The idea of the meeting is to gather a small and heterogeneous group of researchers working on configuration spaces and arrangements of subspaces, with special interest in the interplay between combinatorics and topology, in homological stability and in representation theory. An important role will be played by hyperplane and toric arrangements, reflection groups, braid groups and combinatorial models. We plan to have a rather small number of talks in an informal context, so as to allow ample time for discussions and collaboration.
- Pauline Bailet, Universität Bremen
On the monodromy of Milnor fibers of line arrangements with a sharp pair
Abstract:
Let A={H_{1},…,H_{d},H_{∞}} ⊂ ℙ_{ℝ}^{2} be a projective line arrangement with defining polynomial Q(x,y,z). Let A={H_{1},…,H_{d}} ⊂ ℝ^{2} be the deconing of A with respect the the hyperplane at infinity H_{∞}, and M(A)=ℂ^{2} ∖ ∪_{H ∈ A}H_{ℂ} be the complement of the complexified arrangement. The Milnor fiber F=F_{A} of A is the smooth affine hypersurface defined as Q^{−1}(1)⊂ ℂ^{3}. Consider the geometric monodromy action on F, given by the multiplication by λ= exp(2√−1π / d+1). This automorphism induces the monodromy operators in homology h_{q}: H_{q}(F,ℂ) → H_{q}(F,ℂ), which are all diagonalizable with eigenvalues the (d+1)−roots of the unity {λ^{k}, 0≤ k ≤ d}. For an eigenvalue β, we denote by H_{q}(F,ℂ)_{β} the β−eigenspace of h_{q}.
The question to determine under which (combinatorial or geometrical) assumptions, a certain eigenvalue β can appear, is still open. In particular, sufficient conditions for an arrangement to be a-monodromic (i.e. H_{1}(F,ℂ)_{β}= 0, for all β ≠ 1) are already known (among others, see the results of S. Papadima, A. Suciu) and are related with the multiplicities of the intersection points of A. However, our understanding of the action of the monodromy on Milnor fibers should be futher improved. For instance, it has been conjectured by M. Yoshinaga in 2013 that, if A has a sharp pair (i.e. the intersections points are all contained in one of the two regions of ℙ_{ℝ}^{2} delimited by a pair of hyperplanes in A), then the only possible monodromy is 3.
It is known that the complement of any complex hyperplane arrangement in ℂ^{l} is a minimal space. This has been refined in the case of complexified real arrangement: M. Salvetti and S. Settepanella constructed a minimal complex C_{*}(S(A)) homotopically equivalent to the complement M(A), by using Forman’s Discrete Morse Theory and the Salvetti’s complex S_{*}(A).
By using this minimal complex with a reduced boundary map given by G. Gaiffi and M. Salvetti, we describe an algorithm that allows us to determine that if A has a sharp pair, then if A is not a-monodromic, only monodromy 3 and 4 can appear. In particular monodromy 4 could appear only if a special configuration of hyperplanes appears. Then in order to prove Yoshinaga’s conjecture it remains to show that for this special configuration monodromy 4 can not appear. (joint work with S. Settepanella)
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- Pavle Blagojević, Freie Universität Berlin
Local multiplicity of continuous maps between manifolds
- Emanuele Delucchi, Université de Fribourg
Combinatorial topology of toric arrangements,
[slides]
Abstract:
Combinatorial algebraic topology is a lively and sprawling field with important applications. Hallmarks of this subject are the construction of discrete models for topological spaces and methods for studying their homotopy type, such as Discrete Morse Theory.
In this talk I will introduce some of the techniques of combinatorial algebraic topology and show how they allow us, for instance, to describe the full integral cohomology algebra of complements of arrangements in the complex torus (thus establishing the counterpart of a celebrated result of Orlik and Solomon for hyperplane arrangements). Hopefully this will give a glimpse of the potential of a combinatorial approach to topological problems, in and beyond arrangements' theory.
The results I’ll discuss were obtained in joint work with Giacomo d'Antonio and Filippo Callegaro.
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- Alex Dimca, Université de Nice - Sophia Antipolis
On free and nearly free plane curves
Abstract:
First we recall the definition of free divisors going back to Kyoji Saito around 1980.
Next we explain a new method for constructing free curve arrangements by using pencils
of curves. Finally the relation between irreducible free curves and rational cuspidal curves
will be explored.
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- Clement Dupont, Max-Planck-Institut für Mathematik, Bonn
Formality of arrangement complements,
[slides]
Abstract:
In this talk I will present a formality theorem for arrangement complements, whose corollaries are the formality of complements of hyperplane arrangements (due to Brieskorn) and the formality of complements of toric arrangements (which was proved in the special case of unimodular toric arrangements by De Concini and Procesi). The main tool of the proof is Deligne’s mixed Hodge theory and the Orlik-Solomon spectral sequence, as described independently by Looijenga, Bibby and the speaker. If time permits, I will say a few words on co-arrangements and their relevance to mixed Hodge theory and the theory of motives.
