Singular structures in Frobenius and tt* geometries

这项研究有两个潜在的想法：什么是真实的，事情在哪里分解？复杂这个词指的是五千年前数学界的无知--复杂的物体往往比真实的物体更容易研究。从这个角度来看，人们应该问这样一个问题，一个给定的物体什么时候实际上是真实的？对于复数，二次方程总是有两个解--这正是复数最先被引入数学的原因。但这并不完全正确--有时这两种解决方案实际上是相同的。这是一个问题的例子--特殊方程只有一个解，而不是两个(或者更确切地说，是一个重复的解)。事物分解的地方，或者等价地，物体是单数的地方，就被称为判别式。另一个例子来自于观察万花筒。如果你把其中一面拆开，你会发现两面镜子彼此成60度角。这个角度很关键--轻轻移动它，你会得到一堆杂乱的图像，漂亮的对称图案就会消失。60(=180/3)是你得到漂亮图案的唯一角度吗？否：90(=180/2)可以，45(=180/4)也可以。事实上，形式180/N的任何角度，对于N=2，3，4，...会产生漂亮对称、万花筒的图案。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就是被称为Coxeter群的对象的例子。这些组描述了物体的对称性。现在想象万花筒里的一点。它在每个镜像中都有一个映像，并且每个映像都有更多的次映像，依此类推。当这些镜面之间的夹角取其中一个特定值时，人们只能得到有限数量的图像：柯克斯特群是有限反射群。我让你想象的原点是在万花筒里。如果该点实际上位于其中一个镜像上，会发生什么情况？通常一个点有一个反射--这就是我们所说的镜像。但镜子上的一个点是它自己的镜像--这个点和它的镜像重合。这样的点将仍然具有有限数量的多个图像(因为它将被反射到另一面镜子中)，但图像的数量将少于内部的普通点。这些镜像点对应于一个被称为判别式的对象。同一个词被使用两次并不是偶然的--这两个领域都有相同的数学基础。那么这两个问题用在哪里呢？取三个数字，并将它们相乘。答案不取决于您进行计算的顺序。乘法的过程是可交换的：AxB=BxA和联合(AxB)XC=Ax(BxC)。但数学家可以把其他东西加在一起，而不仅仅是数字。矢量的大小和方向就像一张覆盖着箭头的天气图，其中箭头的方向显示了风向，箭头的大小显示了风速。我们能否将这些向量相乘，使得乘法既是交换的又是结合的？这个问题的答案是肯定的，这导致了一个被称为弗罗贝尼乌斯流形的数学对象。出于各种原因，弗罗贝尼乌斯流形是令人惊叹的对象。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，弗罗贝尼乌斯流形是复杂的对象，因此人们可以问原始的问题：对象什么时候是真实的？问题出在哪里。Frobenius流形有巨大的对称性，但这些对称性往往是隐藏的，必须仔细提取，这个建议的另一个想法是研究它们的对称性，以及三个概念：现实、对称和奇异对象是如何相互作用的。

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.