I spent four years studying mathematics, at the University of Missouri - Columbia, under Dr. Calin Chindris. I came to graduate school knowing nothing about representation theory. During my second year, I got and introduction to Classical Invariant Theory from Calin, and an introduction to linear representations of finite groups through an expository paper I wrote for a number theory course. I focused on various aspects of representation theory for the remainder of my graduate studies. Beyond the research business, there were a number of other growth experiences described below.
Teaching: Graduate students come to graduate school with all kinds of motives, goals, and desires. I spent two years working an after school program, at a Catholic elementary school. During the summer, I had worked summer programs, through Johns Hopkins, Center for Talented Youth (CTY). Hence, I came to graduate school having worked with young students. I developed an interest, in teaching, and wanted to develop that skill. Little did I know that a number of my colleagues would view teaching as a punsihment.
There were difficulties adjusting to teaching in the collegiate environment. I found that students brought there own expectations and baggagae into the classroom. While I understood this would be the case abstractly, I got a quick education in supporting students, making expectations clear, and helping students work through there own frustrations. What I came to find, was that there are common themes that run through all teaching experiences no matter what the age group or the situation. At least three of these are:
I had the opportunity to discuss the craft, of teaching, with one of the
course coordinators, and a couple of professors. Having individuals
willing to mentor me, in the classroom, and support me while I navigated
new encounters was important. Those interactions and discussions helped
me develop as a teacher, a person, and discover my inner love for helping
others discover their own mathematical story. It was through one of these
discussions that I first said: School should not be the place where creativity comes to die.
Research: Ah, the research business! The excitement comes from
engaging a brand new puzzle, and trying to put it together. While
researchers are often trying to answer questions that no one has the answer to,
it can be equally interesting to open closed problems, and come to an
understanding of the techniques and ideas used to solve said problems.
The people that are around you and discussing results, techniques, and
ideas, with you, are as important as the inquiry itself. As a professor
once said to me, The best situation is when you're around people who
ask you interesting questions.
Research feeds the mind. From an outside perspective, it may seem like mathematicians are studying a random set of questions. This is hardly ever the case. Problems don't arise in a vacuum. They often arise in one of two contexts:
Interdisciplinary Inquiry: When I began, as an undergraduate computer engineering student, all I heard about was the need to work collaboratively and interdisciplinary. There is a great joy, when one gets the opportunity to bring techinques and ideas, from different areas of mathematics, down on a problem. What I came to discover, was that this was not a skill that academia valued. University Deans generally wanted to be able to say that a faculty member is an expert in a particular field, and has written x number of papers, in that area. This was but an instance of what frustrated me about the politics in the academy.
As a mathematician, I want to be thinking about interesting questions. I personally don't care if those questions have a direct application to a "real world" application. I have a deep interest in cryptography, which may be one of the most interdisciplinary areas: history, mathematics, politics, computer science, and engineering. I later developed an interest in bioinformatics and hope to have some time to further explore this area.
I came to find that I found great enjoyment engaging the various areas of mathematics. I feel like I often gained insight exploring ideas, such as group theory, that appear in areas like algebraic topology and differential geometry. Seeing how things are the same, i.e. using ideas I've already been exposed to, and different, that is how an algebraic structure can expose insight into the geometry, at hand. This kind of idea lead me to request an independent study in Morse theory, where I came to a new kind of understanding about differential geometry.