Singular structures in Frobenius and tt* geometries

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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.