Math 629: History of Mathematics
Frank Sottile

• Opening Remarks: The theme this week is the the beginning of modern algebra and the role of mechnics from physics on mathematics, as an algebraist, physics major, and now applied mathemeician, I appreciate these topics. It starts with the revival of elementary number theory, particularly in the work of Fermat. Then we read the story of the start of more advanced number theory in the theory of elliptic integrals and their transformations–this is for background when we discuss complex elliptic curves. While it is slightly out of place, we will also read the chapter on mechanics (perhaps this would have been better to do when we read about the Calculus).
      It is worth noting that in the first half of the 19th century the study of elliptic integrals and elliptic functions was one of the main topics in mathematics. While terminology from that time persists, this is no longer an area of current research, although it informs much of modern mathematics, from algebraic geometry to the proof of Fermat's last Theorem (Wiles' Theorem) to the extraordinarily deep Langlands program that lies at the interface of number theory, analysis, and representation theory.
      The story of Fermat's Last Theorem was big news 20 some years ago. When Andrew Wiles announced his proof (at a conference in Cambridge) and released a manuscript, mathematicians all over the world started to study it for Wiles was a widely-known and highly respected mathematician—and not a crank (There is a continuous rain of papers purporting to prove famous results with elementary means, almost all of which have easily detected flaws; These are typically sent to famous mathematicians. When I was a student at Cambridge there was a bulletin board in the mathematics department common room devoted to letters from mathematical cranks, and I have even received several such manuscripts and letters.) After Wiles' announcement, some faculty at the University of Chicago set up a study group, where we read his paper and lectured on it and the background material for most of a year; this fell apart when Wiles retracted his paper, having found an error that he later corrected with the help of Richard Taylor.
      The chapter on mechanics covers a lot of history, from the late medieval period until 1900. This begins with relations between velocity, acceleration, and distance, and ends with chaotic motion and fluid mechanics.
• Reading:
  • Chapter 11 in Stillwell's book. This covers the revival of number theory and some great little gems, such as Fermat's little theorem (which is the basis for one of the first public-key cryptosystems), as well as his famous Last Theorem (Section 11.3).
  • Chapter 12 in Stillwell's book. Some of the topics in this book are more advanced (e.g. the transformations of elliptic integrals and how it leads to periodic functions with two independent periods), so I'd rather have you read about this than do many problems in this chapter.
  • Chapter 13 in Stillwell's book. Pages 262–263 are a bit cumbersome, and Fulling has a succinct and easy summary of d'Alembert's derivation of the wave equation.
• Assignment: Due Monday, 27 March 2017. (HW 10) To hand in: Email a .pdf to both Bennett Clayton bgclayton@math.tamu.edu and Frank Sottile. fjsteachmath@gmail.com
  1. Exercises 11.2.1, 11.2.2, and 11.2.3 in Stillwell. These give a refreshingly elementary proof of Fermat's Little theorem.
  2. Exercise 11.4.4. For this, recall the parametrization of Pythagorean triples. Use that to deduce Fermat's last Theorem in the case of n=4.
  3. Exercises 13.5.1 ad 13.5.2.
  4. Do the steps in Fulling's explanation of d'Alembert's derivation of the wave equation.