Rational curves and A1-homotopy theory

October 5 to October 9, 2009

at the

American Institute of Mathematics, San Jose, California

organized by

Aravind Asok and Jason Starr

Original Announcement

This workshop will be devoted to studying recent interactions between rational connectivity and the newly developing theory of $A^1$-algebraic topology.

A smooth proper variety over an algebraically closed field k having characteristic 0 is rationally connected if any pair of k-points is contained in a rational curve. Rationally connected varieties have highly non-trivial arithmetic structure. More recently, rationally connected varieties have been shown to have interesting homotopic structure as well: they are connected (in an appropriate sense) from the standpoint of $A^1$-homotopy theory. One expects that techniques of homotopy theory can be applied to study arithmetic and geometry of rationally connected varieties and, conversely, geometric properties of specific rationally connected varieties can provide insights about $A^1$-homotopy theory. The goal of this program is to introduce participants studying arithmetic of rationally connected varieties to the techniques of homotopy theory and vice versa.

During the workshop, we will focus on the following problems.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Rational points on K3 surfaces and derived equivalence
by  Brendan Hassett and Yuri Tschinkel
Splitting Varieties for Triple Massey Products
by  Michael Hopkins and Kirsten Wickelgren,  J. Pure Appl. Algebra 219 (2015), no. 5, 1304-1319  MR3299685
R-equivalence on low degree complete intersections
by  Alena Pirutka
Cohomologie non ramifiée et conjecture de Hodge entire
by  Jean-Louis Colliot-Thélène and Claire Voisin
Stable A^1-homotopy and R-equivalence
by  Aravind Asok and Christian Haesemeyer
Hodge theory and Lagrangian planes on generalized Kummer fourfolds
by  Brendan Hassett and Yuri Tschinkel
Characterizing projective spaces on deformations of Hilbert schemes of K3 surfaces
by  David Harvey, Brendan Hassett, and Yuri Tschinkel