Lehrstuhl-VI

• Lehrstuhl VI
  Algebra / Zahlentheorie
  • Startseite
  • Prof. Dr. Gerhard Röhrle
  • Sekretariat
  • Mitarbeiter
  • Gäste
  • Abschlussarbeiten
  • Ehemalige
• Veranstaltungen
  • SS 2023
  • WS 2022/23
  • Semesterarchiv
• Anreise
  • Flugzeug/Bahn
  • Auto
    

• Fakultät

• Studium

• Wochenplan

• Algebraische Lie Theorie
• Algebraische Gruppen
• Endliche Gruppen vom Lie Typ
• Darstellungstheorie
• Hyperebenen- arrangements

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

More

Vortrag am Montag, 06.02.2023, 14:15 Uhr, in IA 1/177

Vortragender: Lorenzo Giordani (RUB)

Titel: "On the Combinatorics and Cohomology of Wonderful models for subspace arrangements"

Abstract:
A classical problem in the theory of hyperplane arrangements is to understand to what extent the combinatorial information of the arrangement, encoded in the associated matroid or lattice of intersections, determines geometric proprieties of the complement space. P. Orlik, L. Solomon, E. Brieskorn et al. proved that the cohomology ring of the complement space is isomorphic to the so called Orlik-Solomon algebra, which is defined entirely in terms of the underlying matroid. In this seminar, we recall the results on the Orlik-Solomon algebra and present some constructions by C. De Concini and P. Procesi, including their "Wonderful model" and its proprieties. Using the model, they proved that the cohomology ring of the complement space is still determined by the combinatorial data when hyperplanes are substituted by subspaces of arbitrary codimension.

Vortrag am Montag, 30.01.2023, 14:15 Uhr, in IA 1/177

Vortragender: Sven Wiesner (RUB)

Titel: "Techniques from algebraic geometry applied to matroids"

Abstract:
June Huh et al. proved longstanding conjectures about specific sequences associated to matroids which are combinatorial objects. They did so by associating a structure to these matroids on which tools from algebraic geometry can get deployed. In my talk I want to give a short overview about the structures involved and how they derived the results about the matroid from them.

Vortrag am Montag, 09.01.2023, 14:15 Uhr, on Zoom

Vortragender: Prof. Dr. Michael Cuntz (Hannover)

Titel: "On arrangements of hyperplanes from connected subgraphs "

Abstract:
We investigate arrangements of hyperplanes whose normal vectors are given by connected subgraphs of a fixed graph. These include the resonance arrangement and certain ideal subarrangements of Weyl arrangements. We characterize those which are free, simplicial, factored, or supersolvable. In particular, such an arrangement is free if and only if the graph is a cycle, a path, an almost path, or a path with a triangle attached to it. This is joint work with Lukas Kühne.




Zoom Link: https://ruhr-uni-bochum.zoom.us/j/61417572342?pwd=ZThVcDBFSGNMOCtEcHZvejlYcWc0QT09
Meeting ID: 614 1757 2342
Passcode: ArrSym22

Vortrag am Montag, 05.12.2022, 16:00 Uhr, in IA 1/109

Vortragender: Dr. David Stewart (University of Newcastle)

Titel: "A Prolog-assisted search for simple Lie algebras" (jt work with David Cushing and George Stagg)

Abstract:
Prolog is a very unusual programming language, developed by Alain Colmerauer in one of the buildings on the way to the CIRM in Luminy. It is not fundamentally iterative in the way that, for example, GAP and Magma are. Instead it operates by taking a list of axioms as input, and responds at the command line to queries asking the language to achieve particular goals. It gained some notoriety by beating contestants on the game show Jeopardy in 2011. It is also the worlds fastest sudoku solver. I will describe some recent Prolog investigations to search for new simple Lie algebras over the field GF(2). We were able to discover some new examples in dimensions 15 and 31 and extrapolate from these to construct two new infinite families of simple Lie algebras.

Vortrag am Montag, 21.11.2022, 14:15 Uhr, in IA 1/177

Vortragender: Laura Voggesberger (RUB)

Titel: "Nilpotent Pieces in Lie Algebras of Exceptional Type in Bad Characteristic"

Abstract:
This talk will be a trial run for my defense concerning certain structures in algebraic groups and their Lie algebras. In group theory, a big and important family of infinite groups is given by the algebraic groups. These groups and their structures are already well-understood. In representation theory, the study of the unipotent variety in algebraic groups — and by extension the study of the nilpotent variety in the associated Lie algebra — is of particular interest. Let G be a connected reductive algebraic group over an algebraically closed field k, and let Lie(G) be its associated Lie algebra. By now, the orbits in the nilpotent and unipotent variety under the action of G are completely known and can be found for example in a book of Liebeck and Seitz. There exists, however, no uniform description of these orbits that holds in both good and bad characteristic. With this in mind, Lusztig defined a partition of the unipotent variety of G in 2011. Equivalently, one can consider certain subsets of the nilpotent variety of Lie(G) called the nilpotent pieces. This approach appears in the same paper by Lusztig in which he explicitly determines the nilpotent pieces for simple algebraic groups of classical type. The nilpotent pieces for the exceptional groups of type G2 , F4 , E6 , E7 , and E8 in bad characteristic have not yet been determined. In my thesis, I have explored the cases for G2 , F4 , and E6, and will present them in this talk.

