Math 629: History of Mathematics
Frank Sottile

• Opening Remarks:
      Now that you have completed the draft of your term paper, it is now time to arrange peer feedback within your group. I will read each draft, make comments, and send that to you in the next week or so.
  
      The first chapter (21) in our reading concerns the development of Algebraic Number theory. I like this topic; having had an influential undergraduate course in this topic of number fields and their rings of integers that also contained a fair bit of history. This is a very beautiful topic concerning generalizations of the integers. These 'rings of integers' were studied originally as part of attempts to prove Fermat's Last Theorem. If unique factorization were to hold in them, that approach would have proven Fermat's Last Theorem. This program was due in part to Sophie Germain, a pioneering woman in mathematics. The truth being what it is, we had to move along and adapt to the lack of unique factorization. That the theory of ideals was able to restore unique factorization led to a richer and more abstract connection between algebra and geometry. A modern perspective is to create a 'space' whose points are the ideals (actually the prime ideals). This important theme in modern mathematics had its start in the 19th century study of these rings of integers.
  
      Chapter 22 is on Topology. This culminates in a discussion of one of the great recent results in mathematics, the proof of the Poincaré conjecture by Perleman, who along the way proved Thurson's geometrization conjecture. There were at least four Fields medals related to the settling of the Poincaré conjecture. As part of Section 22.3, read up on the description of the real projective plane in Tin Can Topology. You might also think about your Klein bottles, too.
  
      I ask you to read the story of groups in Chapter 23. It is a bit technical, but was a major focus of algebra for about 30 years, and the story of the finite simple groups is one of the more accessible parts of modern mathematics (In contrast: gerbes anyone? or how about geometry over the field with one element? or Drinfeld's chtoucas?). I hope that you appreciate the story of Conway's discovery of his groups, and the relation of these simple groups to exceptional structures like the Golay code or the Leech Lattice (a mathematical news story in the past couple of years concerns sphere packing and the Leech lattice). Finally, the story of the Monster simple group, as well as the Moonshine conjecture (I understand that Stillwell gets the provenance of moonshine wrong here) is quite interesting. It is worth noting that John Thompson and Borcherds both won Fields medals for work in this area. A parting note is that John Horton Conway was an early victum of the coronavirus in early 2020.
  
  
• Reading:
  Chapters 21, 22, 23 of Stillwell.
• Assignment: Due Monday, 22 April. (HW 13)
  Here is a .pdf and a LaTeX source of the assignment.
  To hand in: We are using Gradescope for homework submission.
  1. In Chapter 21, do the exercises 21.2.1, 21.2.2, and 21.7.1.
  2. In Chapter 22, do the exercises 22.2.2, 22.2.3, 22.2.4, 22.2.5 and 22.2.6, the topological proof that there are five Platonic solids.

Last modified: Fri Apr 22 14:58:59 CDT 2022 by sottile