# Mathematical Sciences Research Institute

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# Workshop

Introductory Workshop in Rational and Integral Points on Higher-Dimensional Varieties January 17, 2006 - January 21, 2006
 Registration Deadline: December 20, 2005 over 8 years ago October 17, 2005 over 8 years ago
Parent Program: Rational and Integral Points on Higher-Dimensional Varieties
Organizers Jean-Louis Colliot-Thélène, Roger Heath-Brown, János Kollár, Bjorn Poonen (chair), Alice Silverberg, Yuri Tschinkel
Speaker(s)

## Show List of Speakers

Description

NOTE: This workshop is to be held at the International House Berkeley on the UC Berkeley campus, at 2299 Piedmont Avenue. This week-long workshop will feature expository talks giving the "big picture" regarding the rational and integral solutions to systems of multivariable polynomial equations. When the variety defined by the equation is of the dimension 1, much is known, both in theory and in practice, so the focus of the workshop (and of the semester) will be on the case of dimension > 1, where most of the basic questions are far from being answered, and there is great opportunity for new results. The following people will speak at this workshop: Matthew Baker, Lucia Caporaso, Jordan Ellenberg, David Harari, Brendan Hassett, Roger Heath-Brown, Raman Parimala, Bjorn Poonen, Karl Rubin, Joseph H. Silverman, Trevor Wooley. For a more detailed workshop description and open discussion questions to be discussed, please download the description file in either pdf format or tex format.

This workshop is partially supported by the Number Theory Foundation.

PDF files of slides are available for some talks. To download, right-click the talk title and select "Save Taget As..." or "Save Link As...", or click the link to open the file in a new browser window. TUESDAY JANUARY 17 9:30 Welcome and information 9:45-10:35 Poonen: Introduction to rational points Tea 11:00-11:50 Silverman: Height functions - introduction (Slides are for Jan. 17 & 18 lectures) Lunch 2:00-2:50 Rubin: Mordell-Weil, Selmer, and Shafarevich-Tate groups Tea 3:30-4:20 Caporaso: Uniformity of rational points (Slides are for Jan. 17 & 19 lectures) 4:30-5:20 Wooley: The circle method counts for rational points - The Hasse Principle and Weak Approximation via the major arcs WEDNESDAY JANUARY 18 9:30-10:20 Poonen: Hilbert's Tenth Problem Tea 11:00-11:50 Silverman: Height functions - further topics (Slides are for Jan. 17 & 18 lectures) Lunch 2:00-2:50 Rubin: The Birch and Swinnerton-Dyer conjecture Tea 3:30-4:20 Harari: A few reciprocity laws in number theory 4:30-5:20 Wooley: The circle method counts for rational points - Asymptotic formulae via minor arc estimates THURSDAY JANUARY 19 9:30-10:20 Harari: The Brauer group of a variety Tea 11:00-11:50 Caporaso: Classification of complex projective varieties (Slides are for Jan. 17 & 19 lectures) Lunch 2:00-2:50 Baker: Geometry of points over Qbar of small height, I Tea 4:10-5:00 Heath-Brown: Counting rational points and the Manin conjecture, I (U.C.Berkeley colloquium, in Sibley Auditorium in Bechtel Hall) 5:00 Reception at 1015 Evans Hall, U.C. Berkeley FRIDAY JANUARY 20 9:30-10:20 Harari: Abelian and non-abelian descent Tea 11:00-11:50 Hassett: Rationally connected varieties Lunch 2:00-2:50 Parimala: Arithmetic of linear algebraic groups and their homogeneous spaces, I Tea 3:40-4:30 Heath-Brown: Counting rational points and the Manin conjecture, II SATURDAY JANUARY 21 9:30-10:20 Hassett: Potential density Tea 11:00-11:50 Parimala: Arithmetic of linear algebraic groups and their homogeneous spaces, II Lunch 2:00-2:50 Ellenberg: Counting arithmetic objects Tea 3:20-4:10 Baker: Geometry of points over Qbar of small height, II

Funding & Logistics

## Show Funding

To apply for funding, you must register by the funding application deadline displayed above.

Students, recent Ph.D.'s, women, and members of underrepresented minorities are particularly encouraged to apply. Funding awards are typically made 6 weeks before the workshop begins. Requests received after the funding deadline are considered only if additional funds become available.

## Show Lodging

MSRI has preferred rates at the Rose Garden Inn, depending on room availability. Reservations may be made by calling 1-800-992-9005 OR directly on their website. Click on Corporate at the bottom of the screen and when prompted enter code MATH (this code is not case sensitive). By using this code a new calendar will appear and will show the MSRI rate on all room types available.

MSRI has preferred rates at the Hotel Durant. Reservations may be made by calling 1-800-238-7268. When making reservations, guests must request the MSRI preferred rate. If you are making your reservations on line, please go to this link and enter the promo/corporate code MSRI123. Our preferred rate is $129 per night for a Deluxe Queen/King, based on availability. MSRI has preferred rates of$149 - $189 plus tax at the Hotel Shattuck Plaza, depending on room availability. Guests can either call the hotel's main line at 510-845-7300 and ask for the MSRI- Mathematical Science Research Inst. discount; or go to www.hotelshattuckplaza.com and click Book Now. Once on the reservation page, click “Promo/Corporate Code“ and input the code: msri. MSRI has preferred rates of$110 - \$140 at the Berkeley Lab Guest House, depending on room availability. Reservations may be made by calling 510-495-8000 or directly on their website. Select “I am an individual traveler affiliated with MSRI”.

Additional lodging options may be found on our short term housing page.