Workshop on Arithmetic Geometry and Homotopy Theory
31 May – 1 June, 2012
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
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
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
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 is
the district where the main campus of Imperial College lies, and where the
conference will take place. It is close to several
The lectures will take place in room 130, the Huxley Building, 180 Queen’s Gate, SW7 2AZ, London.
For interested participants we can arrange for ground transportation.
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.