Applications of mathematics in research
Name: rick a cazzato
Date: Around 1995
As a mathematics teacher, one of the most common questions students ask is;
When will I ever use this? I can usually give them a adequate response.
However, I would like to hear your response to this question. I know that
the question is broad, but possible explain your research and how you are
using mathematics for real world applications. I guess what I am asking is
for you to describe your research and how mathematics is related or used
Well, I am a theoretical/computational chemist. I use mathematics every day
in my work, directly. . . mostly calculus, trigonometry, and statistics.
Chemists use trigonometry, together with spectroscopic devices that they
either invent or buy, to determine the structure of molecules. Without
trigonometry, we would understand very little about chemistry! Trigonometry
is also at the heart of a method used to analyze spectral signals coming
from molecules. This method is called "Fourier analysis." Fourier analysis
is ESSENTIAL for all kinds of spectroscopy and many of the new medical
imaging techniques (such as MRI).
We use statistics mostly to analyze the results of experiments, although I
also use them for numerical fitting of calculated numbers.
I use calculus to solve the Schrodinger equation, which is the equation
describing the motions of atoms, electrons, and nuclei in molecules. So, in
the end, mathematics is at the very heart of chemistry!
Well I guess there are different forms of Schrodinger's equation. (smile)
As a mathematician we "start" with an initial-boundary value problem and use
a combination of pde, linear algebra, numerical analysis and harmonic
analysis to determine an efficient solution. That these methods can be
"refined" keeps us in business as well as forcing us to sharpen our skills!
I should tell you that "we" (theoretical chemists) use any form of quantum
mechanics necessary to solve the problem at hand, ranging from HY = EY
through Feynman path integrals through Lie algebras, Fourier analysis and
all the tricks you mentioned.
Whatever it takes to get at molecular properties (including many Cray
hours!) is how we tend to tackle things.
As a theoretical physicist (now masquerading as a theoretical chemist)
almost everything topper mentioned in his list is on my list too. The most
important thing to understand however, is that for us trying to solve real
problems, the techniques we use can be drawn from just about any area of
mathematics, and so understanding of a broad range of mathematics is really
quite necessary. There is a certain sense in which what is known in
mathematics is "absolute truth", and can be relied on to help you just about
anywhere. It is not just physics and chemistry in fact - a lot of new PhD's
are now getting jobs on Wall Street using a wide range of mathematics
applied to the very complex problems of economic and financial prediction.
Mathematics provides a huge toolbox, and you need to practice with the tools
and familiarize yourself with them for a long time before you are really
qualified to apply them to the real world, but the applications are immense
once you have acquired that proficiency.
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Update: June 2012