This is our main research seminar. Talks are given both by guests and members of our group.

Let $\Phi$ be a finite root system. A $\Phi$-graded group is a group $G$ together with a family of subgroups $(U_\alpha)_{\alpha \in \Phi}$ satisfying some purely combinatorial axioms. The main examples of such groups are the Chevalley groups of type $\Phi$, which are defined over commutative rings and which satisfy the well-known Chevalley commutator formula. We show that if $\Phi$ is of rank at least 3, then every $\Phi$-graded group is defined over some algebraic structure (e.g. a ring, possibly non-commutative or, in low ranks, even non-associative) such that a generalised version of the Chevalley commutator formula is satisfied. A new computational method called the blueprint technique is crucial in overcoming certain problems in characteristic 2. This method is inspired by a paper of Ronan-Tits.

Let $G$ be a group acting on a set $\Omega$, preserving some equivalence relation $\sim$, and let $k \ge 2$. A $k$-tuple in $\Omega$ is 'distant' if no two entries come from the same block of $\sim$, and $G$ is '$k$-by-block-transitive' if there are at least $k$ blocks and $G$ acts transitively on the set of all distant $k$-tuples.
I will present a classification of the finite $k$-by-block-transitive actions with nontrivial blocks, such that the action on the set of blocks is faithful. It turns out no such actions occur for $k \ge 3$, but there are some for $k=2$ including an infinite family of sharply $2$-by-block-transitive actions (a geometric construction of which was recently suggested to me by Hendrik Van Maldeghem). I will also say something about the original motivation, which relates to groups acting $k$-transitively on the ends of an infinite tree.

Digital communications relies heavily on the usage of different types of codes. In general, a $d$-code Y is a finite subset in a metric space such that $d$ is the minimum distance that can occur between two distinct elements of $Y$. A central problem is to derive upper bounds for the size of codes. We will first introduce Delsarte's linear programming method, which is a powerful tool to obtain such bounds for different types of codes. Some codes that reach these bounds are related to different types of Steiner systems. We will focus on codes and Steiner systems in polar spaces, which consist of the totally isotropic subspaces of a finite vector space equipped with a nondegenerate form. Using bounds for codes in polar spaces, derived by Delsarte's method, we will give an almost complete classification of Steiner systems in polar spaces. Namely, we will show that such Steiner systems can only exist in some corner cases.

Upper triangular matrices with ones on the diagonal and
entries which are integers (or algebraic integers) arise in many
contexts, e.g. as Stokes matrices in the theory of meromorphic
connections with irregular poles, in many situations in algebraic
geometry (often related to Stokes matrices), especially in quantum
cohomology and the theory of isolated hypersurface singularities, but
also in the theory of Coxeter groups.
Concepts from singularity theory like vanishing cycles, monodromy
groups, Seifert forms, tuples of (pseudo-)reﬂections and
distinguished bases can be derived from upper triangular matrices in
cases beyond singularity theory and are interesting to study.
Additionally, always braid group actions on the matrices and on the
distinguished bases are in the background. They give rise to certain
covering spaces of the classifying space of the braid group. These are
interesting natural global manifolds. Some are well known, others are
new. The talk presents concepts and old and new results. It puts
emphasis on some cases from singularity theory and some 3x3 cases.

A finite graph determines a Kirchhoff polynomial, which is a squarefree, homogeneous polynomial in a set of variables indexed by
the edges. The Kirchhoff polynomial appears in an integrand in the
study of particle interactions in high-energy physics, and this provides
some incentive to study the motives and periods arising from the projective hypersurface cut out by such a polynomial.
From the geometric perspective, work of Bloch, Esnault and Kreimer
(2006) suggested that the most natural object of study is a polynomial determined by a linear matroid realization, for which the Kirchhoff polynomial is a special case.
I will describe some joint work with Avi Steiner, Delphine Pol, Mathias
Schulze and Uli Walther on the interplay between geometry and matroid
combinatorics for this family of objects, and the relationship with logarithmic derivations on the associated hyperplane arrangements.

Non-commutative exceptional curves are generalizations of weighted projective lines of Geigle and Lenzing. In my talk, I am going to give an overview of the theory of these curves, explaining their relation with tame hereditary algebras and canonical algebras of Ringel.

A matroid is realizable if we can obtain its bases from the indices of linearly independent columns of some matrix. For a given matroid $M$, this matrix is not unique. The space of all such matrices can be given the structure of an affine scheme, known as the realization space of $M$. It is known that realization spaces of matroids can be arbitrarily singular, although there are few concrete examples. We use software to study smoothness and irreducibility of realization spaces of rank 3 and rank 4 matroids, isolating examples of singular spaces for $(3,12)$-matroids. As an application, we show that singular initial degenerations exist for the $(3,12)$-Grassmannian.