Vortrag am Montag, 14.11.2022, 14:15 Uhr, in IA 1/177

Vortragender: Sven Wiesner (RUB)

Titel: "Inductive Freeness of Ziegler's Canonical Multiderivations for Restrictions of Reflection Arrangements "

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. Recently Hoge and Röhrle proved an analogue of Ziegler's theorem for the stronger notion of inductive freeness: If A is inductively free, then so is the free multiarrangement (A",k). In 2018 Hoge and Röhrle classified all reflection arrangements which admit inductively free Ziegler restrictions. I will talk about joint work with Torsten Hoge and Gerhard Röhrle where we extended this classification to restrictions of reflection arrangements.

Vortrag am Montag, 24.10.2022, 14:15 Uhr, via Zoom

Vortragender: Dr. Paul Mücksch (Kyushu University)

Titel: "Topology of supersolvable oriented matroids"

Abstract:
A central result in the topology of complex hyperplane arrangements, due to Falk, Randell and Terao, states that supersolvability of the intersection lattice of such arrangements implies that their complements are $K(\pi,1)$-spaces.

The homotopy type of the complement of a complexified real hyperplane arrangement can be modeled by a nice regular CW-complex introduced by Salvetti. The Salvetti complex can be constructed for any oriented matroid -- a combinatorial abstraction of a real hyperplane arrangement.

In my talk, I will present a novel combinatorial way to prove that supersolvability of the geometric lattice of an oriented matroid implies the asphericity of its Salvetti complex. In particular, this extends to the non-realizable case.




Zoom Link: https://ruhr-uni-bochum.zoom.us/j/61417572342?pwd=ZThVcDBFSGNMOCtEcHZvejlYcWc0QT09
Meeting ID: 614 1757 2342
Passcode: ArrSym22

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.

Vortrag am Montag, 17.10.2022, 14:15 Uhr, IA 1/177

Vortragender: Avi Steiner (Universität Mannheim)

Titel: "Symmetrizing" logarithmic derivations with respect to matroid duality"

Abstract:
Of interest to people who study both hyperplane arrangements and commutative algebra are the homological properties of the module of logarithmic derivations of a hyperplane arrangement A. I will introduce the "ideal of pairs", which is a sort of "symmetrization" of this module of logarithmic derivations with respect to matroid duality. This is an ideal which simultaneously "sees" many of the homological properties of both the arrangement and its dual.

Vortrag am Montag, 04.07.2022, 14:15 Uhr, IC 03/647

Vortragender: Prof. Dr. Götz Pfeiffer (Galway)

Titel: "Falling Powers and the Algebra of Descents"

Abstract:
A finite Coxeter group of classical type A, B or D contains a chain of subgroups of the same type. We show that intersections of conjugates of these subgroups are again of the same type, and make precise in which sense and to what extent this property is exclusive to the classical types of Coxeter groups. As the main tool for the proof we use Solomon’s descent algebra. Using Stirling numbers, we express certain basis elements of the descent algebra as polynomials and derive explicit multiplication formulas for a commutative subalgebra of the descent algebra. This is joint work with Linus Hellebrandt.

Vortrag am Montag, 20.06.2022, 14:15 Uhr, IC 03/647

Vortragender: Prof. Dr. Gerhard Röhrle (RUB)

Titel: "Inductive Freeness of Ziegler's Canonical Multiderivations"

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), alo known as the Ziegler restriction. I'll report on recent joint work with Torsten Hoge where we prove an analogue of Ziegler's theorem for the stronger notion of inductive freeness. Namely, 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. I shall discuss several consequences of the theorem for natural classes of inductively free arrangements. Time permitting I shall explain counterparts of the latter kind for the notion of additive and recursive freeness.

"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.

Vortrag am Montag, 09.05.2022, 14:15 Uhr, via Zoom

Vortragender: Eirini Chavli (Stuttgart)

Titel: "Complex Hecke algebras are real"

Abstract:
Iwahori Hecke algebras associated with real reflection groups appear in the study of finite reductive groups. In 1998 Broué, Malle, and Rouquier generalized in a natural way the definition of these algebras to complex case. However, some basic properties of the real case are also true for Hecke algebras in the complex case. In this talk we will talk about these properties and their state of the art.


