`abe@imi.kyushu-u.ac.jp`

Hyperplane arrangements are finite sets of linear hyperplanes in a vector space, whose origin is reflecting hyperplanes of Weyl groups. The polynomial vector fields tangent to a hyperplane arrangement is called the module of logarithmic vector fields, and an arrangement is free if the module of logarithmic vector fields is a free module. Whether the freeness depends only on the combinatorial data of arrangements or not, is not yet solved, and called Terao's conjecture. Recently, several developments on this conjecture are achieved and it becomes easy to check the combinatorial dependency of a given free arrangement. We explain them, and if time permits, a new algebra called the Solomon-Terao algebra is also introduced, that connects the algebra of hyperplane arrangements and cohomology rings of regular nilpotent Hessenberg varieties.