Tag Archives: quadratic forms

Extracurricular Mathematics II: Electric Boogaloo

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, \underline{\textrm{Primes of the Form}\,x^2 + ny^2}.  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.

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Extracurricular 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: \underline{\textrm{Primes of the Form}\,x^2 + ny^2}.  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.

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