COURSES

The CIMPA school will consist of following courses

Course 1: On the monodromy of Milnor fibers of hyperplane arrangements, given by Clément Dupont (University of Montpellier, France)

Abstract: This is an introductory course, related to the monodromy operators and Milnor fiber of an arrangement. The Milnor fiber F and the monodromy operators h^m: H^m(F;C) → H^m(F;C) will be carefully defined, and the main examples discussed in detail. Several approaches for the computation of the Betti numbers b_i(F) and of the eigenvalues of the monodromy operators h^m will be discussed as well.

Course 2: Arrangements, wonderful models and toric varieties, given by Graham Denham (University of Western Ontario, Canada)

Abstract: This is an introductory course centred around some geometric aspects of complex hyperplane arrangements. By viewing the complement M(A) as a subvariety of a complex torus in a toric variety, we see a number of interesting spaces constructed from a hyperplane arrangement, such as De Concini-Procesi's wonderful compactification. The course will give a working introduction to the combinatorics of matroids and of toric varieties, leading to the notion of the Bergman fan and a tropical linear space. We will consider some modern applications.

Course 3: On free hyperplane arrangements and Terao’s conjecture, given by Masahiko Yoshinaga (Hokkaido University, Sapporo, Japan)

Abstratc: This is an introductory course related to the Terao's conjecture. The notion of a free singularity and of a free hypersurface will be introduced, and a special attention will be given to the analogies and differences between the local analytic case and the global graded algebraic case. The case of line arrangements in P² will be discussed in detail, in particular the cases of no more than 13 lines, when the conjecture is known to hold.

Course 4: On arrangement groups and associated invariants, given by Daniel  Cohen (Louisiana State University, Baton Rouge, USA)

Abstract: This is an inductory course discussing "arrangement groups", fundamental groups of complements of complex hyperplane arrangements, various invariants of these groups, and the interplay among them.

Course 5: Matroids, arrangements, and representation theory, given by Max Wakefield (The United States Naval Academy, Annapolis, USA)

Abstract: This is an introductory course, centered on two aspects, and related to problem of Kazhdan-Lusztig polynomials. First, the combinatorial natural of most of the questions in this theory, which is best formalized by using the matroids. Then, the fact that many important classes of hyperplane arrangements come from complex reflection groups acting on a vector space. Indeed, the hyperplanes of the arrangements are just the reflecting hyperplanes of the group action. A discussion of the Braid Arrangement, of the Monomial Arrangement and of the Full Monomial Arrangement will illustrate this aspect.

Course 6: Toric arrangements, given by Christin Bibby (University of Michigan, Ann Arbor, USA)

Abstract: This is an advanced course in which instead of looking at hyperplanes in an affine space Cⁿ or in a projective space Pⁿ, ones looks at affine subtori in an algebraic torus (C^*)ⁿ or at abelian subvarieties (say subtori) in an abelian variety A, say an algebraic compact torus). Many results from the case of hyperplane arrangements extend to these new situations, but sometimes new techniques and new ideas are involved.