Motivic invariants and categorification

该提案旨在发现几何、代数和理论物理中的弦理论中的新结构。从一些已经被很好理解的经典情境开始，我们的目标是在两个方向上进行概括：我们可以使经典情境具有动机性，或者我们可以对其进行分类。这些都是专业术语，所以类比可能会有所帮助。我们已经理解的东西，经典数学，就像墙上一个二维的影子，是由某个三维物体投下的。我们的目标类似于理解这个三维物体，探索额外的三维空间的含义，然后通过将阴影视为一个更复杂的三维物体的投影，看看我们能从阴影中发现什么新东西。在数学和物理学中，数学结构的维度都有很好的概念——例如，在物理学中，一个n维场论是一个量子理论，它将n维物体的映射量子化到某个时空。经典量子理论过于简化，将粒子视为在时空中运动的点（0维物体），0维场论也是如此。弦理论认为粒子是在时空中运动的一维弦环，一维场论也是如此；物理学的最新发展（m理论）认为高维膜在时空中运动。分类的思想是以某种系统的方式，用（n+1）维的结构来代替一个问题中的n维数学结构，这样当你减少1维时，你就能重新得到原来的n维结构——就像从一个2维的阴影，到投射它的3维物体。在几何学中，一个不变量通常是一个数，用来计算某一类物体。但由于我们想要计数的对象类别通常是无限的，所以计数必须以一种复杂的方式完成。如果以正确的方式计算对象，则不变量可能具有一些特殊属性——例如，当您变形底层空间时，它可能保持不变。这种东西让数学家们兴奋不已，因为它表明不变量测量的是更深层次的底层结构，我们想知道它是什么。例如，镜像对称是来自物理学的一系列猜想，这些猜想正在慢慢得到证明。其中一个中心主张是一个惊人的不变量等式：在空间X中计算曲线的不变量应该等于在不同空间Y中计算其他东西的不变量，因为X和Y的量子理论是相关的。从表面上看，这就像说量子理论要求冈比亚长颈鹿的数量和赞比亚斑马的数量相同一样奇怪。不变量是对空间中的点进行计数的东西。它可以是一个数字（整数），或者更一般的东西。空间的不变量是有动机的，当你把空间切成两部分时，这个不变量是这两部分不变量的和。最基本的是欧拉特征，但也有许多其他更复杂的动机不变量。几何中研究的一些不变量（例如，出现在弦理论中的Calabi-Yau 3-fold的Donaldson-Thomas不变量）使用欧拉特征来进行实际计数。我们可以尝试定义一个新的不变量它计算相同的东西，但是使用一些其他的动机不变量而不是欧拉特征。这就是我们所说的动机泛化。新的不变量应该更丰富，具有更多的结构和信息。它们也可能使新事物成为可能。作为一个应用，我们希望帮助物理学家更多地了解弦理论到底是什么。弦理论（最终形式）可能是宇宙背后的数学，几十年来一直是新数学的丰富来源，但它的大部分仍然是一个谜。

The proposal aims to discover new structures in geometry, and algebra, and string theory in theoretical physics. Beginning with some classical situation which is already well understood, we aim to generalize it in two directions: we can make the classical situation motivic , or we can categorify it.These are technical words, so an analogy may help. The thing we already understand, the classical mathematics, is like a 2-dimensional shadow on the wall, cast by some 3-dimensional object. Our goals are analogous to understanding this 3-dimensional object, exploring the implications of the extra third dimension, and then seeing what new things we can find out about the shadow by viewing it as the projection of a more complex 3-dimensional object.In both mathematics and physics, there are good notions of the dimension of a mathematical structure - for instance, in physics an n-dimensional field theory is a quantum theory which quantizes maps from n-dimensional objects into some space-time. Oversimplifying rather, classical quantum theory regards particles as points (0-dimensional objects) moving in space-time, so is a 0-dimensional field theory. String theory regards particles as 1-dimensional loops of string moving in space-time, so is a 1-dimensional field theory; more recent developments in physics (M-theory) consider higher dimensional membranes moving in space-time.The idea of categorification is to replace n-dimensional mathematical structures by (n+1)-dimensional structures in a problem, in some systematic way, so that you get the original n-dimensional structure back again when you reduce dimension by one - like passing from a 2-dimensional shadow, to the 3-dimensional object that casts it.In geometry, an invariant is usually a number which counts some class of objects. But because the classes of objects we want to count are usually infinite, this counting has to be done in a complicated way. If you count the objects in just the right way, your invariant may turn out to have some special properties - for instance, it may be unchanged when you deform the underlying space. This kind of thing makes mathematicians excited, as it suggests the invariant is measuring some deeper underlying structure, and we want to know what this is. For example, mirror symmetry is a circle of conjectures coming from physics, which are slowly being proved. One central claim is a surprising equality of invariants: invariants counting curves in a space X should be equal to invariants counting something else on a different space Y, because the quantum theories of X and Y are related. On the face of it, this is as bizarre as saying that quantum theory requires the numbers of giraffes in the Gambia, and of zebras in Zambia, to be the same.An invariant is something which counts the points in a space. It could be a number (integer), or something more general. An invariant of spaces is motivic if, when you cut the space into two pieces, the invariant is the sum of the invariants of the pieces. The most basic is the Euler characteristic , but there are also many other more complicated motivic invariants.Some of the invariants studied in geometry (for instance, Donaldson-Thomas invariants of Calabi-Yau 3-folds, which appear in string theory) use Euler characteristics to do the actual counting. One can try to define a new invariant which counts the same things, but using some other motivic invariant instead of Euler characteristics. This is what we mean by a motivic generalization. The new invariants should be richer, with more structure and information. They may also make new things possible.As one application, we hope to help physicists understand a bit more about what string theory actually is. String theory (in its final form) may be the mathematics underlying the universe, and has been a fertile source of new mathematics for decades, but much of it is still a mystery.