New Techniques in Birational Geometry

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The conference "New Techniques in Birational Geometry" will be held in the Department of Mathematics at Stony Brook University on April 7-11, 2015; it will be preceded by a one-day mini-school for graduate students and postdoctoral researchers. The conference is centered around recent advances on the "rationality problem" in algebraic geometry. Algebraic geometry studies systems of polynomial equations -- ubiquitous in mathematics, science, and engineering -- by looking at the geometry of the set of solutions. A system of polynomial equations is called "rational" (respectively "unirational") if there is a polynomial function whose outputs always give solutions of the system, and such that a general solution occurs among the outputs of this function exactly once (respectively only a finite number of times). This property is of great practical value, but unfortunately it is notoriously difficult to tell whether a given system is rational or unirational. The conference will bring together experts to discuss recent progress on this open problem; it will foster interactions between experts in different fields, and it will train young mathematicians in these important techniques. This award supports participation, primarily by junior researchers, in the conference.Specifically, the conference will explore four themes related to rationality questions:(1) Algebraic cycles and Hodge theory(2) Derived categories(3) Bridgeland stability conditions(4) Birational geometryThe conference will feature presentations by leading experts in these areas. Throughout the history of algebraic geometry, the "rationality problem" has been a touchstone, motivating major progress in Hodge theory (the Clemens-Griffiths theorem disproving the Lüroth conjecture), the minimal model program (the Iskovskikh-Manin theorem and Mori's proof of the Hartshorne conjecture), invariant theory (Saltman's negative solution of Noether's problem), and étale cohomology (the Artin-Mumford theorem solving the stable Lüroth problem). Although the problem has many direct applications, its main value has been inspiring new techniques and serving as a point of contact between different areas. There have been major recent advances on the rationality problem in several parts of algebraic geometry, and this is a rare moment of opportunity to bring together researchers with complementary expertise. Indeed, a complete solution, for instance in the case of cubic fourfolds, likely requires multiple approaches, and there are few mathematicians expert in all these areas. There will be a number of activities specifically for junior participants (such as a one-day mini-school introducing some of the topics), and an effort will be made to recruit a diverse body of participants. More details about the conference can be found at its website: http://www.math.sunysb.edu/AlgebraicGeometry/Birational2015/.