Topicbild

Bildarchiv des Mathematischen Forschungsinstituts Oberwolfach



Arbeitsgebiete

  • Algebraische Lie Theorie
  • Algebraische Gruppen
  • Darstellungstheorie
  • Endliche Gruppen vom Lie-Typ
  • Hyperebenenarrangements
  • Spiegelungsarrangements

  • DFG project: Serre's notion of Complete Reducibility and Geometric Invariant Theory (within the DFG Priority Programme in Representation Theory).
     
  • DFG project: Computational aspects of the Cohomology of Coxeter arrangements: On Conjectures of Lehrer-Solomon and Felder-Veselov (within the DFG Priority Programme "Algorithmic and experimental methods in Algebra, Geometry and Number Theory").
     
  • DFG project: Arrangements of complex reflection groups: Geometry and combinatorics (joint with M. Cuntz (Kaiserslautern) within the DFG Priority Programme "Algorithmic and experimental methods in Algebra, Geometry and Number Theory").
     
  • DFG project: Inductive freeness and rank-generating functions for arrangements of ideal type: Two conjectures of Sommers and Tymoczko revisited (within the DFG GEPRIS).
     
  • DFG project: On the Cohomology of complements of complex reflection arrangements (within the DFG GEPRIS).
     
  • Publications on MathSciNet.

  • Recent preprints on the ArXiv.
     
  • Latest preprints:

    A construction of pseudo-reductive groups with non-reduced root system

    with Michael Bate, Damian Sercombe and David I. Stewart
    Abstract:
    We describe a straightforward construction of the pseudo-split absolutely pseudo-simple groups of minimal type with irreducible root systems of type BC_n; these exist only in characteristic 2. We also give a formula for the dimensions of their irreducible modules.
    math.GR/2204.09540

    Inductive Freeness of Ziegler's Canonical Multiderivations

    with Torsten Hoge
    Abstract:
    Let A be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction A'' of A to any hyperplane endowed with the natural multiplicity k is then a free multiarrangement (A'',k). The aim of this paper is to prove an analogue of Ziegler's theorem for the stronger notion of inductive freeness: if A is inductively free, then so is the multiarrangement (A'',k). In a related result we derive that if a deletion A' of A is free and the corresponding restriction A'' is inductively free, then so is (A'',k) -- irrespective of the freeness of A. In addition, we show counterparts of the latter kind for additive and recursive freeness.
    math.CO/2204.09540

    On Formality and Combinatorial Formality for hyperplane arrangements

    with Tilman Möller and Paul Mücksch
    Abstract:
    A hyperplane arrangement is called formal provided all linear dependencies among the defining forms of the hyperplanes are generated by ones corresponding to intersections of codimension two. The significance of this notion stems from the fact that complex arrangements with aspherical complements are formal. The aim of this note is twofold. While work of Yuzvinsky shows that formality is not combinatorial, in our first main theorem we prove that the combinatorial property of niceness of arrangements does entail formality. Our second main theorem shows that formality is hereditary, i.e. is passed to restrictions. This is rather counter-intuitive, as in contrast the known sufficient conditions for formality, i.e. asphericity, freeness and niceness (owed to our first theorem), are not hereditary themselves. We also demonstrate that the stronger property of k-formality, due to Brandt and Terao, is not hereditary.
    math.CO/2202.09104

    Invariants and semi-invariants in the cohomology of the complement of a reflection arrangement

    with J. Matthew Douglass and Götz Pfeiffer
    Abstract:
    Suppose V is a finite dimensional, complex vector space, A is a finite set of codimension one subspaces of V, and G is a finite subgroup of the general linear group GL(V) that permutes the hyperplanes in A. In this paper we study invariants and semi-invariants in the graded QG-module H^*(M(A)), where M(A) denotes the complement in V of the hyperplanes in A and H^* denotes rational singular cohomology, in the case when G is generated by reflections in V and A is the set of reflecting hyperplanes determined by G, or a closely related arrangement. Our first main result is the construction of an explicit, natural (from the point of view of Coxeter groups) basis of the space of invariants, H^*(M(A))^G. In addition to providing a conceptual proof of a conjecture due to Felder and Veselov for Coxeter groups, this result extends the latter to all finite complex reflection groups. Moreover, we prove that determinant-like characters of complex reflection groups do not occur in H^*(M(A)). This extends to all finite complex reflection groups a result proved for Weyl groups by Lehrer.
    math.RT/2009.12847

    A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups

    with T. Abe, C. Stump and M. Yoshinaga
    Abstract:
    Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration of that module. Our approach is based on a detailed analysis of a flat connection applied to the primitive vector field. This generalizes and unifies analogous results for real reflection groups.
    math.DG/1809.05026


  •  

    Former and current PhD students:
    Simon Goodwin (University of Birmingham)
    Michael Bate (University of York)
    Russell Fowler (Npower)
    Glenn Ubly (NHS)
    Sebastian Herpel (Barmenia Allgemeine Versicherungs-AG)
    Peter Mosch (Volkswohl Bund)
    Nils Amend (INTER Krankenversicherung AG)
    Anne Schauenburg (Aldi International Services)
    Maike Gruchot (Barmenia Krankenversicherung AG)
    Tilman Möller
    Falk Bannuscher
    Sören Böhm
    Sven Wiesner


    Editorial Activity:

    Advanced Studies: Euro-Tbilisi Mathematical Journal .

    Computer Programs