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Introductory Workshop in Rational and Integral Points on Higher-Dimensional Varieties
Jan 17, 2006 to Jan 21, 2006

Organizer(s)

Jean-Louis Colliot-Thélène, Roger Heath-Brown, János Kollár, Bjorn Poonen (chair), Alice Silverberg, Yuri Tschinkel
To apply for funding, you must register by Mon, Oct 31 2005.
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

To apply for funding, you must register by Mon, Oct 31 2005. Click to Register
Students, recent Ph.D.'s, women, and members of underrepresented minorities are particularly encouraged to apply. Funding awards are made typically 6 weeks before the workshop begins. Requests received after the funding deadline are considered only if additional funds become available.
Parent Program(s):
Rational and Integral Points on Higher-Dimensional Varieties


Questions about this workshop should be sent either by email to
or by regular mail to:
Introductory Workshop in Rational and Integral Points on Higher-Dimensional Varieties
Mathematical Sciences Research Institute
17 Gauss Way, Berkeley, CA
94720-5070.
USA

The Institute is committed to the principles of Equal Opportunity and Affirmative Action.



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