The focus of my research is on the combinatorics of arrangements of hyperplanes, and in particular on its connection with the topology of the arrangement’s complement.

• Here is a detailed research statement.

BRIEF SUMMARY OF RESEARCH PLAN:

One of the main questions in the theory of hyperplane arrangements asks how far the topology of the complement of a set of hyperplanes in complex space is determined by its combinatorial data. Both historically and recently, major advances were inspired by the theory of Artin groups.
Artin groups (of finite type) can be seen as fundamental groups of the complement of the set of complexified reflecting hyperplanes associated to a (finite) Coxeter Group. Moreover, these spaces are aspherical (or Eilenberg - Maclane spaces), as was shown by Deligne, answering a problem posed by Brieskorn. Because of the combinatorial nature of Deligne’s argument, since then one of the main open problems about hyperplane arrangements is whether the property of having an aspherical complement is determined by the combinatorial data of the arrangement.
Garside groups are a natural generalization of Artin groups and can be defined by combinatorial data (namely, a labelled lattice satisfying certain conditions). Recently, attention was paid to the remarkable fact that any finite type Artin group can indeed be generated by two different such combinatorial data sets. One of these describes the ‘classical’ presentation of Artin groups, and the lattices that are involved here (i.e., the weak ordering of the corresponding Coxeter group) are special cases of posets that can be defined geometrically for any real arrangement - and when these posets are lattices, then the complexification of the arrangement is aspherical. Some light was shed also on the so-called ‘dual’ Garside structure of finite type Artin groups, which can be described combinatorially by the lattice of generalized noncrossing partitions. This structure can be generalized to the case of unitary reflection groups following Bessis, who uses the involved lattices to give an argument (inspired by Deligne’s) to prove asphericity of the complement of any finite, real or unitary reflection arrangement.
However, unlike the case of real arrangements and ‘standard’ Garside structures, there is not yet a geometric or combinatorial description of the posets giving the ‘dual’ Garside structures that would allow to extend the concept to general complex arrangements. Having handled the complexified case, my research aims also towards the completion of this picture.
For example, let us mention that the same posets seem to appear also in the (up to now) little related question of “minimal models” for arrangement complements, as (subsets of the) posets of cells of some CW-complexes that can be described in combinatorial terms. We will explain how this might help clarify the general picture.