### Wissenschaftliche Arbeitsgebiete

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

### Vortrag im Oberseminar Arrangements and Symmetries

Vortrag am Montag, 29.04.2024, 16:15 Uhr, in IA 1/135

Vortragender: Daniel Bath (KU Leuven)

Titel: "Bernstein—Sato polynomials of Hyperplane Arrangements in C^3"

Abstract:

The roots of Bernstein—Sato polynomial of a hypersurface manages to simultaneously contain most classical singularity invariants. Alas, computing these roots is mostly infeasible. For hyperplane arrangements in C^3, one can hope the roots of its Bernstein—Sato polynomial are combinatorially determined. Alas, things are more subtle. Walther demonstrated two arrangements with the same intersection lattice but whose respective Bernstein—Sato polynomials differ by exactly one root. We will show this is the only pathology possible. For arrangements in C^3, we prove that all but one root are (easily) combinatorially determined. We also give several equivalent criterion for the outlier, -2 + (2/deg), to in fact be a root of the Bernstein—Sato polynomial. These involve local cohomology data of the Milnor algebra and the non-formality of the arrangement.

This is an application of a study of Bernstein—Sato polynomials for a larger class of C^3 divisors than just arrangements. We will discuss Bernstein—Sato polynomials at large, our general strategy for divisor class, our main results, and how the promised formula for hyperplane arrangements appears.

### Vortrag im Oberseminar Lie-Theorie

Vortrag am Montag, 15.04.2024, 14:15 Uhr, in IA 1/75

Vortragender: David Schwein (University of Bonn)

Titel: "Tame supercuspidals at very small primes"

Abstract:

Supercuspidal representations are the elementary particles in the representation theory of reductive p-adic groups. Constructing such representations explicitly, via (compact) induction, is a longstanding open problem, solved when p is large. When p is small, the remaining supercuspidals are expected to have an arithmetic source: wildly ramified field extensions. In this talk I'll discuss ongoing work joint with Jessica Fintzen that identifies a second, Lie-theoretic, source of new (tame!) supercuspidals: special features of reductive groups at very small primes. We'll summarize some of these features and explain how they contribute to the construction of supercuspidals.

### Vortrag im Oberseminar Arrangements and Symmetries

Vortrag am Montag, 22.01.2024, 14:00 Uhr, in IA 1/177

Vortragender: Johannes Schmitt (Universität Siegen)

Titel: "Symplectic reflections and quotient singularities"

Abstract:

Let V be a finite-dimensional vector space over the complex numbers and let G < GL(V)
be a finite group. It is classically known that the reflections contained in G control
aspects of the geometry of the linear quotient V/G, namely its smoothness (Chevalley-
Shephard-Todd) and the class group (Benson). In this talk, we restrict to symplectic
vector spaces V and groups G < Sp(V) which leave the symplectic form invariant and
study partial symplectic resolutions of V/G. Here we find similar phenomena. We
describe how the existence of a (smooth) symplectic resolution is related to aptly named
symplectic reflections in G by a theorem of Verbitsky, but also how this relationship
is still not completely understood. We further show that the class group of a partial
symplectic resolution is controlled by the symplectic reflections in G.

### Vortrag im Oberseminar Arrangements and Symmetries

Vortrag am Montag, 15.01.2024, 14:00 Uhr, in IA 1/177

Vortragender: Takuro Abe (Rikkyo University)

Titel: "Multi-Euler derivations"

Abstract:

In the study of hyperplane arrangements, the research of the logarithmic derivation modules of them is one of the most important topics. Based on Kyoji Saito's primitive derivation, Terao's investigation of the multi-Coxeter arrangements and Yoshinaga's solving of the Edelman-Reiner conjecture, the study of logarithmic derivation modules of multi-reflection arrangements is also important. Among these researches, the existence of so called the universal derivations, defined by Wakamiko, has been a key of the research, and investigated by Stump, Roehrle, Terao, Yoshinaga, Wakamiko and so on.
In this talk, we generalize this concept of universal derivations to arbitrary arrangements as the multi-Euler derivations, and give a criterion for a derivation to be multi-Euler.

