Math 629: History of Mathematics
Frank Sottile

• Opening Remarks: The rest of our book treats more modern topics in mathematics, focussing on the pure side and ignoring analysis (I think this reflects Stillwell's proclivities). This week, we will read about the beginnings of modern algebra, first the rise of group theory and then hypercomplex numbers.
  
     I found the exercises in the chapter on group theory to be a bit heavy; do read the chapter to get a feeling for the development of that topic and its birth in the theory of equations. Despite what Stillwell says, groups and solving equations do keep coming up in mathematics and its applications. I ran a conference at the ICERM on Galois groups in geometry and applications of mathematics in September 2020. Stillwell is right to note that the classification of finite groups is no longer a central theme in mathematics.
  
      The chapter on hypercomplex numbers shows how even if you wish something to be so, it may not be be true. Hamilton and others wanted to find a three-dimensional system of numbers that behaved like the real numbers (one-dimensional) and also complex numbers (two-dimensional). Instead, he found a four-dimensional system, called the quaternions, which almost behaves like the others, except that it is noncommutative. It is hard to overstate the importance of this discovery. You all know that many operations, from multuplying matrices to dressing one's self, depend upon the order in which they are applied (socks first and shoes second is quite different from shoes first and socks second!). This noncommutativity ended up being crucial for the development of quantum mechanics, which enables, for example, a computer smaller than the size of large building with the capabilities of the laptop I am typing on, not to mention the screen I am looking at, and etc. Also, the unit quaternions form a multiplicative group which is equivalent to the group of rotations in R³. Consequently, every person working in computer animation uses them, and their multiplication is hard-wired into all graphics chips in computers and cell phones. My knowing this led to my collaborating with two applied mathematicians on a paper inspired by a problem in determining the structure of a certain class of proteins, and then to one of my more influential papers in 20-teens. The last sections go into overdrive where he talks about the octonians. Read them, but I did not assign exercises from them.
  
      This week is supposed to be a bit light, to give you more time to finish the draft of your term paper.
• Reading:
  1. Chapters 19 and 20 in Stillwell. Note that in Stillwell just after Exercise 20.5.3 there is a typo in the definition of the cross product. The last term is (u₁ v₂ - u₂ v₁)k.
• Assignment: due Tuesday 9 April 2024. (HW 12)
  Here is a .pdf and a LaTeX source of the assignment.
  No group assignment this week: Between finishing the draft of you;r term paper and the upcoming peer review, I want to go a bit lighter on you. To hand in: We are using Gradescope for homework submission.
  1. Do exercises 19.1.2, 19.1.3, 19.1.4, and 19.1.5. Each is a single-step deduction from the previous; I am hoping this is not too onerous.
  2. Also do exercise 19.3.6.
  3. In Chapter 20, do the exercises 20.5.1, 20.5.2, 20.5.3, and 20.5.4.

Last modified: Thu Apr 21 15:56:27 CDT 2022 by sottile