Proofs from the Book

Paul Erdos used to remark that a particularly elegant of beautiful proof of a mathematical theorem was one for the "Book." This reference is to a hypothetical book Erdos figured God was keeping with only the very best proofs. Oh, if one could only be so lucky as to read this book! As things worked out two mathematicians took his book concept to heart and, after careful consultation with Erdos, decided to give it their best shot. The following link interviews the two authors and gives an interesting background to their book, "Proofs from the Book", 5th edition. As mentioned at the end off the interview there may be a sixth edition available at some point.


Proofs from the Book, authors interview.

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More to the point:

A conjecture both deep and profound
Is whether a circle is round.
In a paper of Erdős
Written in Kurdish
A counterexample is found.

Simon's Course in Modern Analysis

Professor Barry Simon of Caltech has a series of books on modern analysis. As a prelude to this set he has written an ~130 page 'companion guide' which is available free from the Am. Math. Society at this link. It's a good read for anyone irrespective of level of math skills. Please avail yourself of this free publication! Give yourself a Valentine's Day present.

Now, right out of the gate, Prof. Simon tells us:

"Analysis is the infinitesimal calculus writ large. Calculus as taught to most high school students and college freshmen is the subject as it existed about 1750—I’ve no doubt that Euler could have gotten a perfect score on the Calculus BC advanced placement exam. Even “rigorous” calculus courses that talk about ε-δ proofs and the intermediate value theorem only bring the subject up to about 1890 after the impact of Cauchy and Weierstrass on real variable calculus was felt.

"This volume [vol 1] can be thought of as the infinitesimal calculus of the twentieth century. From that point of view, the key chapters are Chapter 4, which covers measure theory—the consummate integral calculus—and the first part of Chapter 6 on distribution theory—the ultimate differential calculus.
"But from another point of view, this volume is about the triumph of abstraction. Abstraction is such a central part of modern mathematics that one forgets that it wasn’t until Frechet’s 1906 thesis that sets of points with no a priori underlying structure (not assumed points in or functions on Rn) are considered and given a structure a posteriori (Frechet first defined abstract metric spaces). And after its success in analysis, abstraction took over significant parts of algebra, geometry, topology, and logic."

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Books

"1. Summing It Up is the latest of the three, but it’s also the most elementary. It’s an introduction to the subject of modular forms, starting at the very beginning. The first half of the book covers in detail some basic ideas about number theory that can be understood in elementary terms, including things like the problems of counting the ways an integer can be a sum of squares or higher powers, or partitioned as a sum of smaller integers. The second half of the book tries to explain in as simple and concrete terms as possible what a “modular form” is, and what some of the properties of such objects are"

More here: Number theory

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