### Workshop Algebraic Groups and Representation Theory

Es finden am Montag, 04.12.2023, drei Vorträge statt

**Vortrag 1**

2:15 pm, IA 1/177

Vortragender: Alastair Litterick (University of Essex, UK)

Title: "Complete Reducibility: The Good, the Bad and the Conjugacy"

Abstract: The concept of a "completely reducible subgroup" generalises the representation-theoretic notion (the special case of the general linear group) to other reductive algebraic groups. This turns out to have deep interpretations in terms of algebraic geometry and geometric invariant theory. A long-running project seeks to harness complete reducibility to classify reductive subgroups of reductive groups, particularly the exceptional simple groups. As in representation theory, the study becomes more difficult the smaller the prime, with characteristic 2 presenting qualitative difficulties that do not arise elsewhere. We will discuss these, and give an overview of the current state of the classification.

**Vortrag 2**

3:30 pm, IA 1/177

Vortragende: Jacinta Torres (Jagiellonian University in Krakow, Poland)

Title: "Atoms and charge beyond type A"

Abstract: We present a new phenomenon which allows to compute certain atomic and preatomic decompositions of crystals. This phenomenon provides a conceptual explanation, as well as generalization to type C_n, of the preatomic decompositions and the cyclage map in type A_n. We also present the charge statistic for type C_2 crystals, giving positive combinatorial formulas for Kostka-Foulkes polynomials. This is joint work with Leonardo Patimo.

**Vortrag 3**

5:30 pm, IA 1/109

Vortragender: Benjamin Martin (University of Aberdeen, UK)

Title: "Smooth stabilisers for actions of algebraic groups"

Abstract: Let G be a linear algebraic group over an algebraically closed field k of characteristic p≥0, and let X be an affine variety on which G acts. Given a closed subvariety Y of X, we denote by C_G(Y) and N_G(Y) respectively the pointwise and setwise stabilisers of Y, regarded as closed subschemes of G. There are many situations where one wishes to know whether C_G(Y) and N_G(Y) are smooth. Some results are known in specific settings: for instance, if G is connected and reductive and G acts on X:=G by conjugation then C_G(Y) is smooth for any closed subgroup Y of G if p is very good for G, whereas C_G(Y) = Z(G) is not smooth if G= SL_n, Y= G and p divides n.
Smoothness always holds when p=0 by a well-known result of Cartier. Examples like the one above suggest that smoothness should hold as long as p is sufficiently large, but it is not easy to make precise what ``sufficiently large’’ should mean. David Stewart, Lewis Topley and I recently formulated and proved some results along these lines. I will state some of our theorems and discuss the ideas behind the proofs, which combine the Lefschetz principle from model theory with Gröbner basis techniques.

### Vortrag im Oberseminar Lie-Theorie

Vortrag am Montag, 20.11.2023, 16 Uhr, in IA 1/109

Vortragender: Vanthana Ganeshalingam (University of Warwick)

Titel: "Subgroup Structure of Reductive Groups"

Abstract:

This talk will introduce the concept of G complete-reducibility (G c-r) originally thought of by Serre in the 90s. This idea has important connections to the open problem of classifying the subgroups of a reductive group G. I will explain the methodology of the classification so far and the main obstacle which is understanding the non-G-cr subgroups.

### Doppel-Vortrag im Oberseminar Arrangements and Symmetries

**Vortrag 1** am Montag, 13.11.2023, 14:15 - 14:45 Uhr, in IA 1/177

Vortragender: Sven Wiesner (RUB)

Titel: "Free multiderivations on connected subgraph arrangements "

Abstract:

Recently, Cuntz and Kühne introduced (a way to assign) a hyperplane arrangement A(G) associated with a given connected graph G and they classified all graphs G such that A(G) is a free arrangement. I will present joint work with Gerhard Röhrle and Paul Mücksch in which we generalize this result to the multiarrangement case. We give a complete list of graphs which possess at least one non-trivial multiplicity m such that the corresponding multiarrangement (A(G),m) is free.

