British One and Two-pound Coins
The following story came from a talk entitled, Your Humble Servant, Is. Newton. This talk highlighted the interesting and unique pieces of correspondence between Isaac Newton and his peers in the scientific community.
A 1676 letter from Newton to Robert Hooke contained this quote: “If I have seen further it is by standing on ye shoulders of Giants.” The quote is famous enough to have been milled into the edge of the British two-pound coin. The above letter would certainly not be out of place in a British museum, or at The University of Cambridge where Newton studied. Instead, it resides at the Historical Society of Pennsylvania in the Simon Gratz Collection.
Simon Gratz was a Nineteenth Century Philadelphia lawyer, descended from a prominent family with roots in Colonial Philadelphia. He bought and sold autographs and letters from around the world; his collection contains pieces from the late fourteenth century through the nineteenth century, including the famous letter from Sir Isaac Newton. What a quintessentially American story!
This semester at school I am taking Functional Analysis. In Functional analysis, one studies functionals on spaces of functions, as in Real Analysis one studies functions on the space of real numbers. The first class was an opportunity to see some material with which I was already familiar and to reinforce concepts I had learned in a previous class.
Our first topic of study was metric spaces. A metric generalizes the concept of distance between two elements in a space. Pairs of us students were given a space, and we were to propose a metric for it. Our given space was the space of step functions (each element of the space is a step function); our idea was to partition the space into intervals where each step function has a constant value. Then we can find the area of the rectangle between the functions in each interval. Finally, we can add the areas together to get our proposed metric. This seemed correct, but I needed a written description in mathematical language of what we had found to convince me that it was indeed correct.
I realized that I knew how to write a step function: . Writing a second step function in this notation, but using F as the set variable and j as the index, and using the concept of measure (), our metric can be written as follows: . Voila!
One pair of students was given the space of sequences with the condition that for each sequence, the sum of the squares of the terms converged. A sequence in this space is where . This space is called . This was familiar to me as I had studied , the space of Lebesgue integrable functions with 2-norm, in a previous class. It turns out that and are closely related: a relationship that we will explore later in the class.
This has been my journey in mathematics: to learn new concepts and connect them to old concepts. The opportunity to recall step functions, the concept of measure, and the spaces of Lebesgue integrable functions and to apply these concepts to solving a new, albeit contrived, problem made for a fun first class.
John von Neumann, a major contributor in the field of mathematics, once told one of his students, “…in mathematics you don’t understand things. You just get used to them.” This is sometimes how I’ve felt during my first two years of advanced education in Mathematics. I also think I have gained an understanding of some advanced topics in Real Analysis, Abstract Algebra and Algebraic Number Theory, and Measure Theory, thus increasing by a little bit my understanding of Mathematics. But certainly I have had to get used to the above topics and mathematical concepts before I could fully understand and appreciate them.
I’ve managed to get through four sections (out of 12 or 13) of my summer book, . I’ve gotten slowly used to the concepts presented in these first four sections: quadratic forms, the form class group, and genus theory. I also have come to some understanding of these concepts. But I’ve also hit a wall in Section Five.
The main topic introduced in Section Five is class field theory. However, in the presentation of this topic, the reader is assumed to be familiar with – or used to – Galois Theory. Apparently, Galois Theory was developed to find solutions of polynomials of degree three or higher. I have enough background to understand the basic definitions of Galois Theory, but I will need some time to get used to Galois Theory. My summer book also came due at the library, so I think mine was a good stopping point. Thus my summer project has given birth to a side project.
My summer reading, , follows the mathematical discoveries and subsequent works of Joseph Lagrange, Leonhard Euler, Adrien Legendre, and Carl Gauss. My goal in mathematics is to make the kinds of significant discoveries that these mathematicians made.
Euler lived in the 18th century; he proved some theorems about primes of the form which had been conjectured in the 17th century by another famous mathematician, Pierre de Fermat. Lagrange, who succeeded Euler as the director of the Prussian Academy of Sciences, and Legendre both studied quadratic forms and their applications to Euler’s work. One of the applications of the theory of quadratic forms is to generalize Euler’s proofs. However, Legendre developed the theory in a much more modern and natural way. In addition, he made significant developments in the theory of quadratic reciprocity. Finally, Gauss synthesized all of the above ideas. He is widely considered to be the finest mathematician ever.
I am fascinated by who these famous mathematicians were, whereas during my engineering studies I was more interested in the how and why questions. For example, there is a famous result in algebraic number theory credited to Ernst Kummer: the number field generated by the 23rd root of unity does not have unique factorization while the number fields generated by the nth root of unity for , had all been found to have unique factorization. My question was, “how the heck was Kummer able to figure this out?” to which my professor responded, “…he was some kind of genius.”
I may not be a genius in the same class as Kummer and Gauss, but I wouldn’t be studying mathematics if I didn’t enjoy it or if I didn’t think I could be successful in it. I hope one day, I will be in a position to make a significant mathematical discovery on par with the heroes of mathematics.
This summer, I am working on two math projects outside of school. I am doing this because there is so much to learn in mathematics, and because it is fun!
The first project is a book: . One of my professors at CSU Channel Islands recommended it as a “lovely” book. I am reading through it and solving the problems at the end of each chapter. An exciting thing I learned: the group of reduced positive definite quadratic forms with a given discriminant, D, is isomorphic to a subgroup of the ideal class group of the quadratic number field with the same discriminant, D. This was a surprising result because it was not immediately clear to me how quadratic forms, the subject of chapter two of Primes…, were connected to quadratic number fields, a topic we learned in Algebraic Number Theory spring semester.
I am also developing ideas for a paper to be published in The College Mathematics Journal. The theme is the mathematics of planet Earth. I need to choose a topic soon; I am leaning towards an environmental topic like ecology or wind energy because I deal with those in my day job. Actually, I’ve developed some methods at work for modeling wind energy production. I hope to incorporate these in my paper.
The above two projects will keep me busy and out of trouble during the summer. Hopefully, I’ll learn some more new things. Feel free to share below what projects you have taken on this summer.