets).
3. Resonance varieties where does not vanish.
4. Propagation theorem for .
5. Combinatorial description of using Vinberg's classification of generalized Cartan matrices.
6. Relations of with geometry of line arrangements; nets and multi-nets in complex projective plane.
7. and Hesse-like pencils of curves.
Conferencier: Alexander Suciu
Titre du cours: Braid groups, monodromy and
topology of arrangement complements
Conferencier: Hiroaki Terao
Titre du cours: What make hyperplane arrangements
free?
Although the free arrangements were introduced
more than twenty years ago, they are still somewhat
mysterious. Recently algebraic/differential geometric
techniques were applied to get a quite intriguing
sufficient and necessary condition for
a hyperplane arrangement to be free. The condition
remains valid over any field. As an application,
M. Yoshinaga proved the Edelman-Reiner conjecture for the
extended Shi/Catalan arrangements. In this talk,
I will discuss how much we can say at this moment
about what makes hyperplane arrangements free.
Conferencier: Dan Cohen
Titre du cours: Cohomology of Local systems
The lecture begins with three equivalent definitions of a local system on the complement of an arrangement. Then homology of the complement with coefficients in the local system is defined, and theorem of Hattori concerning the local system homology of a general position arrangement is stated. Features of Hattori's complex, and similar complexes arising in the context of super solvable arrangements are discussed. The lecture concludes with a discussion of the relationship between the cohomology of the Orik-Solomon algebra and the local system cohomology of the complement.
Conferencier: Michael Falk
Titre du cours: Large Geometry and combinatorics
of resonance weights
Abstract (2-3 lectures): Let
be an
arrangement of hyperplanes in
with Orlik-Solomon
algebra
. The
resonance
variety of
is the set
of one-forms
annihilated by some which is not a multiple of .
Resonance varieties are relevant to many different aspects of
arrangements theory: local system cohomology and hypergeometric
functions, Alexander invariants and characteristic varieties,
nonlinear fibrations, classification of OS algebras and homotopy
types, modules of logarithmic forms.
We will present the current state of knowledge concerning over various coefficient rings, in terms of the underlying matroid and the geometry of the arrangement. Our main focus will be the case . In this case it is no loss to assume is a line arrangement in Over fields of characteristic zero, the strong results of Libgober and Yuzvinsky, using the Vinberg-Kac classification of Cartan matrices, yield a combinatorial interpretation in terms of multi-nets, and a geometric interpretation in terms of pencils of plane curves that ``interpolate" the arrangement. Over fields of positive characteristic is seen to be the ruled variety carried by a special-position intersection of Schubert varieties in the Grassmannian of lines in Over rings of non-prime characteristic one sees more strange phenomona which is empirically related to translated tori in characteristic varieties.
For there are partial generalizations of the results above, mostly involving -generic arrangements. In addition, we will see a relationship between elements of and the critical loci of rational functions with zeros and poles along the arrangement.
The subject is imbued with beautiful examples, which will feature prominently in the lectures.
Conferencier: A. Muhammed Uludag
Titre du cours: Configuration spaces of arrangements
The configuration space of ordered points on the projective line
is the pure braid space
The configuration space of ordered hyperplanes on the projective space is the space
In order to get interesting analogues of braid groups ``higher braid groups'' we must impose further restrictions on the admissible sets of hyperplanes. A natural choice is to consider the configuration space of hyperplanes in general position. The corresponding groups were studied by Manin-Schehtman, their presentations were studied by H. Terao and R.J. Lawrence in the pure case. These groups are highly sophisticated.
The lecture will end with the question: Is the generiticity assumption the only plausible restriction to get a good analogue of the braid group?
Conferencier: Eva Maria Feichtner
Conferencier: Daniel Matei "Artin groups, Bestvina-Brady groups and Arrangements of Hypersurfaces ">