Spring School on Complete Reducibility
RuhrUniversity Bochum, 9–13 April 2018
This spring school is aimed at introducing graduate students and junior researchers to the notion of Gcomplete reducibility. Introduced by J.P. Serre in the 1990s, this theory is centrally important in the modern approach to studying linear algebraic groups. The theory generalises the concept of complete reducibility from representation theory, and connects subgroups of linear algebraic groups with tools from geometry and geometric invariant theory.
Academic Programme
The principal event of the spring school will be a lecture series by Ben Martin, University of Aberdeen.
Additional talks will be given by Michael Bate (York), Maike Gruchot (Bochum), Alastair Litterick (Bielefeld/Bochum) and David Stewart (Newcastle).
Excursion and Dinner
The spring school will also feature a conference dinner, at a fixed cost of €25 per person, and an excursion to the Zeche Zollverein, also at €25 per person (including transport). Payment will be collected during registration on Monday morning.
Online Registration
Online registration is now closed. Please direct further enquiries to: Frau Claudia Knauber.
Accommodation
We have reserved a number of rooms from 8th14th April at the Ibis Bochum Zentrum, at €66.64 per night. To take advantage of this price, participants must contact the hotel by phone (+49 234 33311) or by email (H1440@accor.com), quoting "Spring School on Complete Reducibility".
Organising Committee
Scientific advisers
Schedule
Talks will begin the morning of Monday 9th April and finish 12:30 on Friday 13th.
Monday 9th  Tuesday 10th  Wednesday 11th  Thursday 12th  Friday 13th  

