Homotopical Group Theory and Topological Algebraic Geometry

University of Copenhagen, Department of Mathematical Sciences (Denmark)

June 16-20, 2008
(in connection with the conference at the MPI Bonn June 23-27, 2008)

This workshop is aimed at graduate students and young researchers, in part as preparation for the conference at the Max Planck institute the following week. The workshop, as well as conference, focuses on the new interactions of Algebraic Topology with Group Theory, Algebraic Geometry and Mathematical Physics which come from looking at these fields through the eye of a homotopy theorist. It celebrates one of the contributors to the subject by honoring the 60th birthday of Haynes Miller.

NEW (July 9,2008): Workshop pictures!!

Scientific Program

Minicourse by Bill Dwyer (Notre Dame): Homotopical group theory (10 lectures of 45 minutes each).
Minicourse by Paul Goerss (Northwestern): Topological algebraic geometry (10 lectures of 45 minutes each).
In addition to the lectures there will be scheduled problem sessions each day on both topics. Several of the MPI speakers will be in residence to answer questions (Behrens, Lurie...)

Lecture notes further down the page!

Getting to CabInn Scandinavia and the Univ. Copenhagen Math Department/H.C.Ørsted Institute (in "Universitetsparken")

To get to CabInn Scandinavia from the airport, take the frequent metro from the airport terminal and get off at the stop "Forum" (17 minutes, 10 stops; every metro from the airport stops at "Forum"). CabInn Scandinavia is now very close. For a map and more detailed directions see here. You need to buy a 3 zone ticket in the airport terminal before boarding the metro (or buy and stamp a 10 journey 3-zone multijourney ticket).

Getting from CabInn Scandinavia to the Math Department/The H.C. Ørsted Institute (HCØ) is a pleasant 30 minutes walk (metro/bus is also an option). See the directions page for a map.

Within HCØ, all lectures take place in the central Auditorium 1. A map of the HCØ indicating the location of Auditorium 1 can be found here
A more fancy google map, also showing bus stops and lunch places can be found HERE
(but don't worry, there will also be signs, and 120 topologists running around).

Tentative daily schedule monday-friday (all activities in main Auditorium 1)

09:00-10:45 Dwyer HGT lectures
10:45-11:15 Coffee break
11:15-12:00 Independent study time
12:00-13:00 HGT exercise session
13:00-14:00 Lunch
14:00-15:45 Goerss TAG lectures
15:45-16:15 Coffee break
16:15-17:00 Independent study time
17:00-18:00 TAG exercise session
18:00-18:30 "Fyraftensbajer" (aka øl, pivo, cerveza..)

(18:30-01:00 study; 01:00-08:00 sleep; repeat...)
(Monday at 16:15 Y. Manin will give a general-audience talk about Euler products during the independent study time.)

Course outlines

Homotopy theory and its application to classifying spaces (Bill Dwyer)

Synopsis: The lectures will introduce several tools that are widely used in homotopy theory and discuss the role that some of these tools play in the analysis of classifying spaces. Much of the course will focus on unstable homotopy theory, and the intention will be to emphasize common ideas rather than special cases. The topics covered will include many if not all of the following.

(1) Homotopy theories, model categories, and examples (chain complexes, topological spaces, simplicial sets, topological groups, simplicial groups, spectra, diagrams).
(2) Functors, derived functors, and examples (tensor, hom, abelianization, fixed point sets, orbits, limits, colimits, left and right Kan extensions, ends and coends). Initial and terminal functors. Function spaces.
(3) Completions, localizations, and examples (p-completion, rationalizaton, general homology localization, localization with respect to a map). The arithmetic square and its extensions.
(4) Lannes theory, both algebra (the functor T) and geometry (towers and their limits). Cohomology of function spaces and homotopy fixed point sets.
(5) K(\pi,1)'s and maps between them. More general maps between classifying spaces. Compact Lie groups, finite loop spaces, p-compact groups.
(6) Homology decompositions, fusion systems, and p-local finite groups.

Prerequisites: It will be assumed that the members of the audience have a background in the basics of homotopy theory.
NEW (6/2008): Lecture notes can be found here. The slides of the lectures can be found here.

Topological algebraic geometry (Paul Goerss)

Synopsis: I would like to give a concise introduction to derived algebraic geometry over the moduli stack of formal groups. Put another way, I would like to discuss when continuous families of Landweber exact homology theories can be lifted to families of structured ring spectra. I would also like to explain why we care: this is part of long standing program (going back to Morava and others) of using arithmetic algebraic geometry to understand phenomena in stable homotopy theory. While very much a developing theory, the work of Hopkins, Miller, Lurie, Behrens, Lawson, and others over the last ten or so years have given us very precise tools. The theory and practice of these tools is the emphasis of these lectures.
Topics we be a subset of the following list, depending on time and the background of the audience.

