Spaces of curves and their interaction with diophantine problems
A conference in algebraic geometry
the NSF, Columbia University, Stony Brook University, Rice University and Courant Institute.
The conference will be held at Columbia University.
Dates of the conference:
1-5 June 2009.
Schedule of talks!
Confirmed participants: (in random order)
- Max Lieblich.
Title: A counter counterexample example.
- Rahul Pandharipande.
Title: The moduli space of stable quotients.
- Chenyang Xu.
Title: Strong Rational Connectedess and Its Applications.
Title: Diophantine subsets of function fields of curves.
- Jean-Louis Colliot-Thelene.
Title: Local-global principles for homogeneous spaces of linear algebraic
groups over function fields of p-adic curves
(joint work with Parimala and Suresh).
- Hélène Esnault.
Title:Hodge level and rational points (joint on one hand with
O. Wittenberg and on the other with P. Berthelot and K. R\"ulling).
- Izzet Coskun.
Title: The birational geometry of moduli spaces.
- Amanda Knecht.
Title: Del Pezzo Surfaces Over Finite Fields.
- Ana-Maria Castravet.
Title: Rational curves of minimal degree on higher Fano manifolds
(joint work with Carolina Araujo).
- Damiano Testa.
Title: Big rational surfaces.
- Bjorn Pooonen.
Title: Undecidability of polynomial equations over C(t_1,t_2).
- Jordan Ellenberg.
Title: Analytic number theory and spaces of rational curves.
- Roya Beheshti-Zavareh.
Title: The geometry of spaces of rational curves on hypersurfaces.
- Carolina Araujo.
Title: Cohomological characterizations of projective spaces and hyperquadrics.
- Aravind Asok.
Title: Rational connectivity and A^1-homotopy theory.
- David Harbater.
Title: Quadratic forms and a local-global principle (joint with Julia Hartmann and Daniel Krashen).
- Ivan Cheltsov.
Title: Exceptional del Pezzo surfaces.
- Brendan Hassett.
Title: Rational curves on K3 surfaces (joint with F. Bogomolov and
- Matt DeLand.
Title: Nonsingular cubic hypersurfaces in P^9 are rationally simply connected.
- Olivier Wittenberg.
Title: Existence of zero-cycles on fibrations over number fields.
- All participants (those that were noticed by the organizers in any case)
- Armand Brumer, Fordham University
- Chuck Weibel, Rutgers University
- Hélène Esnault, Universität Duisburg-Essen
- Izzet Coskun, Universtiy of Illinois at Chicago
- David Harbater, University of Pennsylvania
- Roya Beheshti-Zavareh, Washington University
- Jordan Ellenberg, University of Wisonsin - Madison
- Rahul Pandharipande, Princeton University
- Luis Garcia, Columbia University
- Davesh Maulik, MIT
- Linda Chen, Swarthmore
- Aravind Asok, UCLA
- Damiano Testa, Oxford University
- Ivan Cheltsov, University of Edinburgh
- Chenyan Wu, Columbia University
- Ana-Maria Castravet, University of Arizona
- Anand Deopurkar, Harvard University
- Hu Yong, Universitee Paris-Sud 11
- Jason Starr, Stony Brook Universty
- Jean-Louis Colliot-Thelene, CNRS/Universitee Paris-Sud FRANCE
- Ljudmila Kamenova, Stony Brook University
- Mingmin Shen, Columbia University
- Fei Xu, Rice University
- Rebecca Bellovin, Stanford University
- Benjamin Waters, Rice University
- Alon Levi, Columbia University
- Joe Ross, Columbia University
- Matt DeLand, Columbia University
- Alice Rizzardo, Columbia UNiversity
- Alena Pirutka, Ens Paris, Paris 11
- Michael Thaddeus, Columbia University
- Robert Friedman, Columbia University
- Max Lieblich, Princeton University
- Rob Findley, Stony Brook University
- Andrew Obus, Upenn/Columbia University
- Abhinav Kumar, MIT
- Amanda Knecht, University of Michigan
- Yi Zhu, Stony Brook University
- Yu Yasufuku, CUNY Graduate Center
- Yusuf Mustopa, Stony Brook University
- Zhiyu Tian, Stony Brook University
- Akshay Venkatesh, Stanford UNiversity
- Yanhong Yang, Columbia University
- Ben Bakker, Princeton University
- Melissa Liu, Columbia University
- Chenyang Xu, MIT
- Brendan Hassett, Rice University
- Olivier Wittenberg, CNRS-ENS
- Maksym Fedorchuck, Columbia University
- Alexei Oblomkov, Princeton University
- Nathan Grieve, Queen's University
- Jerry Hu, University of Houston-Victoria
- Jacob Tsimerman, Princeton University
- Yongqiang Zhao, University of Wiscconsin Madison
- Derek Garton, University of Wisconsin-Madison
- Ali Altug, Princeton University
- János Kollár, Princeton University
- Thibaut Pugin, Columbia University
- Hang Xu, Columbia University
- Jorge Pineiro, Bronx Community Colledge
- Bhargav Bhatt, Princeton University
- Darren Glass, Gettysburg College
- Ishai Dan-Cohen, UC Berkely
- Tony Várilly-Alvarado, UC Berkely - Rice University
- Aise Johan de Jong, Columbia University
- Sho Tanimoto, New York University
- Carolina Araujo, IMPA (Brazil)
- More local people
- If your name should be here and it isn't let me know.
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.
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!
This is really tricky. We have booked a block of rooms in the
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.