**Vortrag 2** am Montag, 13.11.2023, 14:45 - 15:15 Uhr, in IA 1/177

Vortragender: Lorenzo Giordani (RUB)

Titel: "Cohomology Rings of toric wonderful Models "

Abstract:

One of the leading motifs in the theory of arrangements is to understand the interplay of algebraic, geometric and topological properties with combinatorial ones, meant as properties of the intersection structure of the arrangements. An approach in this direction, initiated by DeConcini and Procesi in the nineties, is the introduction of ``wonderful'' compactifications for complement spaces of subspace arrangements.
The study of wonderful models has ensured the combinatorial nature of relevant topological invariants of complement spaces, namely: cohomology, rational homotopy type, and mixed Hodge structure. Projective wonderful models were extended to the case of toric arrangements of arbitrary codimension. Their cohomology has then been studied by DeConcini and Gaiffi in the ``well-connected'' case, in which properties holding naturally in the linear case are imposed.
I will report on work in progress with Roberto Pagaria and Viola Siconolfi, where we remove the hypothesis of well-connectedness, and offer a different presentation for the cohomology ring of the model. Our approach is closer to the combinatorial picture: we adapt the notions of blowups for semilattices and nested set complex introduced by Feichtner and Kozlov to the poset of layers of toric arrangements.

### Vortrag im Oberseminar Arrangements and Symmetries

Vortrag am Montag, 06.11.2023, 14 -16 Uhr, in IA 1/177

Vortragender: Torben Wiedemann (Bielefeld/Giessen)

Titel: "Root Graded Groups"

Abstract:

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.

### Vortrag im Oberseminar Arrangements and Symmetries

Vortrag am Montag, 23.10.2023, 14:00 Uhr, in IA 1/177

Vortragender: Graham Denham (University of Western Ontario)

Titel: "Kirchhoff polynomials configuration hypersurfaces"

Abstract:

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.

### Spring School on "Real, Complex, and Symplectic Reflection Groups"

From Monday 6 March until Friday 10 March, 2023 we will host a Spring School on "Real, Complex, and Symplectic Reflection Groups", on the Ruhr-University Bochum Campus.

This spring school aims at introducing junior researchers to a variety of topics in the theory of reflection groups, reflection arrangements, and related topics. The school will consist of three lecture series given over four days, followed by a day of conference-style talks.

Lecture series will be given by Götz Pfeiffer (Galway), Ulrich Thiel (Kaiserslautern), and Masahiko Yoshinaga (Osaka). Additional talks will be given by Eirini Chavli (Stuttgart), Michael Cuntz (Hannover), Paul Mücksch (Kyushu), and Tan Tran (Bochum).

Organisers: Georges Neaime, Gerhard Röhrle, Christian Stump

### Alexander von Humboldt Fellowship awarded to Dr Sercombe

Dr Sercombe has received a two year Alexander von Humboldt Fellowship to conduct independent research at Ruhr-University Bochum.

Dr Sercombe's field of research is algebraic groups over an arbitrary field k. His project within his Alexander von Humboldt Fellowship centers around the problem of classifying maximal subgroups of a simple k-group G up to conjugacy by some element of G(k), in terms of Tits indices. This project follows on from his PhD work, which looked at maximal subgroups of maximal rank in G.

In addition Dr Sercombe has recently become interested in more general group schemes, in particular pseudo-reductive groups. He will be looking to generalise some of the aforementioned ideas to pseudo-simple k-groups.

Dr Sercombe works in the research group of Prof. Dr Gerhard Röhrle.