1. Schemes, stacks, and algebraic stacks
2. Structured ring spectra and derived algebraic geometry
3. Formal groups, p-divisible groups, and their role in homotopy theory
4. Derived stacks over the moduli stack of formal groups; Lurie's Theorem
5. Deformations of formal groups and the Hopkins-Miller theorem
6. Examples: the Elliptic Case (Topological Modular Forms)
7. Examples: Shimura varieties (Topological Automorphic Forms)

Prerequisites: See here for some recommended reading!.
Lecture 1
Lecture 2
Lecture 3
Lecture 4
Lecture 5
Lecture 6
Lecture 7
Lecture 8
Lecture 9+10
All lectures as one file

California: A musical tribute to Haynes Miller

Registration & application for funding

THE REGISTRATION DEADLINE HAS NOW PASSED. See further down the page for funding decisions.
A list of registred participants is available from the Bonn Conference website. (It should say "Yes" in the column "Reg. CO" if you are registered for Copenhagen.)


Accomodation in Copenhagen will be at CABINN SCANDINAVIA. The price is DKK 585/night for a single room and DKK 645/night (approx EUR 86) for a double room, including breakfast.

A block of rooms has been reserved for 7 nights from Sunday June 15, 2008 to Sunday June 22, 2008. Note: Talks starts Monday morning and end Friday afternoon; there will be a social program Saturday.

**PARTICIPANTS ARE RESPONSIBLE FOR BOOKING THEIR OWN ACCOMODATION AT CABINN SCANDINAVIA** by calling (+45) 35 36 11 11 and citing the booking reference #100253.

Hotel space at Cab Inn only guaranteed until FEBRUARY 15, 2008. As of late February there are still rooms available at CabInn. But we cannot guarantee future availability, so please book today.

The website of CabInn is: www.cabinn.com, and click under CAB INN SCANDINAVIA. Warning: that CabInn Scandinavia is not the only CabInn hotel in the greater copenhagen area.

The address is:
CAB INN Scandinavia
Vodroffsvej 55
DK-1900 Frederiksberg C
Tlf: (+45) 35 36 11 11
Fax: (+45) 35 36 11 14
Email: scandinavia@cabinn.com

THE FUNDING DEADLINE HAS PASSED. Please see below for funding information, for those who applied.

Funding decision

If you applied for funding you should have received an email from the Max Planck Institute with information.
Copenhagen Funding to those who applied: We can offer 6 nights of shared double occupancy accomodation at CabInn Scandinavia, or 7 nights in a 3- or 4-person room. For single occupancy we can reimburse 3 nights if needed, but strongly encourage you to use alternatively funding, since this will free up funds for younger participants (120 people applied for funding). You are responsible for booking your own room at CabInn Scandinavia using reference number #100253, and finding your own roommates. Rooms booked any other way cannot be refunded. The granted nights will be deducted from your bill at checkout; please also write an email to homotopical2008@math.ku.dk stating that you want to accept the funding. THE OFFICIAL HOLD-DATE ON THE ROOMS HAS PASSED SO IF YOU HAVE NOT BOOKED ALREADY CALL TODAY! We cannot guarantee availability. A list of participants is linked from the conference homepage http://www.ruhr-uni-bochum.de/topologie/conf08. If you cannot find a roommate you can write to homotopical2008@math.ku.dk, and we will try to offer a suggestion. (Note: Requests concerning the conference in Bonn should be directed to MPI instead.)

Getting to and from Copenhagen

Arrival day is Sunday June 15, 2008. Lectures start Monday June 16, 2008 in the morning and end Friday June 20 in the afternoon. There will be optional social activities Saturday June 21, 2008 for the people staying until Sunday.
For people traveling onwards to Bonn, there are frequent flights between Copenhagen (CPH) and Cologne/Bonn (CGN), Frankfurt (FRA), or Dusseldorf (DUS). Overseas participants attending both events may want to consider buying an "open-jaw" ticket arriving in Copenhagen Sunday June 15 and departing from e.g. Frankfurt Saturday June 29, and then buy the one-way Copenhagen-Bonn on June 21 or 22 separately.
(See here for a listing of the route network of low-cost carriers within Europe.)


The conference is partially supported by the DFG Graduiertenkolleg 1150 "Homotopy and Cohomology" , the Max Planck Institute Bonn., and the US National Science Foundation.
The workshop is supported by the University of Copenhagen through a grant from the researcher training program "Forskerskole i Matematik og anvendelser" and a Faculty of Science grant, as well as a EURYI award from the ESF and Roemer award from the Danish Natural Science Research Council.


The workshop is organized by the Copenhagen Topology Group, in connection with the conference at MPI organized by Ando, Grodal, Laures, and Shipley.