For a fun intro to modern math I highly recommend: https://www.amazon.com/Love-Math-Heart-Hidden-Reality/dp/046....

This paper touches on a lot of the aspects in that book. With it and some undergrad math one can appreciate the brilliance of the paper.

Also a good reading from earlier from the same author: https://archive.is/qk5GC

Shapes just come about when you are talking about objects with certain types of symmetries. One of the main "attacks" on polynomials is to realize that their roots are related to eachother, in terms of total, product, difference, etc.

The simplest of these methods is just Vieta's formulas:

https://en.wikipedia.org/wiki/Vieta%27s_formulas

I highly recommend Edward Frenkel's book, Love and Math. It was a great read about the mathematics itself, and also his troubles as a Jewish academic in Soviet Russia.

Love and Math: https://amzn.to/2qcqUb4

My memory isn't as good as it used to be. I thought the article was about Edward Frenkel at first who wrote the fantastic novel Love and Math [1]. It's about his life growing up in Soviet Russia as a Jewish mathematician. He was good friends with the famed physicist Murray Gell-Mann who discovered a theory of hadrons, leading to the quark model we have today [2]. In the novel he talks about some of the groundbreaking work he has done. But he's mostly known for his cinema work on the emotional fullness of mathematics and physics. I would nominate him for a Nobel if I could. But Fields medals are about pure mathematical achievement, right? So I'm not sure he would qualify.

[1] http://amzn.to/2HKD3Mc [2] https://en.wikipedia.org/wiki/Eightfold_Way_(physics)

A similar story[1] was told by the mathematician Edward Frenkel. He was a Jew in Russia and took an exam with "hard" math problems designed to prevent his admission into Moscow State University. It's possible the problems shown in the paper are the ones Frenkel saw since he was a teenager during that time period. (The Cornell paper does not specifically mention Frenkel.)

[1]http://www.amazon.com/Love-Math-Heart-Hidden-Reality/dp/0465...

Love and Math: The Heart of Hidden Reality - by Edward Frenkel http://www.amazon.com/Love-Math-Heart-Hidden-Reality/dp/0465...

Such a great book for people who love math. The end of the book gets a little bit hairy with more complex subjects but it is a great story from Mr Frenkel of going from school to working on the Langlands Project - toted as the rosetta stone for math. Defiantly a must read for anyone who wants to get into mathematics as a career

My wife gave me this great book for Christmas:

  Love and Math: The Heart of Hidden Reality
  by Edward Frenkel

It begins with the author's struggle to learn the math behind quantum physics in spite of cold-war era soviet educational obstacles and leads bit by bit into the Langlands program, drawing connections between group theory, number theory and harmonic analysis.

It's definitely my favorite book of 2013.

http://www.amazon.com/Love-Math-Heart-Hidden-Reality/dp/0465...

<blockquote>The statement of the Shimura-Taniyama-Weil conjecture must have sounded crazy to its creators. . . . the idea that this was true. . . must have sounded totally outrageous at the time. This was a leap of faith, in the form of a question that [Taniyama] posed at the International Symposium on Algebraic Number Theory held in Tokyo in September 1955.

I've always wondered: what did it take for him to come to <em>believe</em> that this wasn't crazy, but real? To have the courage to say it publicly?

We'll never know. Unfortunately, not long after his great discovery, in November 1958, Taniyama committed suicide. He was only thirty-one. To add to the tragedy, shortly afterward the woman whom he was planning to marry also took her life, leaving the following note:

<blockquote>We promised each other that no matter where we went, we would never be separated. Now that he is gone, I must go too in order to join him.</blockquote>

. . . In his thoughtful essay about Tayniyama, Shimura made this striking comment:

<blockquote>Though he was by no means a sloppy type, he was gifted with the special capability of making many mistakes, mostly in the right direction. I envied him for this, and tried in vain to imitate him, but found it quite difficult to make good mistakes. (94) </blockquote></blockquote>

—Edward Frenkel, <a href="http://www.amazon.com/Love-Math-Heart-Hidden-Reality/dp/0465... and Math: The Heart of Hidden Reality</em></a>, which is recommended.

What mistakes have you made lately?