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Motivic invariants and categorification

动机不变量和分类

基本信息

批准号：
EP/I033343/1
负责人：
Dominic Joyce
金额：
$ 236.96万
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2011
资助国家：
英国
起止时间：
2011 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FI033343%2F1
关键词：
Motivic invariants categorification

项目摘要

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.

该计划旨在发现几何学、代数学和理论物理学中的弦理论的新结构。从一些已经被很好理解的经典情境开始，我们的目标是把它概括成两个方向：我们可以把经典情境变成动机，或者我们可以把它分类。这些都是技术词汇，所以类比可能会有所帮助。我们已经理解的经典数学，就像墙上的二维阴影，是由一些三维物体投射出来的。我们的目标类似于理解这个三维物体，探索额外的第三维的含义，然后通过将阴影视为更复杂的三维物体的投影，看看我们可以发现关于阴影的新东西。在数学和物理学中，都有关于数学结构的维度的好概念-例如，在物理学中，n维场论是一种量子理论，其将从n维物体到某个时空的映射量子化。经典量子理论将粒子视为在时空中运动的点（0维物体），这是一种过度简化的理论，0维场论也是如此。弦理论把粒子看作是在时空中运动的弦的一维环，因此是一维场论;物理学的最新发展（M理论）考虑在时空中运动的高维膜。分类的思想是用问题中的（n+1）维结构代替n维数学结构，以某种系统的方式，这样当你把维度减1的时候，你就可以得到原来的n维结构了--就像从一个2维的阴影，到投射它的3维物体。在几何学中，不变量通常是一个数，它可以计数某类物体。但是因为我们要计数的对象的类通常是无限的，所以这种计数必须以一种复杂的方式进行。如果你以正确的方式计算对象，你的不变量可能会有一些特殊的属性-例如，当你变形底层空间时，它可能是不变的。这让数学家们兴奋不已，因为它表明不变量正在测量一些更深层次的结构，我们想知道这是什么。例如，镜像对称是一个来自物理学的圆形几何，它正在慢慢被证明。一个核心主张是不变量的惊人相等：空间X中的不变量计数曲线应该等于不同空间Y上的不变量计数曲线，因为X和Y的量子理论是相关的。从表面上看，这就像量子理论要求冈比亚的长颈鹿和赞比亚的斑马的数量相同一样奇怪，不变量是指计算空间中的点。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motivic，如果当你把空间分成两部分时，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motivic不变量。一些几何学中研究的不变量（例如，出现在弦理论中的Calabi-Yau 3-folds的Donaldson-Thomas不变量）使用欧拉特征来进行实际计数。人们可以尝试定义一个新的不变量，它计算相同的东西，但使用一些其他的motivic不变量，而不是欧拉特征。这就是我们所说的动机概括。新的不变量应该更丰富，具有更多的结构和信息。作为一个应用，我们希望能帮助物理学家更好地理解弦理论到底是什么。弦理论（最终形式）可能是宇宙的数学基础，几十年来一直是新数学的丰富来源，但其中大部分仍然是一个谜。