Summer School

Summer School - Quadratic Forms in Chile 2018

the week prior to the conference: 2nd - 5th of January 2018

{Where and When ?}
The four day school  Quadratic Forms over Function Fields will be held in Santiago, at the Universidad de Santiago de Chile , 
Tuesday 2nd of January - Friday 5th of January 2018, the week just prior to the conference.
{For Whom ?}
The school is primarily (but not exclusively) aimed both at international graduate students who will attend the subsequent conference in Talca, as well as at local graduate stundents or postdocs with background in algebra or geometry (not necessarily quadratic forms). Ideally, the school serves as a warm up and als as a teaser for the conference. 
The first day and a half will be dedicated to introducing the basics of the theory of quadratic forms. For students with a background in quadratic forms (and who cannot arrive by January 2), it should be no problem to join the school starting from the afternoon of January 3.
{about What ?}

There have been several recent results on the isotropy for quadratic forms over function fields, most notably, a local-global principle over function fields of p-adic curves, which will be discussed in the summer school. With the intention to put emphasis on methods over results, and in order to plan a realistic schedule that leaves enough time for exercises and individual work, we decided to restrict the focus of the summer school to a few "smaller" results whose proofs are not too exhaustive, but which exemplify the use of various techniques from algebraic geometry and topology to solve arithmetic questions over function fields. 

Course 1 (by Karim J. Becher, Universiteit Anterpen, Belgium): Quadratic forms over fields and local rings

The course will introduce the basic notions corresponding to quadratic forms over fields and local rings, such as isotropy, the notion of residue forms of unimodular forms over a local ring, as well as lifting of isotropy of the residue form to the completion of the ring. Moreover, ground work for the other two courses will be layed, for example the relation between orderings of fields, sums of squares, quadratic forms and valuations, or the relation between the isotropy of 3 and 4-dimensional quadratic forms and the splitting of certain quaternion algebras. Also, quadratic forms over particular base fields will be discussed without complete proves (e.g. Tsen-Lang theory for complex function fields, local-global principles over number fields or p-adic function fields). 

Course 2 (by David Grimm, Universidad de Santiago de Chile): Sums of squares in function fields of real curves and surfaces

The explicit objective of this course is the revise some better and also lesser known results on sums of squares in and quadratic forms over function fields of real varieties, such as upper and lower bounds for the so called Pythagoras number; in particular for algebraic (and arithmetic) surfaces over the real numbers - (the Pythagoras number beeing the smallest natural number n needed to express any totally positive element as a sum of n squares). 
These will be consequences of implicite objectives of the course: Relating arithmetic objects and techniques for function fields (such as valuations, orderings, sums of squares) to geometric objects and techniques (such as blowing ups, divisors, double covers, generic hyperplane sections, possibly vector bundles and their chern classes). 

Course 3 (by Asher Auel, Yale University, USA): Quadratic forms over function fields of complex curves and surfaces

The main objective of this course is to construct counterexamples to the existence of a local-global principle for isotropy of quadratic forms over function fields of complex algebraic varieties, with emphasis on the case of curves and surfaces. The construction of quadratic forms that are locally isotropic but globally anisotropic will involve input from the Picard group, in the case of curves, and the Brauer group and Hodge theory, in the case of surfaces. These methods invoke many of the arithmetic and geometric concepts outlined in Course 2, and will be further developed in Course 3. 

Special lecture 1 (by David Leep; University of Kentucky, USA): t.b.a.

More information to come. 

Special lecture 2 (by Jean-Louis Colliot-Thélène , Université Paris 11, France): t.b.a.

Title : The Brauer group and beyond

The Brauer group of varieties may detect nonrationality of vareties. It may also prevent local-global principles for rational points of projective varieties and for integral points of affine varieties. There are ``higher'' analogues of the Brauer group. I shall describe various techniques to compute the Brauer group and these higher variants.
{How to register ?}
If you also register for the conference in Talca, you can register for the school in Santiago in the same email message. Or, you can also register for the school in Santiago separately by writing an email message with the subject line "school registration" to the official USACH mail address Esta dirección de correo electrónico está protegida contra spambots. Usted necesita tener Javascript activado para poder verla. of the summer school. There will be no registration fee. Note that lunch will not be provided, though we are currently working on getting special lunch offers at a nice nearby restaurant for participants of the school.
{Where to stay ?}
There are several hostels within 20 minutes walking distance to the USACH campus in the historic and quiet quarter "Yungay". However, the metro system in Santiago is quite good, and thus if you stay anywhere near the green or red metro line (not too far away from the center) you should be able to get to the school in 30 minutes (closest metro stops: "Estacion Central" on red line or "Quinta Normal" on green line). 
If you want us to help you find accommodation in Santiago, let us know in the registration mail, also indicate if you are potentially interested in sharing an apartment (on AirBnB) with other participants, so that we can team you up and rent the place for you (in this case we would ask you to please send a copy of your flight tickets).
{What else ?}
If this is your first time to Chile, we strongly recommend using the week-end between the school and the conference to visit the beautiful city of Valparaiso and also enjoy the nearby beaches of Vina del Mar (personal note: if you are into surfing, then the bay of Concón in the north of Vina del Mar is the right beach for you - steady waves good for beginners; material can be rented from the surf schools there).

Universidad de Talca