Counting curves with symmetry

枚举代数几何，又称。“曲线计数”在过去的30年里一直是代数几何的主要部分。它指的是通过计算曲线（有时是更一般的对象）可以位于空间内的方式的数量而获得的空间的微妙不变量。这个主题起源于理论物理学（弦理论）的思想，在那里曲线可以被认为是弦的世界片。对于任何给定的空间，都有无限多的曲线计数不变量，并且数字的集合被编码到一个称为空间的分区函数的单个函数中。计算空间的配分函数（特别是对于某种称为紧致卡-丘三重的六维空间），是代数几何和物理学的主要目标。这个项目的目标是丰富我们对具有额外对称性（包括某些称为“导出”对称性的“隐藏”对称性）的空间的曲线计数不变量的理解。广义地说，人们可以计算在对称性下不变的曲线。这将导致新的曲线计数不变量，这反过来又会让我们更好地理解原始不变量。初步调查表明，对于某些空间，与这些新的曲线计数理论相关的配分函数将由模形式给出-数论中出现的非凡函数，已经研究了一百多年。这在三个非常不同的学科之间提供了强有力的联系：几何学，物理学和数论。在这个主题中的圣杯之一是获得一个完整的和明确的公式，任何紧凑的卡-丘三重的配分函数。这个项目的一个目标就是要做到这一点：找到一个明确的公式，为配分函数，在模形式方面，一个非常特殊的卡-丘三重，即舍恩流形。这个想法是，这个空间有大量的对称性，包括上面提到的许多“隐藏”对称性。这一点，沿着新的几何思想的介绍，应该使我们能够给一个完整的计算，这个分区功能，并表示它的模块形式。还没有人能够计算出紧致卡-丘三重体的配分函数（弦理论中最相关的几何）。拥有这样一个函数将使我们对相应的物理理论有很好的了解，从而对几何学和物理学产生重大影响。

Enumerative algebraic geometry, a.k.a. "curve counting", has been a major part of algebraic geometry for the last 30 years. It refers to subtle invariants of spaces obtained by counting the number of ways that curves (or sometimes more general objects) can sit inside a space. The subject has its origin in ideas coming from theoretical physics (string theory) where the curves can be thought of as the world-sheets of strings. There is an infinite number of curve counting invariants for any given space, and collection of numbers is encoded into a single function called the partition function of the space. Computing the partition function of a space (especially for a certain kind of six dimensional space called a compact Calabi-Yau threefold), is a major goal of both algebraic geometry and physics. The goal of this project is to enrich our understanding of curve counting invariants for spaces which have extra symmetries (including certain "hidden" symmetries called "derived" symmetries). Broadly speaking one can count curves which are invariant under the symmetry. This will lead to new curve counting invariants, which will in turn give us greater understanding of the original invariants. Preliminary investigations suggest that for certain spaces, the partition functions associated to these new curve counting theories will be given by modular forms --- extraordinary functions that arise in number theory and have been studied for well over a hundred years. This provides a powerful link between three very different subjects: geometry, physics, and number theory. One of the holy grails in this subject is to obtain a complete and explicit formula for the partition function of any compact Calabi-Yau threefold. One aim of this project is to do exactly that: find an explicit formula for the partition function, in terms of modular forms, of a very special Calabi-Yau threefold, namely the Schoen manifold. The idea is that this space has an enormous number of symmetries, including many of the "hidden" symmetries alluded to above. This, along with new geometric ideas introduced, should allow us to give a complete computation of this partition function and express it in terms of modular forms. No one has every been able to compute the partition function of a compact Calabi-Yau threefold (the geometry most relevant in string theory). Having such a function would give us great insight into the corresponding physical theory and thus would have a significant impact in both geometry and physics.