### Oberwolfach Seminar June 2022

"G-Complete Reducibility, Geometric Invariant Theory and Spherical Buildings"

5 June - 11 June, 2022

Summary:

The "Oberwolfach Seminar 2223b" is in a core area of algebraic group
theory and at the
interdisciplinary cross roads of algebra and representation theory on
the one hand, geometry
and geometric invariant theory on the other. The notion of G-complete
reducibility for
subgroups of a reductive algebraic group G was introduced by J-P. Serre
in the 1990s as a
natural generalization of the notion of complete reducibility in
representation theory (which
corresponds to the case where G is the general linear group). Since its
introduction, this notion has been widely studied, both as an
important concept in its own right, with applications to the structure of
linear algebraic groups, and also as a useful tool with applications in
representation theory,
geometric invariant theory, the theory of buildings, and number theory.

The aim of this Oberwolfach seminar is to introduce participants to
G-complete reducibility and explain some of its many applications across
pure mathematics
— participants will learn some rich and deep modern algebra, and leave
equipped with an
understanding of how this mathematics continues to be applied to solve a
diverse range of
problems, particularly in the theory of algebraic groups.

Please see the detailed program and a recommended reading list on the
website of the seminar.

### Mathematisches Forschungsinstitut Oberwolfach

Logarithmic Vector Fields and Freeness of Divisors and Arrangements: New perspectives and applications (online meeting)

24 January - 30 January 2021

Organizers:

Takuro Abe, Fukuoka

Alexandru Dimca, Nice

Eva-Maria Feichtner, Bremen

Gerhard Röhrle, Bochum

### Promotionen in 2021

Unser wissenschaftlicher Mitarbeiter Falk Bannuscher hat im November 2021 seine Dissertation erfolgreich verteidigt und seine Promotion zum "Dr. rer. nat." mit Bravour abgeschlossen.

Herr Bannuscher hat seine Dissertation zum Thema: "On G-complete reducibility over non-perfect fields" zum Begriff von Serre von G-vollständiger Zerlegbarkeit verfasst.

Der Lehrstuhl für Algebra und Zahlentheorie gratuliert Herrn Bannuscher zu dieser herausragenden Leistung herzlich und wünscht ihm für seine weitere Karriere alles Gute.

### Alexander von Humboldt Fellowship awarded to Dr Tran

We are very pleased to announce that Dr. Tan Nhat Tran has been awarded a competitive two-year Alexander von Humboldt Fellowship for Postdoctoral Researchers, hosted by Professor Gerhard Röhrle, with a research topic of 'Tutte polynomial and its applications to arrangement theory'.

Tutte polynomial is one of the most-studied invariants of graphs. This two-variable polynomial encodes a substantial amount of the combinatorial information of a graph, and specializes to several important graph polynomials (including the chromatic, flow and reliability polynomials). Significant features of the Tutte polynomial have also been shown in diverse areas of mathematics and physics, for instance, it appears as the Jones and homfly polynomials in knot theory, and as the Ising and Potts model partition functions in statistical mechanics.

From arrangement theory's viewpoint, the Tutte polynomial is unquestionably important because of the pervasiveness of its extensions from graphs to other objects that have richer combinatorial and topological properties. These extensions find applications to three broad types of arrangements: hyperplane, toric and subgroup arrangements. Dr. Tran's research focuses on seeking general frameworks to study arrangements and their Tutte-like polynomials globally that may reveal interesting connections between them.

### Promotionen in 2020

Unsere wissenschaftliche Mitarbeiterin Anne Schauenburg hat im August 2020 ihre Dissertation erfolgreich verteidigt und ihre Promotion zum "Dr. rer. nat." mit Bravour abgeschlossen.

Frau Schauenburg hat ihre Dissertation zum Thema: "Ideal Subarrangements of Real Reflection Arrangements and Notions of Freeness" auf einem aktuellen Gebiet der Hyperebenenarrangements verfasst.

