In the spirit and tradition of previous international conferences on the algebraic and arithmetic properties of quadratic forms in Chile that had been organized or promoted by Ricardo Baeza, we plan a one week Conference on Quadratic Forms (and related topics) in Talca. The conference will end with a conference dinner, where we will also take the occasion to celebrate Ricardo's 75th birthday. 


A preceeding postgraduate school on "Quadratic forms over function fields" will take place in Santiago (at the Universidad de Santiago de Chile) from january 2nd to Friday 5th of january 2018, just prior to the conference (see also the item on the side bar).

The conference will last from monday morning, january 8th 2018 until friday night, january 12th 2018 after the conference dinner (cena de despedida).

 schedule   abstracts of talks

 

Click here to download List

 

Eva Bayer-Fluckiger (EPFL, Switzerland) 
Wai Kiu Chan (Wesleyan University, USA)
Jean-Louis Colliot-Thélène (U. Paris Sud, France)
Daniel Krashen (University of Georgia, USA)
David Leep (University of Kentucky, USA)
Guillermo Mantilla (U. los Andes, Colombia)
Alexander Merkurjev (UCLA, USA)
Raman Parimala (Emory University, USA) 
Jean-Pierre Tignol (UCL, Belgium)
Giancarlos Urzúa (PUC, Chile)
Alexander Vishik (University of Nottingham, GB)

Send a mail to Esta dirección de correo electrónico está protegida contra spambots. Usted necesita tener Javascript activado para poder verla.  with the subject line "conference registration". Indicate your name, status and affiliation, and whether you are interested to contribute a talk (with tentative title). 
International participants please also indicate if we should help find accomodation (which dates? travelling alone?), especially whether you are interested in a special offer (40.000 CLP per day single room, 50.000 CLP per day double room) at the main hotel for the time of the conference (www.ecohotel.cl). 

 

Participants arriving to Santiago airport on sunday 7th, send us your flight numbers at your earliest convenience. This way we will organize the transfer from Santiago to Talca, and it also allows us to react in real time to delayed arrivals. 


If you arrive prior to Sunday, 7th of January 2018 to Santiago (or afterwards), you can get in contact with us if you want our help in organizing your travel (by bus) to Talca. The transfer from Santiago Airport will NOT make a detour through Santiago (and getting from Santiago first to the airport outside the city in order to join our transfer would be more time- and money consuming than getting a regular bus from Santiago to Talca) 

We offer assistance for non-Chilean residents in finding and reserving accomodation that is conveniently located with respect to the conference venue. Other than the aforementioned room reserve four star hotel Ecohotel (with special offers for international participants), we can also help in finding housing for smaller budgets, such as cabins for doctoral students, for example. Simply ask us.

Karim Johannes Becher (U Antwerpen, Belgium), David Grimm (USACH, Chile), Detlev Hoffmann (TU Dortmund, Germany), María Inés Icaza (U Talca, Chile), Maximiliano Leyton (U Talca, Chile), Alvaro Liendo (U Talca, Chile).

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.

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.

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. 


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). 


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). 


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. 


More information to come. 

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.

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).

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).