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The most important task of a faculty member at a university is research.
That statement is false at most liberal arts colleges. The people---students, their parents, state taxpayers---who provide the cash to run the university might disagree with the sentiments expressed in that statement. Nevertheless, it is a simple fact that the people within the university who decide on salaries and pay raises use research as their primary criterion. (Cynics have been known to say that it is their only criterion.)
Why is research given priority over teaching? The underlying philosophy is this: The purpose of a university is to increase knowledge. The university achieves this goal by having faculty members who discover new things about the world. In addition, they need to explain what they have found; that is, to teach others about it. Teaching is a necessary part of increasing knowledge, but it plays a role secondary to the actual discovery of new knowledge.
I am a pure mathematician. Loosely translated, that means that I study mathematics for its own sake, purely to increase our knowledge about things mathematical. I don't spend much time worrying about whether the things I discover have any immediate applications.
Some people (mathematicians among them) believe that there is a
branch of mathematics called "Applied Mathematics," separate from all
the other branches of mathematics. Personally, I think that applied
mathematicians are distinguished from pure mathematicians not by the
subject of their inquiries but by their attitudes toward them. There
are applied mathematicians who study analysis because they want to
make it a better tool for the study of molecular dynamics or fluid
flows. There are pure mathematicians who study analysis simply to get
a deeper understanding of analysis. There are pure mathematicians who
study number theory. There are applied mathematicians who study
number theory because they wnat to make it a better tool for the study
of cryptography. In a sense, the word "pure" in the phrase "pure
mathematics" means "unmixed", in the sense of "unmixed with interests
from outside mathematics."
My Area of Research
My principal area of research is arithmetic algebraic geometry. The problems that motivate my research come from the realm of arithmetic or number theory (as it is also known). In its most basic form, number theory is concerned with properties of the integers and rational numbers, with a particular interest in prime numbers. A wide class of problems in number theory are known as diophantine equations. The best known diophantine equation is Fermat's Last Theorem, which was proven after a 350 year wait by Andrew Wiles (with help from Richard Taylor, and building on work by Ken Ribet, Jean-Pierre Serre, and a host of others). The theorem states that there are no solutions in positive integers to the equation
xn + yn = zn
with n > 2. Polynomials like the one in this example are properly the subject matter of the branch of mathematics known as algebra. The techniques that I use in my research are drawn from the field of geometry, especially algebraic geometry, which concerns itself with the geometric properties of the solutions of polynomial equations. (A second set of techniques is drawn from a field called algebraic K-theory, which I am not about to try to explain on a web page. There are limits to everything.) To get a detailed picture of my research, you can consult my list of publications and preprints. If you can't wait for the nitty-gritty details, then you can jump directly to the abstracts of my papers.
For more information, you can consult the following preprint archives:
Or you can explore the following mathematical sites:[Work] [Bioinformatics] [Mathematics] [Life]
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