Spring School on Complete Reducibility

Spring School on Complete Reducibility

Ruhr-University Bochum, 9-13 April 2018

This spring school is aimed at introducing graduate students and junior researchers to the notion of G-complete 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), Alastair Litterick (Bielefeld/Bochum), Gerhard Röhrle (Bochum) and David Stewart (Newcastle). Further details will be provided in due course.

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

To register or make enquiries, please send an email to Claudia Knauber by 5th March 2018: Frau Claudia Knauber . Please include your name, current affiliation and a short description of your research area. Please also state whether you would like to attend the dinner and excursion.

There is no registration fee for the spring school. Unfortunately, we are unable to provide funds for participation.

Accommodation

We have reserved a number of rooms from 8th-14th 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

Preliminary Schedule

Talks will begin the morning of Monday 9th April and finish around midday on Friday 13th. More details will be provided in due course.

The dinner will take place in Bochum City Centre, within walking distance of the Ibis Hotel Zentrum. Further details will be provided closer to the time.

Abstracts

Lecture series by Ben Martin (Aberdeen):
Algebraic groups and complete reducibility

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 GLn(k) is completely reducible if and only if the inclusion map i : H → GLn(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 Kempf-Hesselink-Rousseau construction of optimal destabilising cocharacters, which has found many applications to algebraic groups and Lie theory.

Recommended preparation for the lecture series

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. Springer-Verlag, New York-Heidelberg; 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 Bate-Martin-Röhrle ("A geometric approach to complete reducibility", Invent. Math. 161 (2005), no. 1, 177-218) describes the approach to complete reducibility via geometric invariant theory.

Additional talks

Abstracts to follow in due course.

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, fast-food 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 the University campus on Monday morning. If you wish to attend the excursion and/or dinner, please bring payment with you in cash. Further details will be provided nearer the time.

Getting here 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 main-line train. SkyTrain tickets can be purchased from machines at the station, and a separate ticket is required for the main-line 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 here by rail

Bochum is very well-connected 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 underground service (U-Bahn) U35 will take you directly from Bochum Hbf to the university campus.

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, U-Bahn, S-Bahn 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 (€5.80). To get to and from Düsseldorf, you require a Zone D ticket (currently €15). Note that these tickets do not let you use the IC or ICE train services.

Local Information

Travel between the hotel and university

The easiest way from The Hotel Ibis Zentrum to the University campus is to take the underground (U-Bahn) route U35 from Bochum main station. Blue signs and downward escalators to the U-Bahn are at both ends of the station. You require a Zone A ticket for this journey; these currently cost €2.70 each, or €10 for a ticket which can be stamped four times. Tickets can be stamped on board the U-Bahn.

Food and drink

For lunch, we recommend following a local, who will probably go to the University Mensa or to the Q-West restaurant on campus.

There are many good places to eat in Bochum city centre. We recommend heading to the Bermuda Dreieck which is a ten-minute walk from the main train station, and features an array of restaurants, bars and pubs (all serving pub food). Here is a (by no means) exhaustive list of suggestions within the city centre.

  • Amalfi-Pizzeria (Italian)
  • Avli (Greek)
  • Hans im Glück (Burgers)
  • NhyStar (Vietnamese)
  • Vapiano (Italian)

Shops on campus

The Uni-Center is a small shopping precinct located at the University U-Bahn stop. Turn left out of the station. It includes bakeries, cafes, a cashpoint, a pharmacy and two supermarkets.

List of Participants

  • Falk Bannuscher (Ruhr-University Bochum)
  • Michael Bate (University of York)
  • Maike Gruchot (Ruhr-University Bochum)
  • Alastair Litterick (Bielefeld University / Ruhr-University Bochum)
  • Ben Martin (University of Aberdeen)
  • Gerhard Röhrle (Ruhr-University Bochum)
  • David Stewart (Newcastle University)