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