Alle Mitglieder des Lehrstuhls für Algebra und Zahlentheorie gratulieren Frau Schauenburg zu dieser herausragenden Leistung herzlich und wünschen ihr für die weitere Karriere alles Gute.

### Workshop on Hyperplane Arrangements and Reflection Groups

Leibniz Universität Hannover, Germany · September 23 - 27, 2019

Organizers:

Michael Cuntz (Hannover), Gerhard Röhrle (Bochum) and Christian Stump (Bochum)

Confirmed speakers:

Takuro Abe (Kyushu University)

Alexandru Dimca (Nice University)

Matthew Douglass (NSF)

Eva-Maria Feichtner (Bremen University)

Misha Feigin (University of Glasgow)

Lukas Kühne (Hebrew University of Jerusalem)

Jean Michel (Université Denis Diderot)

Tilman Möller (Bochum University)

Paul Mücksch (Bochum University)

Bernhard Mühlherr (Giessen University)

Luis Paris (Université de Bourgogne)

Götz Pfeiffer (National University of Ireland)

Mario Salvetti (Università di Pisa)

Anne Shepler (University of North Texas)

Jiro Sekiguchi (Tokyo University of Agriculture and Technology)

Michele Torielli (Hokkaido University)

Masahiko Yoshinaga (Hokkaido University)

### Summer School - Perspectives in Linear Algebraic Groups

From Monday 19th August until Friday 23rd August, 2019 we will host a Summer School on
"Perspectives in Linear Algebraic Groups", on Ruhr-University Bochum Campus.

This summer school aims at introducing junior researchers to a variety of
topics in linear algebraic groups over arbitrary fields, with a focus on
pseudo-reductive groups and unusual behavior over fields which are not
algebraically closed.

The school will consist of four lecture series given over four days,
followed by a day of conference-style talks.

Lecture series will be given by Ben Martin (University of Aberdeen),
Bernhard Mühlherr (University of Giessen), Gopal Prasad (University of Michigan)
and Bertrand Rémy (Université Paris-Saclay). Additional talks will be given by
Philippe Gilles (Université Lyon 1, France), David Stewart (Newcastle University, UK),
Adam Thomas (Birmingham, UK) and Tomohiro Uchiyama (Soka University, Japan).

### Promotionen in 2019

Unsere wissenschaftlichen Mitarbeiter Maike Gruchot und Tilman Möller haben beide ihre Dissertationen erfolgreich verteidigt und ihre Promotion zum "Dr. rer. nat." mit Bravour abgeschlossen.

Frau Gruchot untersuchte in ihrer Dissertation zum Thema: "A Relative Variant of Complete Reducibility" eine Verallgemeinerung des Begriffes von Serre von G-vollständiger Zerlegbarkeit.

Herr Möller widmete sich in seiner Arbeit "Combinatorial properties of hyperplane arrangements and reflection arrangements" mehreren aktuellen Themen aus der Theorie der Hyperebenenarrangements.

Der Lehrstuhl für Algebra und Zahlentheorie gratuliert den beiden zu ihren herausragenden Leistungen recht herzlich und wünscht ihnen für ihre weitere Karriere alles Gute.

### Alexander von Humboldt Fellowship awarded to Dr Uchiyama

We are very pleased to announce that Dr Tomohiro Uchiyama has been awarded a competitive six-month Alexander von Humboldt Fellowship for Postdoctoral Researchers, hosted by Professor Gerhard Röhrle, with a research topic 'Complete reducibility, geometric invariant theory, and spherical buildings'.

Dr. Uchiyama's research focuses on subgroup structures of reductive algebraic groups (matrix groups). In particular, he studies Serre's notion of complete reducibility of subgroups of reductive groups via geometric invariant theory (a branch of algebraic geometry) and the theory of spherical buildings (highly symmetric combinatorial objects).
In this research project the recently proved 50-years-old center conjecture of Tits comes into play. He also investigates various relations between complete reducibility and pseudo-reductivity due to Conrad-Gabber-Prasad.