Zugangsdaten
https://ruhr-uni-bochum.zoom.us/j/62326053276?pwd=TlVPNzAxdkt2NzcycVNwTXhQSW9Ldz09
Meeting ID: 623 2605 3276
Passcode: arrsym22

Vortrag am Montag, 02.05.2022, 14:15 Uhr, via Zoom

Vortragender: Prof. Apoorva Khare (Indian Institute of Science, Bangalore)

Titel: "Higher order Verma modules, and a positive formula for all highest weight modules"

Abstract:
We study weights of highest weight modules $V$ over a Kac-Moody algebra $\mathfrak{g}$ (one may assume this to be $\mathfrak{sl}_n$ throughout the talk, without sacrificing novelty). We begin with several positive weight-formulas for arbitrary non-integrable simple modules, and mention the equivalence of several "first order" data that helps prove these formulas. We then discuss the notion of "higher order holes" in the weights, and use these to present two positive weight-formulas for arbitrary modules $V$. One of these is in terms of "higher order Verma modules", and we end by explaining BGG resolutions and Weyl-Kac type character formulas, for these modules in certain cases. (Joint with G.V.K. Teja and with Gurbir Dhillon.)


Zugangsdaten
https://ruhr-uni-bochum.zoom.us/j/6340579550?pwd=Mk5BQVZySW5JYVpCeXkyM2tFMWRqZz09
Meeting ID: 634 057 9550
Passcode: alt

Vortrag am Montag, 02.05.2022, 16:15--17:45, on Zoom

Vortragender: Giovanni Paolini (Caltech)

Titel: "Dual Coxeter groups of rank three"

Abstract:
In this talk, I will present ongoing work aimed at understanding the noncrossing partition posets associated with Coxeter groups of rank three. In particular, I will describe the combinatorial and geometric techniques used to prove the lattice property and lexicographic shellability. These properties can then be used to solve several problems on the corresponding Artin groups, such as the K(π,1) conjecture, the word problem, the center problem, and the isomorphism between standard and dual Artin groups. Joint work with Emanuele Delucchi and Mario Salvetti.


Zugangsdaten
https://ruhr-uni-bochum.zoom.us/j/62326053276?pwd=TlVPNzAxdkt2NzcycVNwTXhQSW9Ldz09
Meeting ID: 623 2605 3276
Passcode: arrsym22

Vortrag am Montag, 25.04.2022, 14-16 Uhr, IC 03/647

Vortragender: Dr. Paul Mücksch (MPI Bonn)

Titel: "On formality for hyperplane arrangements"

Abstract:
An arrangement of hyperplanes is called formal provided all linear dependencies among the defining linear forms of the hyperplanes are generated by ones corresponding to intersections of codimension two. This notion turns out to be necessary at the one hand for the apshericity of the complement of a complex arrangement due to work by Falk and Randell. One the other hand it is also necessary for the freeness of the module of logarithmic vector fields thanks to a result by Yuzvinsky.
In joint work with T. Möller and G. Röhrle we extend the above line of results by showing that the combinatorial property of factoredness implies formality. Furthermore, we study formality with respect to the standard arrangement constructions of restriction and localization and comment on the behavior of the stronger property of k-formality introduced by Brand and Terao.

Vortrag am Montag, 11.04.2022, 14:15 Uhr, IA 1/75

Vortragender: Timm Peerenboom (Bonn)

Titel: "The Affine Grassmannian in Type A "

Abstract:
The Affine Grassmannian associated to a reductive group is an infinite-dimensional analogue of classical (partial) flag varieties. In this talk I will introduce the Affine Grassmannian with its Schubert cell decomposition in type A examples. I will also state the Geometric Satake Equivalence which relates the geometry of the Affine Grassmannian with the representation theory of the Langlands dual group.

Vortrag am Montag, 11.04.2022, 13-14 Uhr, via Zoom

Vortragender: Shuhei Tsujie (Hokkaido University of Education)

Titel: "MAT-free graphic arrangements and strongly chordal graphs"

Abstract:
Recently Cuntz and Mücksch introduced MAT-free arrangements based on the Multiple Addition Theorem (MAT) by Abe, Barakat, Cuntz, Hoge, and Terao. In this talk, we will focus on graphic arrangements. Stanley showed that a graphic arrangement is free if and only if the graph is chordal. We will show that a graphic arrangement is MAT-free if and only if it is strongly chordal. This is joint work with Tan Nhat Tran.


Zugangsdaten
https://ruhr-uni-bochum.zoom.us/j/62326053276?pwd=TlVPNzAxdkt2NzcycVNwTXhQSW9Ldz09
Meeting ID: 623 2605 3276
Passcode: arrsym22

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

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.

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.

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.

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)

More

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).

More

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.

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.

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.

More

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.

More

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.

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.