Singular structures in Frobenius and tt* geometries

Frobenius 和 tt* 几何中的奇异结构

• 批准号: EP/H019553/1
• 负责人: Ian Strachan
• 金额: $ 2.04万
• 依托单位: University of Glasgow
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2010
• 资助国家: 英国
• 起止时间: 2010 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FH019553%2F1
• 关键词: Singular; structures; Frobenius; tt; geometries

There are two underlying ideas in this research: What is real and where do things break down? The word complex refers to the ignorance of the mathematical community half a millennium ago - complex objects are often simpler to study than real objects. From this perspective one should ask the question when is a given object actually real? With complex numbers, quadratic equations always have 2 solutions - this is precisely why complex numbers were first introduced into mathematics. But this is not quite true - sometimes the 2 solutions are actually the same. This is an example of where do things break down - special equations have only one solution not two (or rather, a repeated solution). Where things break down, or equivalently, where objects are singular, is known as the discriminant.Another example comes from looking at kaleidoscopes. If you take one apart you will find two mirrors set at an angle of 60 degrees to each other. This angle is crucial - move it slightly and you will get a jumble of images and the nice symmetric pattern is lost. Is 60 (=180/3) the only angle for which you get a nice pattern? No: 90 (=180/2) will work, as will 45 (=180/4). In fact, any angle of the form 180/N, for N=2,3,4,... will produce is nice symmetric, kaleidoscopic pattern. The mathematics used to describe symmetries is called group theory , and the groups associated to these special angles are examples of objects called Coxeter Groups. These groups describe symmetries of objects.Now imagine one point inside the kaleidoscope. It will have an image in each mirror, and each image has yet more secondary images etc.. When the angle between these mirrors takes one of these special values one only gets a finite number of images: Coxeter groups are finite reflection group. The original point I asked you to imagine was inside the kaleidoscope. What happens if the point is actually on one of the mirrors? Normally a point has a reflection - this is what we mean by a mirror image. But a point on a mirror is its own mirror image - the point and its image coincide. Such a point will still have a finite number of multiple images (because it will be reflected in the other mirror), but the number of images will be less than for an ordinary point inside. These mirror points correspond to a an object known as a discriminant. It is not an accident that this same word has been used twice - the same mathematics underlies both areas. So where are these two questions to be applied?Take three numbers and multiple them together. The answer does not depend on the order you do the calculation. The process of multiplication is commutative: AxB=BxA and associative (AxB)xC=Ax(BxC). But mathematicians can multiple other things together than just numbers. Vectors have size and direction / think of a weather map covered with arrows where the direction of the arrow shows the direction of the wind and the size of the arrow the speed of the wind. Can one multiple such vectors together in such a way that the multiplication is both commutative and associative?The answer to this question is yes , and this leads to a mathematical object known as a Frobenius manifold. Frobenius manifolds are amazing objects, for all sorts of reasons. They sits at the crossroads of pure mathematics and applied mathematics, as well as mathematical and theoretical physics.Basics examples come from the study of kaleidoscopes, as described above. However, Frobenius manifolds are complex objects, and one can thus ask the original questions: when are objects real? and where do things break down . Frobenius manifolds have huge symmetries, but these are often hidden and have to be extracted carefully and another idea of this proposal is to study their symmetries, and how the trio of ideas: reality, symmetry and singular objects, interact.

在这项研究中有两个基本的想法：什么是真实的，以及事物在哪里分解？复杂这个词指的是半千年前数学界的无知--复杂的对象往往比真实的对象更容易研究。从这个角度来看，人们应该问这样一个问题：一个给定的对象什么时候才是真正的真实的？对于复数，二次方程总是有两个解-这正是为什么复数首先被引入数学。但这并不完全正确-有时两种解决方案实际上是相同的。这是一个例子，在那里做的事情打破-特殊方程只有一个解决方案，而不是两个（或更确切地说，一个重复的解决方案）。事物分解的地方，或者等价地，物体是单一的地方，被称为判别式。另一个例子来自于观察万花筒。如果你把一面镜子拆开，你会发现两面镜子成60度角。这个角度是至关重要的-稍微移动它，你会得到一个混乱的图像和漂亮的对称图案丢失。60 °（=180/3）是唯一能得到好图案的角度吗？否：90（=180/2）可以，45（=180/4）也可以。实际上，对于N= 2，3，4，.会产生对称的万花筒图案。用来描述对称性的数学称为群论，与这些特殊角度相关的群是称为考克斯特群的对象的例子。这些组描述了物体的对称性。现在想象万花筒中的一个点。它将在每个镜像中有一个映像，每个映像还有更多的次映像，等等。当这些反射镜之间的角度取这些特殊值之一时，只能得到有限数量的图像：Coxeter群是有限反射群。我让你们想象的最初的点是在万花筒里。如果点实际上在其中一面镜子上会发生什么？通常一个点有一个反射-这就是我们所说的镜像。但是镜子上的一个点是它自己的镜像--这个点和它的镜像重合。这样一个点仍然会有有限数量的多重镜像（因为它会被另一面镜子反射），但是镜像的数量会比内部的普通点少。这些镜像点对应于一个被称为判别式的对象。这个词被使用了两次并不是偶然的--这两个领域都有同样的数学基础。那么，这两个问题应该应用在哪里呢？取三个数字并将它们相乘。答案并不取决于你做计算的顺序。乘法的过程是可交换的：AxB=BxA和结合（AxB）xC=Ax（BxC）。但数学家可以把其他东西相乘，而不仅仅是数字。矢量有大小和方向/想想一张布满箭头的天气图，箭头的方向表示风的方向，箭头的大小表示风的速度。我们能不能把这样的向量相乘，使乘法既可交换又可结合？这个问题的答案是肯定的，这就引出了一个数学对象，称为弗罗贝纽斯流形。弗罗贝纽斯流形是令人惊奇的物体，因为各种各样的原因。它们位于纯数学和应用数学以及数学和理论物理的十字路口。基本的例子来自对万花筒的研究，如上所述。然而，弗罗贝纽斯流形是复杂的对象，因此人们可以问最初的问题：对象何时是真实的？以及在哪里发生故障。弗罗贝纽斯流形具有巨大的对称性，但这些对称性往往是隐藏的，必须仔细提取，这个提议的另一个想法是研究它们的对称性，以及三个想法：现实，对称性和奇异物体，如何相互作用。