The Bellairs Workshop in Number Theory
The Bellairs Institute, McGill University, Barbados, 2014


The cohomology of arithmetic groups and the Langlands program.
May 2-9, 2014

Keynote Speaker: Frank Calegari (Northwestern University)


Organizer: Payman Kassaei (McGill
Co-organizers: Eyal Goren and
Henri Darmon (McGill)

Arrival: Friday, May 2. Departure: Friday, May 9.


The workshop is dedicated
to cohomology of arithmetic groups with a view to applications to the Langlands program. A series of lectures will be given by Frank Calegari on this topic, supplemented by lectures from attending experts, for a total of 20 hours and many more hours of stimulating exchanges. The intention of the workshop is to provide an access point to the field, assuming at least a background comparable to advanced graduate students. Participation is by invitation only; Those interested in participation should contact the organizers to inquire about space availabilities and to make arrangements.


Program

The Bellairs Research Institute

Recreational Activities

The Participants

Getting to Bellairs

Coming prepared

 

The facilities


 

Program

The general schedule is morning lectures by Calegari followed by after-lunch and  after-dinner lectures by participating experts. The program is designed to encourage discussion and new research initiatives by allowing enough free time and having all the participants in the same residence.

 An important  milestone towards proving Langlands's reciprocity conjecture was the work of Wiles (and others) showing that any elliptic curve E over Q is modular (equivalently, automorphic). Despite the immense progress in the field over the past twenty years, the class of natural algebraic varieties that we know to be modular is still extremely limited. For example, consider the problem of proving modularity of a general elliptic curve E over an imaginary quadratic number field F. In this instance, the associated locally symmetric spaces are hyperbolic manifolds of real dimension three and thus are not amenable to the usual techniques of algebraic geometry.  Moreover, the relevant cohomology classes of these manifolds are no longer concentrated in a single degree. All previous generalizations of the Taylor-Wiles method are restricted to contexts in which neither of these difficulties occurs.

  One of the main goals of these lectures is to outline a viable strategy to overcome these issues, with an emphasis on explaining the various outstanding problems and conjectures which still need to be resolved. An overarching theme of our approach is the study of torsion classes in cohomology --- both in the Betti cohomology of arithmetic groups and the coherent cohomology of Shimura varieties --- and their conjectural relationship with the Langlands program.


Friday (2 May)
----arrival

Saturday
9:00-11:30   Frank Calegari: Selmer groups, the Greenberg-Wiles formula, and the Taylor-Wiles method.
19:30-20:30     Jared Weinstein: The geometry of modular curves.
20:00-21:30     Peter Scholze: The geometry of Shimura varieties.

Sunday
9:00-11:30   Frank Calegari: Local-global compatibility for torsion representations in weight one; the doubling method.
15:00-16:30     Peter Scholze: An approach to the vanishing of non-Eisenstein cohomology of Shimura varieties outside the middle degree.
19:00-21:00     Toby Gee: An overview of local-global compatibility.

Monday
9:00-11:30    Frank Calegari: A modification of the Taylor-Wiles method when l_0 > 0.
14:30-16:00      Jack Thorne: Ordinary minimal modularity lifting for GL(n) and Leopoldt's conjecture.
19:00-21:30      Jack Thorne: Taylor-Wiles primes.

Tuesday
9:00-11:15    Frank Calegari: Minimal modularity lifting for low weight Siegel modular forms.
14:30-16:00      David Geraghty: The Kisin modification of Taylor-Wiles.
19:00-22:30    Sug Woo: Automorphic forms and cohomology.

Wednesday
9:00-11:00    Frank Calegari: Non-minimal modularity lifting in weight one.
14:30-16:00      George Boxer: Mod p^n coherent cohomology of automorphic vector bundles on Shimura varieties.
19:00-21:00      Ila Varma: Local-global compatibility for l=/=p non-self-dual Galois representations.
21:00-22:15    David Geraghty: Vexing primes.

Thursday
9:00-11:00    Frank Calegari: The stable cohomology of congruence subgroups and algebraic K-theory.








