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Workshop on
Arithmetic Geometry and Homotopy Theory 31 May – 1 June, 2012 |
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Participants Programme Titles and
abstracts Practical
The aim of the
workshop is to bring together researchers working on applications of homotopy
theory to arithmetic geometry, with particular emphasis on recent work on
rational points and the section conjecture.
Organizers: Ambrus
Pál and Alexei Skorobogatov
Contact email:
a.pal[…]imperial.ac.uk
This workshop is
funded by EPSRC and LMS.
Confirmed
participants include:
Ilan Barnea, Stephane
Bijakowski, Kevin Buzzard, Yonathan Harpaz, Andreas Holmstrom, Irene Galstian,
Toby Gee, Rick Jardine, Chris Lazda, Frank Neumann, Behrang Noohi, Ambrus Pal,
Alena Pirutka, Jon Pridham, Gereon Quick, Tomer Schlank, Jack Shotton, Alexei
Skorobogatov, Arne Smeets, Kristian Strommen, Jan Vonk, Kirsten Wickelgren
Thu
09:00-09:30 Registration
Thu
09:30-10:30 Tomer
Schlank Introduction
to etale homotopy
Thu
10:45-11:45 Behrang Noohi Model categories
Thu
12:00-13:00 Rick Jardine
Galois descent
and pro-objects
Thu
15:00-16:00 Ilan Barnea
From weak
fibration categories to model categories
Thu 16:45-17:45 Arturo Pianzola Reductive group schemes,
Torsors, and Extended Affine Lie Algebras
Thu
20:00-
Conference dinner
Fri
09:30-10:30 Jon Pridham
Hodge structures
on homotopy types of quasi-projective varieties
Fri
10:45-11:45 Kirsten
Wickelgren 2-nilpotent
real section conjecture
Fri
12:00-13:00 Yonathan
Harpaz Etale homotopy
and Diophantine equations
Fri
15:00-16:00 Ambrus
Pal The section conjecture and homotopy theory
Fri
16:30-17:30 Gereon Quick
Existence of
rational points as a homotopy limit problem
Title: From weak
fibration categories to model categories
Abstract:
Model categories
provide a very general framework in which it is possible to set up the basic
machinery of homotopy theory. The structure of a model category is very
convenient, however it is not always available. There are situations in which
there is a natural definition of weak equivalences and fibrations, however, the
resulting structure is not a model category. A notable example is the category
of simplicial sheaves over a Grothendieck site where the weak equivalences and
the fibrations are local, in the sense of Jardine. This motivated the search
for a more flexible structure then a model category, in which to do abstract
homotopy theory. In this lecture I will introduce such a structure, called a
"weak fibration category". The novelty of this structure is that it
can be "completed" into a full model category structure, provided we
pass to the pro category. Applying this result to the weak fibration category
of simplicial sheaves mentioned above, gives a new model structure on the
category of pro simplicial sheaves. This model structure turns out to be very
convenient for the study of etale homotopy and homotopical obstructions to
rational points, as was introduced by Pal and Harpaz-Schlank.
Title: Etale homotopy
and Diophantine equations
Abstract:
In 1969
Artin and Mazur defined the etale homotopy type Et(X) of a scheme X as a way to
homotopically realize the etale topos
of X. In this talk we will describe an alternative construction of
Et(X). This construction is enabled by endowing the category of pro-simplicial
sets with an appropriate model structure which was recently constructed by
T. Schlank and I. Barnea. One advantage of this construction over the classical
one is that it upgrades Et(X) from a pro-homotopy type to a pro-simplicial set.