### Summer School – New Perspectives in Hyperplane Arrangements

In September 2018 Tilmann Möller, Paul Mücksch, Gerhard Röhrle, and Anne Schauenburg held a summer school directed focusing on new developments in the theory of hyperplane arrangements and related topics.

The school was aimed towards Ph.D. students and young Postdocs in the field. The international set of speakers and the participants came from all over the globe.

The school consisted of six series of minicourses on several topics followed by a day of conference-style talks from international experts. The summer school was funded by RUB Research School.

### Spring School on Complete Reducibility

In April 2018 we will host a spring school, aimed at introducing graduate students and junior researchers to the notion of G-complete reducibility, featuring a lecture series by Professor Ben Martin (Aberdeen), and a number of additional lectures.

### Sloan Visiting Professor Cohen

Professor Daniel Cohen from Louisiana State University in Baton Rouge, LA, USA is currently visiting the Department of Mathematics. Professor Cohen's visit to Bochum is supported by the Sloan Visiting Professor scheme (SVP) of the Mathematisches Forschungsinstitut Oberwolfach (MFO).

Professor Cohen's research focus is in topology, with connections to other areas such as combinatorics and geometric group theory. He is interested in hyperplane arrangements, configuration spaces, braid groups, and related objects. In addition to, and sometimes in relation with, these subjects, portions of his recent work have been on topological problems motivated by the motion planning problem from robotics.

Professor Cohen will present his work in two lectures during the week long visit. They are titled "Pure braid groups and direct products of free groups" and "Topological complexity of surfaces and their configuration spaces" which are held Monday 22.1.18, at 14:15 and Wednesday 24.1.18, at 14:15, respectively. Both venues take place in Wasserstrasse 221, Room 4/20. All are welcome to attend.

During his visit Professor Cohen is hosted by Professor Gerhard Röhrle.

### Alexander von Humboldt Fellowship awarded to Dr Litterick

We are very pleased to announce that Dr Alastair Litterick has been awarded a competitive two-year Alexander von Humboldt Fellowship for Postdoctoral Researchers, hosted by Professors Christopher Voll (Bielefeld University) and Gerhard Röhrle (Ruhr-University Bochum), with a research topic of 'Subgroup structure and generation properties in reductive groups'. Reductive groups are an important class of mathematical objects which lie at the intersection between algebra and geometry. Because of their multi-faceted nature, they are important throughout mathematics and beyond, arising naturally, for instance, in branches of theoretical physics such as string theory.

Dr Litterick's research focuses on algebraic properties of these groups. Although the groups themselves are easy to describe, several aspects of their subgroup structure remain mysterious. In particular, he is concerned with understanding subgroups which are themselves reductive group, and with subgroups containing elements of specified forms. Current approaches to these problems involve many diverse areas of pure mathematics, including algebraic geometry, computational algebra, geometric group theory, geometric invariant theory, Lie theory, cohomology theory, representation theory, and still more. Working in the diverse groups of Professors Voll and Röhrle, Dr Litterick intends to pursue all of these avenues towards the solutions of these problems.

## Archiv der laufenden Ankündigungen

Briefanschrift | Paketanschrift | Fon/Fax/Mail |
---|---|---|

Ruhr-Universität Bochum Fakultät für Mathematik Prof. Dr. Gerhard Röhrle IB 2/133, Fach 69 D-44780 Bochum |
Ruhr-Universität Bochum Fakultät für Mathematik Prof. Dr. Gerhard Röhrle IB 2/133, Fach 69 Universitätsstraße 150 D-44801 Bochum |
Tel.: +49(0)234-32-28304 Fax: +49(0)234-32-14025 E-Mail: gerhard.roehrle@ruhr-uni-bochum.de |