The workshop "Counting Points on Varieties" consisted of two parts.

The first week, April 14--17, was a so called Stieltjesweek, aimed at high-level masters students, graduate students, and beginning postdocs. There were three mini-courses: one on the Birch and Swinnerton-Dyer conjecture, another on Zeta functions, Laplacians, and étale cohomology, and a third on the Batyrev-Manin conjecture. Each morning there was a one-hour lecture on each of the three topics, while the participants worked on exercises during the afternoon, which they presented to each other, each day at 16.00.

 

Tim Dokchitser taught the course on the Birch and Swinnerton-Dyer conjecture, one of the famous one-million dollar millennium problems. It relates the rank of the Mordell-Weil group of an elliptic curve to the order of vanishing of its associated L-series. He focused mostly on the parity conjecture, which follows from the Birch and Swinnerton-Dyer conjecture. There is no reason to believe the parity conjecture is easier to prove, but it is more easily accessible, especially in a short course.

 

Ted Chinburg taught the course on Zeta functions, Laplacians, and étale cohomology. He presented a wide variety of related problems, including the question whether you can hear the shape of a drum.

 

Ronald van Luijk taught the course on the Batyrev--Manin conjecture, which predicts the asymptotic growth of the number of rational points of bounded height on certain varieties in terms of the bound.

 

The courses were well received by the students, who almost all (around 40) participated very actively in the exercises and the presentation of the solutions. Thursday night there was a well-attended dinner where all students from various countries (Netherlands, Germany, England, France, Switzerland, Tunesia, Turkey) were able to integrate

in a nonmathematical manner.

 

Based on the responses from the students, we consider the week a big success.

 

The second part was a week-long research workshop.  About 20 talks were given by experts in one of several fields, all related to the “counting points on varieties'' theme. In total about 60 people attended the workshop, amongst them were many participants of the first, instructional, week and other young mathematicians.

 

The speakers were very well aware of the broad nature of this workshop and consistently went through a lot of effort to communicate to the whole audience, and not just to those most familiar with their topic. Also this made it easier for the attending students to connect some of these talks with what they had learnt in the first week of the workshop.

 

With only four one-hour talks per day, the program was designed to allow ample space for informal discussion during the breaks. This opportunity was used intensively and it was not uncommon to find mathematicians of the highest reputations (Stephen Lichtenbaum, Jean-Pierre Serre) discuss mathematics with young mathematicians during the breaks.

 

Almost everybody was present for the conference dinner which was held on Wednesday evening, on a boat making a tour of the Groene Hart.

 

The conference ended with two talks on Friday morning, given by renowned mathematicians Carl Pomerance and Manjul Bhargava. These talks formed also the beginning of the day-long festivities organized for Hendrik Lenstra's sixtieth birthday. On top of the workshop participants, about 50 mathematicians mostly from the Netherlands came to Leiden to attend those talks.