New Techniques in Birational Geometry

“双几何新技术”会议将于2015年4月7日至11日在石溪大学数学系举行；在此之前，将为研究生和博士后研究人员提供为期一天的迷你学校。会议的中心是代数几何中“合理性问题”的最新进展。代数几何研究多项式方程系统——在数学、科学和工程中无处不在——通过观察解集的几何形状。如果存在一个多项式函数，其输出总是给出该系统的解，并且在该函数的输出中恰好出现一次通解（分别只有有限次），则该多项式方程系统被称为“有理”（分别为“酉”）。这个属性具有很大的实用价值，但不幸的是，要判断一个给定的系统是理性的还是非理性的，是出了名的困难。会议将汇集专家讨论这一悬而未决问题的最新进展；它将促进不同领域专家之间的互动，并将在这些重要技术方面培养年轻数学家。该奖项主要支持初级研究人员参加会议。具体来说，会议将探讨与合理性问题相关的四个主题：(1)代数循环和霍奇理论(2)派生类别(3)桥地稳定性条件(4)两国几何。会议将由这些领域的主要专家发表演讲。在代数几何的整个历史中，“合理性问题”一直是一个试金石，推动了霍奇理论（克莱门斯-格里菲斯定理证明了l<s:1>罗斯猜想）、最小模型程序（Iskovskikh-Manin定理和Mori对Hartshorne猜想的证明）、不变理论（Saltman对Noether问题的负解）和<s:1>上同调（Artin-Mumford定理解决了稳定l<s:1>罗斯问题）的重大进展。尽管该问题有许多直接应用，但其主要价值在于激发新技术，并作为不同领域之间的联系点。最近在代数几何的几个部分的合理性问题上取得了重大进展，这是一个难得的机会，将具有互补专业知识的研究人员聚集在一起。事实上，一个完整的解决方案，例如在立方四倍的情况下，可能需要多种方法，并且很少有数学家在所有这些领域都是专家。将有一些专门针对青少年参与者的活动（例如介绍一些主题的为期一天的迷你学校），并将努力招募不同群体的参与者。有关会议的更多细节，请访问其网站：http://www.math.sunysb.edu/AlgebraicGeometry/Birational2015/。

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/.