There is so much mathematical knowledge.
I like to delude myself into thinking that I am reasonably mathematically mature.
I have essentially completed a BS in mathematics (I am still finishing the degree, but all of my work left is for my other major, physics). I know all the standard differential and integral calculus, I can do a lot of ordinary differential equations on the spot, I understand some of the more common partial differential equations (Heat equation, Laplace's and Poisson's equations, Schroedinger's equation, the Wave equation). I know the standard introductory material of statistics, and a good deal of probability theory. I have a firm grasp on linear algebra and vector spaces, and even a very rudimentary understanding of tensor analysis. I am currently trying to get better at topology and complex analysis, as well as solidify my understanding of vector analysis and higher algebra.
But wait! Flo comes along on K5 talking about Atiyah-Singer, and I have never heard of this theorem. What am I, an idiot?
So I just started trying to get the gist of Geometry of Physics by Frankel, and this seems right up Frankel's alley. So I go to the index, sure enough, Atiyah-Singer is on page 465 of a 600 page book. A book which constitutes a year's worth of study of differential geometry at the graduate level. And Frankel has this to say about the theorem:
The Atiyah-Singer index theorem is a vast generalization of (17.27)[Gauss-Bonnet theorem] replacing the tangent bundle by other bundles ..., the Gauss curvature by higher-dimensional curvature forms ..., and replacing the Laplacian by other elliptic differential operators associated with the bundle in question. The Atiyah-Singer theorem must be considered a high point of geometrical analysis of the twentieth century, but it is far too complicated to be considered in this book. (Emphasis Frankel's).
My understanding of the concepts expressed is vague, but it seems clear that there is much more studying to be done (and this near the end of a graduate course on the subject!) to understand Atiyah-Singer.
At what point does the body of mathematical knowledge become so large as to make it very difficult to further? A BS in mathematics teaches one, almost without exception, math that is all hundreds of years old. And the average BS in mathematics has mathematical analytical skills that really dwarf what the common person has. Yet, said student is a couple hundred years behind the state of the art.
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