This was achieved before by Friedlander in a different approach. However, the
current construction enjoys certain additional properties. In particular it
generalizes naturally to the relative setting X->S. This results in
a relative etale homotopy type, Et_/S(X), which is a
pro-object in the category of simplicial etale sheaves over S (using again the
model structure of Schlank and Barnea). It turns out that the relative homotopy
type can be especially useful in studying the sections of the map X->S. In particular this notion can be used in
order to obtain homotopy-theoretic obstructions to the existence of a section,
as well as homotopy-theoretic classification of sections. In this lecture we
will describe and exemplify these constructions in the special case where
S=Spec(K) is the spectrum of a number field K (in which case sections
correspond to rational points) and in the case where S=Spec(O_K) is the
spectrum of a number ring (in which case sections correspond to integral
points). Furthermore we will explain the connection between these
homotopy-theoretic constructions and the relevant parts of the classical
arithmetic theory, like the Brauer-Manin obstruction and Grothendieck's section
obstruction. This is joint work with T. Schlank.
Title: Galois descent
and pro-objects
Abstract:
The
Lichtenbaum-Quillen conjecture says that the algebraic K-theory and the etale
algebraic K-theory of fields coincide outside of a finite range of degrees, in
the presence of suitable torsion coefficients. This conjecture is now known to
be a consequence of the Bloch-Kato conjecture, by a result of Suslin and
Voevodsky. Earlier attempts to prove Lichtenbaum-Quillen involved a Galois
cohomological descent technique. These attempts invariably failed because the
relation between "finite" descent and Galois descent was not properly
understood. This talk will describe a local homotopy theory for pro objects in
simplicial presheaves which can be applied in this context. It will be shown
that finite descent plus the existence of a certain pro-equivalence implies
Galois descent for simplicial presheaves on the etale site of a field.
Title: Model
categories
Title: The section
conjecture and homotopy theory
Abstract: One of the
main motivations to study rational points on algebraic varieties via homotopy
theory is Grothendieck’s celebrated section conjecture. In my talk I will try
to give an overview of this subject.
Title: Hodge
structures on homotopy types of quasi-projective varieties
Title: Existence of
rational points as a homotopy limit problem
Abstract: We discuss
different ways to relate the existence of rational points for varieties over a
field to comparison maps between fixed points and homotopy fixed points of the
etale homotopy types of these varieties.
Title: Introduction
to etale homotopy
Title: 2-nilpotent
real section conjecture
Abstract: Sullivan's
conjecture, proven by Haynes Miller and Gunnar Carlsson, relates the fixed
points to the homotopy fixed points of p-group actions on finite complexes.
Applying this result to algebraic curves defined over R with the action of
complex conjugation gives the real analogue of Grothendieck's section
conjecture predicting that the rational points on curves over finitely
generated fields are determined by maps between etale fundamental groups. By
examining the symmetric powers of curves, we show a 2-nilpotent section conjecture
over R: for a curve X over R such that each component of its normalization has
real points, pi_0(X(R)) is determined by the maximal 2-nilpotent quotient of
the topological fundamental group or etale fundamental group with its Z/2
action. This implies that the set of real points equipped with a real tangent
direction of a smooth compact curve X is determined by the maximal 2-nilpotent
quotient of the absolute Galois group of the function field, showing a
2-nilpotent birational real section conjecture.
WIFI connection is
available in the Huxley Building. If you bring your laptop with you, you will
be able to connect to the internet there. Ask for the password at the
registration at the beginning of the workshop.
South Kensington
South Kensington is
the district where the main campus of Imperial College lies, and where the
conference will take place. It is close to several
Directions
The lectures will
take place in room 130, the Huxley Building, 180 Queen’s Gate, SW7 2AZ, London.
Google map.
Transportation
For interested
participants we can arrange for ground transportation.
Hotels
Park International
Hotel,
Public transportation
Money
The currency in the
United Kingdom is GBP (pound sterling, symbol: £).
The approximate
currency rate is: 1 US$ ~ 0.63GBP or 1 Euro ~ 0.82GBP. See the currency converter for up-to-date rates. There are
exchange booths near Gloucester Road underground station.
Food and coffee
There are many
restaurants and coffee shops on Gloucester Road (to the west) and on Old
Brompton Road (to the south), especially in the vicinity of the two underground
stations. There are restaurants, coffee shops and high street shopping on High
Street Kensington (to the north), too.