Groups of automorphisms of regular rooted trees are a rich source of examples with interesting properties in group theory, and they have been used to solve very important problems. The first Grigorchuk group, defined by Grigorchuk in 1980, is one of the first instances of an infinite finitely generated periodic group, thus providing a negative solution to the General Burnside Problem. It is also the first example of a group with intermediate growth, hence solving the Milnor Problem. Many other groups of automorphisms of rooted trees have since been defined and studied. Important examples are the Gupta-Sidki $p$-groups, for $p$ an odd prime, and the second Grigorchuk group. These are again finitely generated infinite periodic groups and they belong to the large family of the so-called Grigorchuk-Gupta-Sidki groups (GGS-groups, for short).
Some research has been done regarding the lower central series of groups of automorphisms of regular rooted trees and more specifically of some particular GGS-groups, but the knowledge is scarce. The aim of this talk is to present some of the techniques and tools that we have developed and are still developing in order to understand the lower central series of some of the GGS-groups that act on the $p$-adic tree, and how do we use them to completely determine the lower central series of some GGS-groups such us the $p$-Fabrykowski-Gupta groups.
This is a joint work with G. Fernández-Alcober and M. Noce.

A group is invariably generated if there is some generating set S such that upon replacing any number of elements of S with arbitrary conjugates of these elements, it still results in a generating set for the group. All finite groups are invariably generated, but this is not the case for infinite groups. For example, Kantor, Lubotzky and Shalev showed that a finitely generated linear group is invariably generated if and only if it is virtually solvable. In this talk, we investigate invariable generation among some classical examples of branch groups, such as the Grigorchuk-Gupta-Sidki groups; this is joint work with Charles Cox. Recall that branch groups are well-studied groups acting on rooted trees, that first arose as explicit examples of infinite finitely generated torsion groups, and since then have established themselves as important infinite groups, with numerous applications within group theory and beyond.

We call a poset factorable if its characteristic polynomial has all positive integer roots. Inspired by inductive and divisional freeness of a central hyperplane arrangement, we introduce and study the notion of inductive posets and their superclass of divisional posets. It then motivates us to define the so-called inductive and divisional abelian (Lie group) arrangements, whose posets of layers serve as the main examples of our posets. We will present three main results. Our first result is that every divisional poset is factorable. Our second result shows that the class of inductive posets contains strictly supersolvable posets, the notion recently introduced due to Bibby and Delucchi (2022). This result can be regarded as an extension of a classical result due to Jambu and Terao (1984), which asserts that every supersolvable hyperplane arrangement is inductively free. Our third result is an application to toric arrangements, which states that the toric arrangement defined by an arbitrary ideal of a root system of type $A$, $B$ or $C$ with respect to the root lattice is inductive. This is joint work with R. Pagaria, M. Pismataro and L. Vecchi (Bologna).

Digital communications relies heavily on the usage of different types of codes. In general, a $d$-code Y is a finite subset in a metric space such that $d$ is the minimum distance that can occur between two distinct elements of $Y$. A central problem is to derive upper bounds for the size of codes. We will first introduce Delsarte's linear programming method, which is a powerful tool to obtain such bounds for different types of codes. Some codes that reach these bounds are related to different types of Steiner systems. We will focus on codes and Steiner systems in polar spaces, which consist of the totally isotropic subspaces of a finite vector space equipped with a nondegenerate form. Using bounds for codes in polar spaces, derived by Delsarte's method, we will give an almost complete classification of Steiner systems in polar spaces. Namely, we will show that such Steiner systems can only exist in some corner cases.

Date | Time | Room | Speaker | Title | Abstract |
---|---|---|---|---|---|

11 Oct 2023 | 14:15 - 15:45 | U2-232 | Torben Wiedemann | Root Graded Groups | |

18 Oct 2023 | 14:15 - 15:45 | U2-232 | Colin Reid (Muenster) | Multiple transitivity except for a system of imprimitivity | |

25 Oct 2023 | 14:15 - 15:45 | U2-232 | Charlene Weiß (Paderborn) | Cancelled | |

8 Nov 2023 | 14:15 - 15:45 | U2-232 | Claus Hertling (Mannheim) | Upper triangular matrices and induced structures: vanishing cycles, monodromy groups, distinguished bases, braid group orbits, moduli spaces | |

15 Nov 2023 | 14:15 - 15:45 | U2-232 | Graham Denham (London, ON) | Kirchhoff polynomials configuration hypersurfaces | |

22 Nov 2023 | 14:05 - 15:45 | U2-232 | Igor Burban (Paderborn) | Introduction to the theory of non-commutative exceptional curves | |

29 Nov 2023 | 14:15 - 15:45 | U2-232 | Dante Luber (FU Berlin) | Singular Matroid Realization Spaces | |

6 Dec 2023 | 14:15 - 15:45 | U2-232 | Mikel Garciarena | The Lower Central Series of certain Grigorchuk-Gupta-Sidki groups | |

13 Dec 2023 | 14:15 - 15:45 | U2-232 | Anitha Thillaisundaram (Lund) | Invariable generation of certain branch groups | |

20 Dec 2023 | 00:00 - 00:00 | Gemma Crowe (Heriott-Watt) | TBA | ||

10 Jan 2024 | 14:15 - 15:45 | U2-232 | Tan Nhat Tran (Hannover) | Inductive and divisional posets | |

24 Jan 2024 | 14:15 - 15:45 | Charlene Weiß (Paderborn) | Codes and Steiner Systems | ||

31 Jan 2024 | 14:15 - 15:45 | U2-232 | Burban/Krause | TBA |

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