Friday (9 May)
----departure

















 


The Participants


Massimo Bertolini
George Boxer
Miljan Brakocevic
Bryden Cais
Ana Caraiani
Henri Darmon
Daniel Disegni
Frank Calegari
Toby Gee
David Geraghty
Eknath Ghate
Eyal Goren
Florian Herzig
Payman Kassaei
Marc Masdeu
Stefano Morra
Patrick Allen
Vincent Pilloni
Robert Pollack
David Savitt
Peter Scholze
Sug Woo Shin
Joel Specter
Naser Talebi Zadeh
Jack Thorne
Ila Varma
Nike Vatsal
Jared Weinstein


Group Photo

 The Bellairs Research Institute
Here is a link to the Institute's website. It is located in St. James, which is just to the north of Holetown. The exact location is
13.192104, -59.640130 (I think... you can plug these coordinates in  http://maps.google.ca)

 


Getting to Bellairs
Here you can find maps of Barbados, as well as much information about tourism. The local airport is in Bridgetown, Barbados (code BGI), while the Bellairs institute is more towards the north of the island, on its west coast. The easiest is to share a taxi from the airport to the institute. Most taxi drivers will know where is the Bellairs research institute; if not, tell them is in St. James, north to Holetown. When you pass the St. James Parish and the Folkstone marine park look for the Bellairs research institute on the left (sea side) of the road. Hook up at the airport with other participants based on arrival information to reduce the cost.


View Larger Map

 


The facilities

  • The institute is well equipped for marine research. One can rent air tanks there very cheaply. Regardless, it is an excellent place for scuba diving and swimming, but you should bring your own gear (good quality mask and snorkel can be purchased for a reasonable price at a local sport store).
  • Most people take their breakfast at the institute, but this is optional. Dinner at the institute is mandatory and is charged separately at $25US. This is not the cheapest, but the food is great and local and in generous portions and it is a nice way for us to get together. For lunch everyone is on his/her own. Usually people get together and go to one of the restaurants in Holetown. There is a kitchen there where, if needed, you can store food and cook for yourself if you cannot eat the food served.
  • There is a flat rate of $15US for use of internet for the whole week. There is wireless in the Institute and also a few terminals and a printer (not for heavy jobs!). Laptop is the easiest way to be connected and to call home (using Skype, say). There is a public phone at the institute. 
  • Though Barbados in the large is a safe place, there have been cases of thefts from rooms in the institute. Some prudence is recommended: do not leave valuables unattended. Lock your room at night and do not leave valuables near the window (allowing someone from the outside to get to them). The institute has a safe where you can deposit your valuables.
  • There is a modest size blackboard, and there is a data projector (to use with a laptop).
  • Please wash your body from sand very carefully using the taps outside (in the main yard, for example). Sand can cause very difficult problems of blockage at the showers.
  • The institute accepts Visa, Mastercard, cash and checques, which gives flexibility in settling your bill.
  • The prices of room are as per the posted rate on the Institute website. Postdocs are considered research faculty and students as students. The price depends on the specific room. Students, as a rule, will be staying at the more affordable rooms.
  • Breakfast at a cost of $7 will be served on Monday morning (it includes toast, cereal, fruit, tea, coffee, juice and something hot like pancakes, eggs or bacon). After that we will  ask people to choose either to have breakfast at the institute for the rest of the week, or not (at the same cost). One has the option of buying groceries and preparing breakfast independently, however what they offer seem to me a rather fair deal and the only option people may wish to take is to have their breakfast outside the institute altogether (there are no obvious choices, though).

 


Recreational Activities

  • Here is a link to a tourist information site.
  • The institute is located 1 minute walk (literally) from the beach. It is a beautiful beach with a coral reef. The Folkstone marine park is next door.
  • Some world-class cricket (an acquired taste...).
  • Bridgetown is a fun city to visit.
  • Crane beach on the east-coast is great fun (waves and white sands).

 


Coming prepared

  • The local currency is Bajan dollars. US dollars are accepted everywhere, at a fair rate, and it is easy to withdraw money from any ATM. You can change some money at the airport before exiting, though the taxi will accept US dollars.
  • Barbados is hot and sunny. Some information about the weather can be found here and claims almost constant temperature during night and day at the range of 25-30 Celsius. In particular, sunscreen is a must and one should plan on summer-like weather with the possibility of showers and occasional chilly evenings. A mosquito repellent is also highly recommended.
  • The institute is in St. James, which is a short walking distance to Holetown, where there is an excellent supermarket, banks, pharmacies and souvenir shops and so on, as well as cheap to expensive (and excellent) restaurants, cafes and bars. If you have a special diet (kosher, halal, vegan, ...) it is worth doing some research beforehand and making sure you have what you need. Dinners in the institute will not have special dishes prepared for people with special diet. Life, in general, gravitates between the institute and Holetown. For an occasional trip to Bridgetown the public buses are a fun experience and cheap, and shared taxis are a reasonable expense. Normally, one would not rent a car for the stay.
  • Stationary is not provided.
  • Coffee lovers should be mindful that the local coffee is instant coffee and bring with them ground coffee and filters.
  • Here is some screenshot of the area with indication where you can get basic stuff like water, sunscreen, etc..