Spaces of curves and their interaction with diophantine problems

A conference in algebraic geometry

Organizers:

Supported by: the NSF, Columbia University, Stony Brook University, Rice University and Courant Institute.

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Place: The conference will be held at Columbia University.

Dates of the conference: 1-5 June 2009.

Schedule of talks!

Confirmed participants: (in random order)

Topic of the conference: Here is the abstract of the FRG proposal under whose auspices this conference is held:

This project addresses the geometry of spaces of rational curves on smooth projective varieties, with a view toward understanding the structure of rational points for varieties defined over function fields. Consider a rationally-connected variety: Which homology classes contain free rational curves? Very free rational curves? Is the space of such curves connected? Irreducible? Rationally connected? Of general type? Is there a workable notion of `rational simple connectedness' and is this a birational property? How can we distinguish unirational varieties as a subclass of rationally-connected varieties? These questions are related to fundamental problems in Diophantine geometry over function fields: Does a rationally-connected variety over C(t) satisfy weak approximation? Can the hypothesis of the Tsen/Lang Theorem over C(s,t) be formulated geometrically? For rationally-connected varieties over C(s,t), to what extent do cohomological obstructions govern the existence of rational points? This award will support research on systems of polynomial equations with coefficents varying in parameters. Our goal is to solve these equations with rational functions that depend on these parameters. The case of a single equation (or of several independent equations) was addressed in the mid 20th century; the feasibility of finding a solution depends on the degree of the equation, the number of free variables, and the number of varying parameters. Recently, a comprehensive geometric approach was developed when there is just one varying parameter. However, for multiple (not necessarily independent) equations in two varying parameters much remains to be understood. This work will also have broader impacts on the education of graduate students and postdoctoral fellows, the development of web-based collaboration tools, and the promotion of robust academic networks linking universities across the country.

Application to participate: Please email Johan de Jong (dejong youknowwhat math.columbia.edu) to apply for a spot in the conference and possible reimbursements. In case of graduate students: Please have your advisor write a short letter supporting your application.

Educational component: We are solliciting topics from junior particpants as topics for (up to 3) lectures by senior participants in the evenings. The idea is to sketch proofs of foundational results, or of relatively recent important results in the area to younger researchers. Please email your suggestions for topics to the organizers, especially Johan de Jong and Jason Starr. In addition, if you are a senior investigator, please sign up to give one of these lectures!

Hotels accommodations: This is really tricky. We have booked a block of rooms in the Milburn hotel which is nearby. Of course this is very limited. You can look for hotels yourself or contact the organizers (Johan) to see if there is space left in the reserved block.