CONFERENCE IN HONOUR OF PETER ORLIK

to be held at the Fields Institute

**Alejandro Adem**, University of Britsh Columbia

*Simplicial Spaces of Homomorphisms*

Let $\Gamma^q$ denote the $q$-th stage of the descending central series of
the free group on $n$ generators $F_n$. For each $q$ and every topological group
$G$, a simplicial space $B_*(q,G)$ is constructed where $B_n(q,G) = Hom(F_n/\Gamma^q,G)$
and the realizations $B(q,G)=|B_*(q,G)|$ filter the classifying space $BG$.
Cohomology calculations are provided for compact Lie groups and homotopy properties
of $B(q,G)$ are considered for finite groups, including their description as
homotopy colimits. The spaces $B(2,G)$ (built out of commuting elements) are
described in detail for transitively commutative groups. Stable homotopy decompositionsof
the $B(q,G)$ are also given.

This is joint work with Fred Cohen and Enrique Torres.

**Chris Brav**, Queen's University

*The projective McKay correspondence*

Alexander Kirillov Jr. has described a McKay correspondence for finite subgroups of $PSL_{2}({\mathbb C})$ which associates to each `height' function an affine Dynkin quiver, together with a derived equivalence between equivariant sheaves on $\mathbb{P}^{1}$ and representations of this quiver. The equivalences for various height functions are related by reflection functors for quiver representations.

We develop an analogous story for the cotangent bundle of $\mathbb{P}^1$, in which each height function gives a derived equivalence between equivariant sheaves on the cotangent bundle and modules over the preprojective algebra of an affine Dynkin quiver. These various equivalences are related by the spherical twists of Seidel-Thomas, which take the place of the reflection functors for $\mathbb{P}^{1}$.

Since there is a well-known equivalence between equivariant sheaves on $\mathbb{C}^2$ and modules over the preprojective algebra, our construction also provides a bridge between Kirillov's correspondence for $\mathbb{P}^{1}$ and the usual McKay correspondence for $\mathbb{C}^2$.

**Fred Cohen**, University of Rochester

*On generalized moment-angle complexes, their properties, and applications*

Properties of the stable structure of generalized moment-angle complexes as well as applications will be developed. Connections to questions in homotopy theory as well as engineering will also be developed.

This lecture is based on joint work with Anthony Bahri, Martin Bendersky and
Sam Gitler.

**Mike Davis**, Ohio State University

*Compactly supported cohomology of buildings*

In 1976 Borel and Serre calculated the compactly supported cohomology of affine
buildings. I will explain an analogous formula for general buildings (where
we are interested in nonclassical buildings which are neither affine nor spherical).

This is joint work with Jan Dymara, Tadeusz Januszkiewicz, John Meier and Boris
Okun.

**Corrado de Concini, **University of Rome

*Vector partition functions and index of transversally elliptic operators
*

We introduce a space of functions on a lattice $\Lambda$ which generalizes
the space of quasi--polynomials satisfying the difference equations associated
to cocircuits of a sequence of vectors $X$ in $\Lambda$. This space $ \mathcal
F(X)$ contains the partition function ${\mathcal P }_X $. We prove a ``localization
formula'' for any $f$ in $\mathcal F(X)$. Inparticular, this implies that the
partition function ${\mathcal P}_X $ is a quasi--polynomial on the sets ${\mathfrak
c}-B(X)$ where ${\mathfrak c}$ is a big cell.

Let $G$ be the compact torus dual to the lattice $\Lambda$. We consider the
complex representation $M$ of $G$ having $X$ as list of weights.

We determine the range of the index map from $G$-transversally elliptic operators
on $M$ to generalized functions on $G$and to prove that the index map is an
isomorphism on its image which is the $R(G)$ module spanned by $ \mathcal F(X)$.
This is a setting studied by Atiyah-Singer which is in a sense {\it universal}
for index computations.

The talk will report on joint work with C. Procesi and M. Vergne.

**Igor Dolgachev**, University of Michigan

*Moduli spaces of hyperplane arrangements*

I will discuss some open questions about the moduli spaces of hyperplane arrangements with fixed topology modulo projective transformations. Some of these questions are the rationality questions, compactness and period mappings.

