Math 629: History of Mathematics
Frank Sottile

• Opening Remarks: I have not so much to say this week; it is busy here in Sweden, as a conference starts tomorrow at the Kunglia Technische Hochschule on "Tropical geometry and amoebas in higher dimension". There will be 60-100 mathematicians, and I give the first talk tomorrow morning, on "Describing Amoebas". While I finished marking your papers (I read each at least three times), I haven't yet figured out how to scan them and email you the scans. I am working on this.
  
      The first chapter concernes the development of Algebraic Number theory. 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. 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 facrtorization 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 has its start in the 19th century study of these rings of integers.
  
      For Chapter 22 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. 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 (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.
  
  
• Reading:
  Chapters 21, 22, 23 of Stillwell.
  Read Bell's chapters on Dedekind and Kummer, and on Poincaré
• Assignment: Due Monday, 23 April. (HW 15) Email a .pdf to Mehrzad Monzavi Mehrzad@math.tamu.edu.
  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, 22.2.4, 22.2.5 and 22.2.6, the topological proof that there are five Platonic solids.
  3. Write one or two paragraphs around the topic of the proof of the Poincaré Conjecture.