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- Dmitry Kozlov, Universität Bremen
Topological and combinatorial methods in Theoretical Distributed Computing
Abstract:
In the first half of the talk I will give a very compressed introduction into parts of Theoretical Distributed Computing from the point of view of mathematician. I will describe how to construct simplicial models whose combinatorics contains important information about computability and complexity of standard distributed tasks. In the second part, I will outline our recent progress on estimating the complexity of the so-called Weak Symmetry Breaking task, where we are able to derive some quite surprising results. At the technical core of our argument we need to construct complete matchings on specific graphs associated to the so-called iterated standard chromatic subdivision of a simplex. The talk is based on our monograph "Distributed Computing through Combinatorial Topology" (with Herlihy and Rajsbaum), as well as on a recent series of preprints.
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- Eva Maria Feichtner, Universität Bremen
Bergman fans - a link between arrangement theory and tropical geometry
Abstract:
Tropicalizations of arrangement complements turn out to be rational polyhedral fans
whose link at the origin is homeomorphic to the order complex of the respective
intersection lattice. On the level of matroids, the so-called Bergman fans are
discrete-geometric constructions that allow to recover the matroid. Proving
the latter requires an intriguing mix of discrete-geometric and tropical techniques.
Part of this material is joint work with Michael Falk.
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- Ivan Marin, Université d'Amiens
Flat manifolds and crystallographic groups from braid and reflection groups
- Luca Moci, Université Paris 7 "Diderot"
Colorings and flows for cellular complexes
Abstract:
In a recent series of papers by various authors, the theory of colorings and flows on graphs has been extended to the higher-dimensional case of CW-complexes. We will survey on this theory and show how the arithmetic Tutte polynomial naturally comes into the play. Joint work with E. Delucchi.
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- Giovanni Paolini, Scuola Normale Superiore
On the classifying space of an Artin monoid,
[slides]
Abstract:
In this talk we show that, if X is the standard CW model for the classifying space of an Artin monoid, it is possible to obtain the Salvetti complex as the Morse complex of a suitable acyclic matching on X. As a consequence we obtain a new proof of a theorem of Dobrinskaya which gives a reformulation of the K(π,1) conjecture for Artin groups.
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- Elia Saini, Université de Fribourg
Arrangements with up to 7 hyperplanes,
[slides]
Abstract:
To every matroid we associated a topological space called its reduced realization space. By means of symbolic computation we prove that for any matroid with ground set of up to 7 elements this topological space is either empty or connected. As an application we show that, in any rank, complex central hyperplane arrangements with up to 7 hyperplanes and same underlying matroid are isotopic. In particular, the diffeomorphism type of the complement manifold and the Milnor fiber and fibration of these arrangements are combinatorially determined.
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- Simona Settepanella, Hokkaido University
Strata of discriminantal arrangements
Abstract:
We give explicit description of the combinatorics of codimension two strata of Manin-Schechtman arrangements. As applications we discuss connection of these results with Gale transform, the study of the fundamental groups of the complements to discriminantal arrangements and study of the space of generic arrangements of lines in projective plane. (joint work with A. Libgober)
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- Dev Sinha, University of Oregon
Cohomology of symmetric and alternating groups
Abstract:
We present recent calculations of the mod-two cohomology of symmetric and alternating groups. The ingredients are:
- considering all cohomology together, and then having two products and a coproduct defining an (almost) Hopf ring.
- Fox-Neuwirth cell structures.
- restriction to subgroups and invariant theory.
We introduce each of these, which have relevance to representation theory and invariant theory as well as analogues for Coxeter groups and general linear groups over finite fields. Then we show how together they give efficient, combinatorially rich presentations of cohomology. [hide abstract]
- Michele Torielli, Università di Torino/Hokkaido University
Homotopy type, Orlik-Solomon algebra and Milnor fiber of supersolvable arrangements
Abstract:
In this talk I will give a very natural description of the bijections between the minimal CW-complex homotopy equivalent to the complement of a supersolvable arrangement A, the nbc-basis of the Orlik-Solomon algebra associated to A and the set of chambers of A. I will use these bijections to get results on the first (co)homology group of the Milnor fiber of A. (joint work with S. Settepanella) [hide abstract]
- Christine Vespa, Université de Strasbourg
Stable homology of automorphism groups via functor homology
Abstract:
Functor homology (i.e. homological algebra in functor categories) on suitable monoidal categories allows us to compute some stable homology of automorphism groups with twisted coefficients. In this talk, I will give a brief overview of the known results for several families of groups (symmetric groups, general linear groups, orthogonal and symplectic groups) and develop some recent results obtained for automorphism groups of free groups. If time permits I will present work in progress for other families of groups. [hide abstract]
Participants are not required to register. Nevertheless we will be grateful to you if you acknowledge your participation in advance contacting one of the organizers.