**Michael Falk**, Northern Arizona University

*On vanishing products in Orlik-Solomon algebras*

We use elementary tropical geometry to derive necessary and sufficient conditions
for a collection of p linearly independent logarithmic one-forms with poles
along an arrangement A to have vanishing wedge product. This result is applied
to derive equivalent combinatorial conditions describing some resonant weights
for p-generic arrangements, generalizing the neighborly partition/Q-matrix description
in the p=1 case. Time permitting, we will describe implications for critical
loci of A-master functions.

This is a report on work in progress, growing out of ongoing joint work with
D. Cohen, G, Denham, and A. Varchenko on critical loci.

**Eva-Maria Feichtner, **University of Bremen

*Arrangements and tropical geometry*

Tropical geometry is a multi-faceted field emerging at the crossroads of algebra,
geometry and combinatorics. Given their place in the mathematical landscape,
arrangement theory and tropical geometry have a number of meeting points and
ample opportunities to enrich each other. We discuss some of their interactions
- from the self-evident to the unexpected.

**Joel Kamnitzer**, University of Toronto

*Braid group actions on derived categories of coherent sheaves with applications
to knot homology*

I will consider constructions (joint with Sabin Cautis) of braid group actions on derived categories of certain moduli spaces of vector bundles on curves. The motivation is an attempt to give a geometric construction of knot homology theories. I will explain some background on the general topic of braid group action on triangulated categories -- including the work of Seidel-Thomas, Chuang-Rouquier and other.

**Toshitake Kohno**, The University of Tokyo

*Bar complex of Orlik-Solomon algebra and rational universal holonomy maps*

The purpose of this talk is to present an overview on basic properties of the bar complex of Orlik-Solomon algebra, which provides iterated integrals of logarithmic forms depending only on homotopy classes of loops. In the case of braid arrangement I will explain a method to construct a universal holonomy map defined over the field of rational numbers based on Drinfel'd associator.

**Daniel Matei**, IMAR

*Fundamental groups of smooth algebraic varieties*

We start by surveying the restrictions known to date imposed on a fundamental group of a complex quasiprojective manifold by the algebraic structure. We then move to investigate, using characteristic and resonance varieties, quasiprojectivityof various classes of groups such as the Artin groups, their normal subgroups, and certain generalizations of both. This will lead us to consider the finiteness properties of quasiprojective groups.

Recent work of M. Yoshinaga showed that in some instances certain higher homotopy
groups of arrangements map onto non-resonant homology. This is in contrast to
the usual Hurewicz map to untwisted homology, which is always the zero homomorphism
in degree greater than one. In this work we examine

this dichotomy, generalizing both results.

**Jan-Erik Roos**, Stockholm University

*Complex Hyperplane Arrangements having Unexpected Homological Properties.*

Let $R$ be the Orlik-Solomon algebra (over $\bf{Q}) of a central complex hyperplane
arrangement, let $R^!$ be the Koszul dual of $R$ and let $R(z)$ and $R^!(z)$
be the corresponding Hilbert series. We always have that $R(z)$ is a polyomial
and in some cases (e.g. braid arrangements) we have that $R^!(z)=1/R(-z)$ which
is in particular a rational function. In [1] we recently found two cases with
8 hyperplanes in $\bf{C^3}$ where $R^!(z)$ is a {\it transcendental} function.
Even more recently we have analyzed the recent tables of Serenevy [2] and found
a few other transcendental cases for 9 and 10 hyperplanes. All this is related
to

general homological results about the Yoneda Ext-algebras (and their underlying
graded Lie algebras) of the Orlik-Solomon algebras.

[1] Jan-Erik Roos, The Homotopy Lie Algebra of a Complex Hyperplane

Arrangement Is Not Necessarily Finitely Presented, Experimental

Mathematics, vol. 17, issue 2, pages 129-143, (2008)

[2] Dean Serenevy (2008)

{\tt http://dean.serenevy.net/code/projects/arrangement}

**Mario Salvetti**, University of Pisa

*A method for local system cohomology of an Hyperplane complement*

We use a recent construction in Combinatorial (or Discrete) Morse theory to
produce algebraic complexes which (efficiently) calculate local system cohomology
of the complement of a real hyperplane arrangement.

**Hal Schenck**, University of Illinois at Urbana-Champaign

*"The Orlik-Terao algebra and 2-formality"*

In 1994, Orlik and Terao introduced a commutative analog of the Orlik-Solomon
algebra to answer a question of Aomoto, and showed the Hilbert series depends
only on the intersection lattice L(A). A few years earlier, Falk and Randell
had introduced the property of 2-formality; Yuzvinsky showed that 2-formality
is not a combinatorial invariant. We show that 2-formality is detected by the
quadratic component of the (non-Artinian version) of the Orlik-Terao algebra.
We also show that the syzygies of a rank 2 local flat in this algebra are resolved
by an Eagon-Northcott complex.