09:00  09:30  Registration  
09:30  10:30  Martin I  Martin III  Martin V  Litterick  Bate 
10:30  11:30  Coffee break  Coffee break  Coffee break  Coffee break  Coffee break 
11:30  12:30  Martin II  Martin IV  Martin VI  Gruchot  Martin (additional) 
12:30  14:00  Lunch  Lunch  Lunch  Lunch  Lunch / end 
14:00  15:00  Coffee break  Coffee break  Coffee break  Coffee break  
15:00  16:00 / end  Discussion group  Discussion group  Excursion  Stewart 
Dinner
The spring school dinner will take place on Wednesday evening, 19:00 at Amalfi Pizzeria (Google Maps), in easy walking distance of the Ibis Hotel Zentrum.
Lecture series: Algebraic groups and complete reducibility
Ben Martin, University of Aberdeen
The notion of a completely reducible subgroup is important in the study of algebraic groups and their subgroup structure. It generalises the usual idea of complete reducibility from representation theory: a subgroup H of a general linear group GL_{n}(k) is completely reducible if and only if the inclusion map i : H → GL_{n}(k) is a completely reducible representation of H. In these lectures I will give an introduction to the theory of complete reducibility, and explain an approach to the subject using geometric invariant theory. In particular, I will sketch the KempfHesselinkRousseau construction of optimal destabilising cocharacters, which has found many applications to algebraic groups and Lie theory.
Accompanying notes
Notes to accompany the lecture course can be found below:
Recommended preparation
I will give a brief review of linear algebraic groups, but it will help if you have met some of the main ideas. The books of Humphreys ("Linear algebraic groups", Graduate Texts in Mathematics, No. 21. SpringerVerlag, New YorkHeidelberg; corrected printing, 1998) and Springer ("Linear algebraic groups", Modern Birkhäuser Classics, 2nd ed.; 2nd printing, 2008) are a good introduction to the subject. In particular, Chapter I of Humphreys gives a helpful summary of the necessary algebraic geometry. The books of Geck ("An introduction to algebraic geometry and algebraic groups", OUP, Oxford, 2013) and Carter/Segal/Macdonald ("Lectures on Lie groups and Lie algebras", London Mathematical Society Student Texts, Cambridge University Press, 2009) are aimed at undergraduates. Sections 1.1 to 1.3 of Newstead ("Introduction to moduli problems and orbit spaces", Tata Institute of Fundamental Research Publications, Volume 17, 2011) gives a quick and approachable introduction to geometric invariant theory. Part II of Serre's 1998 Moursund lectures ("The notion of complete reducibility in group theory",http://arxiv.org/abs/math/0305257) gives a very readable account of some topics in complete reducibility. The paper of BateMartinRöhrle ("A geometric approach to complete reducibility", Invent. Math. 161 (2005), no. 1, 177218) describes the approach to complete reducibility via geometric invariant theory.
Additional talks
Alastair Litterick  Relative Complete Reducibility, I
The notion of Gcomplete reducibility can be expressed in terms of limits of cocharacters. By considering only the cocharacters of some proper reductive subgroup of G, we obtain a relative notion of relative complete reducibility, first formulated in 2011 by Bate, Martin, Röhrle and Tange. In this talk we will introduce this notion, explain why it is a natural property to consider, and present a new result which lets us generalise many results from the 'absolute' case.
Maike Gruchot  Relative Complete Reducibility, II
Sticking with the theme of the preceding talk, in this talk we again consider the relative notion of complete reducibility, focusing in particular on the many results which generalise from the 'absolute' case. We will also discuss the special case when G is the general linear group, and show that the representationtheoretic interpretation of complete reducibility has an analogue in the relative setting.
David Stewart  Representations of pseudoreductive groups
Pseudoreductive groups are smooth connected linear algebraic groups over a field k whose kdefined unipotent radical is trivial. If k is perfect then all pseudoreductive groups are reductive, but if k is imperfect (hence of characteristic p) then one gets a strictly larger collection of groups. They come up in a number of natural situations, not least when one wishes to say something about the simple representations of all smooth connected linear algebraic groups. Recent work by ConradGabberPrasad has made it possible to reduce the classification of the simple representations of pseudoreductive groups to the split reductive case. I’ll explain how. This is joint work with Mike Bate.
Michael Bate  Complete reducibility and the centre conjecture
In this talk I'll outline the approach to complete reducibility through spherical buildings and show how this gives a third point of view to sit alongside the group theoretic and geometric approaches seen during the course. These relationships are most striking when viewed through the prism of the "Centre Conjecture" due to Tits. I'll talk about this conjecture and some known cases of it, and outline a possible approach to attacking other cases.
Ben Martin  Algebraic groups and a question of Külshammer
Let F be a finite group and let G be a reductive algebraic group over an algebraically closed field of characteristic p>0. Burkhard Külshammer asked whether homomorphisms from F to G are controlled in an appropriate sense by their restrictions to a Sylow psubgroup F_{p} of F. Under quite general hypotheses the answer is yes, but a counterexample was recently discovered by Bate, Martin and Röhrle. Külshammer's question makes sense when we replace F with an arbitrary linear algebraic group H and F_{p} with a maximal unipotent subgroup of H. It turns out that the answer is yes if H is connected and semisimple. I will discuss this and some related results. The proofs involve ideas from geometric invariant theory and the theory of complete reducibility. This is joint work with Daniel Lond.
Arrival
We expect that most participants will arrive on Sunday, 8th April, and will be staying at the Ibis Hotel Zentrum. In this case, you should head directly to the hotel. If you arrive at Bochum by train, you will notice the hotel as soon as you disembark; it is a tall white building with a large, red "Ibis" sign at the top.
Please be aware that most shops are closed on Sundays in Germany, and the Ibis Hotel Zentrum does not have a restaurant for dinner. However, a number of small cafes, fastfood outlets and bakeries do remain open, especially those located in train stations. Most shops do not accept credit cards, so make sure you have Euros on you if you do not own a Eurozone bank card (EC card).
Local registration
Registration and talks will take place at Wasserstrasse 221 on Monday morning. Note: This is not on the main university campus. If you wish to attend the excursion and/or dinner, please bring payment with you in cash.
Getting to Bochum by air
The nearest international airports to Bochum are Düsseldorf (DUS) and Dortmund (DTM). Cologne/Bonn Airport (CGN) is further away, but is on a direct railway line to Bochum. Be warned that Düsseldorf Weeze (NRN) is not easy to reach from Bochum.
From Düsseldorf International (DUS) you must take the 'SkyTrain' to Düsseldorf Flughafen railway station and change onto a mainline train. SkyTrain tickets can be purchased from machines at the station, and a separate ticket is required for the mainline train.
From Dortmund airport (DTM) the easiest route is to take the AirportExpress bus to Dortmund Main Station (Dortmund Hbf), and from there take a train to Bochum. Tickets for the AirportExpress can be purchased on the bus. A rail ticket must be purchased separately. Warning: do not take the 'AirportShuttle' as this goes to a small local train station that is not useful for Bochum.
Getting to Bochum by rail
Bochum is very wellconnected by rail. Frequent trains run from the main station (Bochum Hbf) to many other major cities. The main station is located in the city centre, and is immediately next to the Ibis Hotel Zentrum.
The basic type of ticket is a 'VRR' ticket, which can be purchased from Deutsche Bahn ticket machines. Once stamped, this ticket will let you use all buses, UBahn, SBahn and local trains (RE/RB) for at least the next 90 minutes. The price of the ticket depends on how far you want to go. To get to and from Dortmund, you require a Zone B ticket (currently (€5.90). To get to and from Düsseldorf, you require a Zone D ticket (currently €15.30). Note that these tickets do not let you use the IC or ICE train services.
Local Information
Travel between the hotel and talk venue (Offcampus)
All talks will take place at Wasserstrasse 221 (Google Maps), away from the University Campus. From the hotel, one can either follow the 30minute walk shown on the map, or take the underground (UBahn U35) from Bochum main station to Wasserstrasse. For the UBahn, blue signs and downward escalators to the UBahn are at both ends of Bochum main station. You require a 'VRR Zone A ticket' for this journey; these currently cost €2.80 each, or €10.20 for a ticket which can be stamped four times. Tickets can be stamped on board the UBahn.
Food and drink
For lunch, we recommend following a local. For dinner, there are many good places to eat in Bochum city centre. We recommend heading to the Bermuda Dreieck which is a tenminute walk from the main train station, and features an array of restaurants, bars and pubs.
List of Participants

Chris Attenborough (University of York)

Falk Bannuscher (RuhrUniversity Bochum)

Leon Barth (RuhrUniversity Bochum)

Michael Bate (University of York)

Alexandre Borovik (University of Manchester)

Melvin Dauter (TU Kaiserslautern)

Ögmundur Eiriksson (Bielefeld University)

Lukas Gohla (TU Dresden)

Maike Gruchot (RuhrUniversity Bochum)

Ulrike Hansper (Bielefeld University)

Mikko Korhonen (École Polytechnique Fédérale de Lausanne)

Denis Kuz (RuhrUniversity Bochum)

Alastair Litterick (Bielefeld University / RuhrUniversity Bochum)

Ben Martin (University of Aberdeen)

Tilman Möller (RuhrUniversity Bochum)

Paul Mücksch (RuhrUniversity Bochum)

Gerhard Röhrle (RuhrUniversity Bochum)

Nithi Rungtanapirom (Goethe University Frankfurt)

Yuri Santos (Bielefeld University)

Anne Schauenburg (RuhrUniversity Bochum)

David Stewart (Newcastle University)

Michele Zordan (Katholieke Universiteit Leuven)