Workshop on
Arithmetic Geometry and Homotopy Theory 31 May Š 1 June,
2012 

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 P‡l and Alexei
Skorobogatov
Contact email: a.pal[É]imperial.ac.uk
This workshop is funded by EPSRC and
LMS.
Confirmed participants include:
Ilan Barnea, 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, Kirsten Wickelgren
Thu
09:0009:30 Registration
Thu
09:3010:30 Tomer
Schlank TBA
Thu
10:4511:45 Behrang Noohi TBA
Thu
12:0013:00 Rick Jardine
Galois
descent and proobjects
Thu
15:0016:00 Ilan Barnea
From
weak fibration categories to model categories
Thu
20:00
Conference dinner
Fri
09:3010:30 Jon Pridham
Hodge
structures on homotopy types of quasiprojective varieties
Fri
10:4511:45 Kirsten Wickelgren 2nilpotent
real section conjecture
Fri
12:0013:00 Yonathan
Harpaz Etale homotopy and Diophantine equations
Fri
15:0016:00 Ambrus
Pal TBA
Fri
16:3017:30 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 HarpazSchlank.
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 prosimplicial 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 prohomotopy type to a prosimplicial 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
proobject 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 homotopytheoretic obstructions to the existence of a section,
as well as homotopytheoretic 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
homotopytheoretic constructions and the relevant parts of the classical
arithmetic theory, like the BrauerManin obstruction and Grothendieck's section
obstruction. This is joint work with T. Schlank.
Title: Galois descent and proobjects
Abstract:
The LichtenbaumQuillen conjecture
says that the algebraic Ktheory and the etale algebraic Ktheory 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
BlochKato conjecture, by a result of Suslin and Voevodsky. Earlier attempts to
prove LichtenbaumQuillen 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 proequivalence implies Galois descent
for simplicial presheaves on the etale site of a field.
Title: TBA
Abstract:
Title: TBA
Abstract:
Title: Hodge structures on homotopy
types of quasiprojective 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: TBA
Abstract:
Title: 2nilpotent real section
conjecture
Abstract: Sullivan's conjecture, proven
by Haynes Miller and Gunnar Carlsson, relates the fixed points to the homotopy
fixed points of pgroup 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 2nilpotent 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 2nilpotent 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 2nilpotent quotient of the absolute Galois group of
the function field, showing a 2nilpotent birational real section conjecture.
WIFI connection is available in the
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 QueenÕs Gate, SW7 2AZ, London.
Google map.
Transportation
Information coming soon.
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 uptodate 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.