(Joint work with S. Tohaneanu)

**Alexander Suciu**, Northeastern University

* Spectral sequences and homology with local coefficients*

In this talk, I will describe the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CW-complex $X$. The $d^1$ differential is determined by the coalgebra structure of $H_*(X,\k)$, and the $\k\pi_1(X)$-module structure on the twisting coefficients.

This basic tool provides information on the homology of all regular covers of $X$, and leads to new obstructions to formality. It also yields computable upper bounds on the ranks of the cohomology groups of $X$, with coefficients in a prime-power order, rank one local system. When $X$ admits a minimal cell decomposition, we relate the linearization of the equivariant cochain complex of the universal abelian cover to the Aomoto complex, arising from the cup-product structure of $H^*(X,\k)$. In the case when $X$ is the complement of a complex hyperplane arrangement, we recover two well-known results of Dan Cohen and Peter Orlik.

This is joint work with Stefan Papadima (arXiv:0708.4262)

**Hiroaki Terao**, Hokkaido University

*Totally free arrangements of hyperplanes*

A central arrangement $\A$ of hyperplanes in an $\ell$-dimensional vector space $V$ is said to be {\it totally free} if a multiarrangement $(\A, m)$ is free for any multiplicity $ m : \A\rightarrow \Z_{> 0}$. It has been known that $\A$ is totally free whenever $\ell \le 2$. In this talk, we will prove that there does not exist any totally free arrangement other than the obvious ones, that is, a product of one-dimensional arrangements and two-dimensional ones.

(joint work with Takuro Abe and Masahiko Yoshinaga)

**A. Varchenko**, University of North Carolina,
Chapel Hill

*Transversality in Schubert calculus and the Gaudin model*

I will discuss recently discovered connections of quantum integrable systems with Schubert calculus.

**Masahiko Yoshinaga**, Kobe University

*Periods and computational complexity of real numbers.*

The periods, introduced by Kontsevich and Zagier, form a class of complex numbers which contains all algebraic numbers and several transcendental quantities. Little has been known about qualitative properties of periods. In particular, no concrete non-periods have been known. In this talk, we compare the periods with hierarchy of real numbers induced from computational complexities.

**Thomas Zaslavsky**, Binghamton University

*Topological hyperplane arrangements*

A topological hyperplane (which we call a topoplane) is a subspace $H$ of $R^d$
which is topologically like a true hyperplane. An arrangement of topoplanes
is a finite set $\mathcal A$ of topoplanes such that, for each pair $H, H'$
that intersect, the intersection is a topoplane in $H$ and in $H'$. (Actually,
this is a slight

oversimplification.) Therefore, when $H$ and $H'$ intersect, there are two possibilities:
they cross, or they do not cross.

Main examples:

(1) An arrangement of hyperplanes.

(2) An arrangement of affine pseudohyperplanes (representing an

affine oriented matroid).

(3) A convex section of a hyperplane arrangement.

In all these, intersecting topoplanes always cross.

Main Question: Is the number of regions (connected components of the complement of the arrangement) given by the same formula as for an arrangement of hyperplanes? The answer: Yes, if every region is a topological ball. But we do not yet know whether regions must be balls.

Second Question: How different is a topoplane arrangement $\mathcal A$ from
an affine arrangement of pseudohyperplanes? Given some nice property, we may
ask whether every $\mathcal A$ can be `rearranged' into an $\mathcal A'$ such
that $\bigcup \mathcal A = \bigcup \mathcal A'$ (that is, they have the same
regions) and $\mathcal A'$ has the nice property. For instance, consider the
property that every intersecting pair of topoplanes crosses. Then the answer
is: Yes in

the plane, yes in all dimensions if there are no multiple intersections, but
in general no.

Third Question: If intersecting hyperplanes in $\mathcal A$ always cross, is the arrangement a convex section of a hyperplane arrangement or close topological generalization? Las Vergnas says this is true in the plane but not in higher dimensions.

Fourth Question: Are there simple properties that imply all topoplanes cross? For instance, if there are enough topoplanes?

This is joint work with David Forge.