I have been studying hyperplane arrangements. Hyperplane arrangements
appear many branches of mathematics including combinatorics, representation
theory, topology, etc. I am mainly interested in two topics:
I am also interested in computational complexity of real numbers,
especially for the so called "periods" (introduced by Kontsevich and Zagier),
the real numbers which have integral expressions.
- Freeness of logarithmic vector fields: Freeness of Coxeter arrangements
(by Kyoji Saito) and Terao's Factorization theorem are the results I like
most over all mathematics. Freeness is also related to splitting problems
of vector bundles over projective spaces into sum of line bundles.
In my thesis I proved Horrocks type freeness criterion for arrangements, and
apply it to prove that extended Catalan and Shi arrangements are free.
Guiding idea is "Combinatorial phenomina of arrangements
should be explained by some
geometric properties of logarithmic vector fields."
- Minimality and application to topology:
Dimca-Papadima, and Randell proved that the complemental space to
a complex hyperplane arrangement is homotopic to a minimal CW complex,
i.e., the finite CW complex satisfying the number of k-dimensional cells
equals to k-th Betti number (for k=0, 1, 2,...) This results unifies
several formar results by Hattori, Falk, Suciu, etc. The minimality is
(as far as I know) peculiar to hyperplane arrangements. I have been
trying to understand attaching maps of the minimal CW complex for
real arrangements in terms of real structures. Recently I am
applying them to computing Betti numbers of Milnor fiber of arrangements.
List of papers and
I welcome students who are interested in (at least one of)
research areas: combinatorics, algebraic geometry and topology.
(21 Feb. 2013)