Thematic Program on Calabi-Yau Varieties: Arithmetic, Geometry, and Physics

该提案将有助于支持“弦理论的模块化形式”研讨会，该研讨会定于2013年9月16日至20日在加拿大多伦多菲尔兹研究所举行。该研讨会是菲尔兹研究所关于卡拉比-丘品种专题项目的一部分。美国国家科学基金会的支持将补充现有的资金，并将用于支付在美国的参与者的一些旅行费用。菲尔兹研究所是一个重要的国际研究中心，在组织类似活动方面有着丰富的经验。研讨会的重点是在许多不同的背景下模形式的出现：在卡拉比-丘流形的物理学中，例如为某些物理量生成函数；在Calabi-Yau流形的算术中，作为一般朗兰兹哲学的一部分，研究在数域上定义的Calabi-Yau变种的伽罗瓦表示；在Calabi-Yau变量的几何不变量结构中，如Donaldson-Thomas不变量。所有这些都是非常活跃的研究课题，研讨会旨在综合这些观点带来的不同观点。Calabi-Yau流形在数学和物理的交叉领域起着至关重要的作用。虽然它们最初是作为几何中的重要对象出现的，但卡拉比-丘流形在弦理论中自然出现后才真正引人注目。弦理论用一个在时空中运动的小环弦取代了传统的点粒子概念。为了使弦理论与量子力学兼容，时空必须是十维的。由于时空看起来是四维的，人们期望其中的六个维度是一个非常小的“卷曲”的几何物体。这些对象是卡拉比-丘流形。他们对物理学的介绍导致了大量新的数学。Calabi-Yau流形的一个重要性质是它们都具有非常微妙的平坦性（满足所谓的里奇平坦性），并且它们可以用多项式方程组来定义。因此，它们既可以被视为代数几何对象，也可以被视为算术对象，如果多项式方程在数字域中具有系数，则可以被视为算术对象。在卡拉比-丘流形的物理、代数-几何和算术特征之间已经发现了密切的联系。本次研讨会的目的是探索这些关系，在这三种不同观点的研究人员之间建立新的联系。该奖项由代数和数论以及几何分析项目共同资助。

This proposal will help support the workshop on "Modular forms around string theory", scheduled for September 16-20 2013 at the Fields Institute in Toronto, Canada. The workshop is part of the thematic program on Calabi-Yau varieties at the Fields Institute. The NSF support will complement the existing funding and will be used to cover some of the travel expenses of the participants based in the United States. The Fields Institute is a major international research center with extensive experience in organizing similar events. The workshop in question focuses on the appearance of modular forms in many different contexts: in the physics of Calabi-Yau manifolds, such as generating functions for certain physical quantities; in the arithmetic of Calabi-Yau manifolds, as part of the general Langlands philosophy in the study of Galois representations of Calabi-Yau varieties defined over number fields; in the structure of geometric invariants on Calabi-Yau varieties, such as Donaldson-Thomas invariants. All of these represent extremely active topics of research, and the workshop aims to synthesize the different points of view brought by each of these perspectives.Calabi-Yau manifolds play a vital role at the crossroads of mathematics and physics. While they first arose as important objects in geometry, Calabi-Yau manifolds really came to prominence after they appeared naturally in string theory. String theory replaces the traditional notion of the point particle with a small loop of string, moving through space-time. To make string theory compatible with quantum mechanics, space-time must be ten-dimensional. Since space-time appears four-dimensional, one expects six of these dimensions to be a very small "curled up" geometric object. These objects are Calabi-Yau manifolds. Their introduction into physics led to a great deal of new mathematics. A crucial property of Calabi-Yau manifolds is that they both have very delicate flatness properties (satisfying so-called Ricci flatness) and they can be defined using systems of polynomials equations. As a result, they can be viewed as both algebro-geometric and arithmetic objects, the latter if the polynomial equations have coefficients in a number field. Intimate connections have been found between the physical, algebro-geometric and artihmetic features of Calabi-Yau manifolds. The goal of this workshop is to explore these relationships, making new connections between researchers with these three different perspectives.This award is co-funded by the Algebra and Number Theory and the Geometric Analysis programs.