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Financial Mathematics 金融数学
Financial Mathematics is the mathematics of investment, risk, and uncertainty. It is all about deciding what to do today based on uncertain knowledge of the future. Most of life is about that, but financial math is a bit narrower in that it deals with decisions with purely financial implications. Examples of financial math questions are: How should financial risk be defined? How should a portfolio be selected to balance risk and return? How much is the option to trade a security at a preset price worth? If you own such an option, what is the best way to extract value from it? How should a portfolio of options be assembled to reduce the risk in one's business activities?
One might expect financial math to be related to economics, finance, and statistics and of course it is. It is perhaps more surprising that it is strongly linked with partial differential equations of classical applied mathematics as well! In fact, most options pricing problems can be formulated as partial differential or partial differential integral equations, often with moving boundaries. Both analytical and numerical techniques are used to solve these challenging and important problems.
Mathematical Biology 生物数学
Biological systems are incredibly complex, and our understanding of that complexity is increasing all the time. Fifty or a hundred year ago, when most biological systems were understood in far less detail, loosely defined mental models of \"how things work\" were enough to describe all we needed to describe in biology. Since then, our understanding of biology has sky-rocketed, and so has the need for sophisticated mathematical tools to understand and model this expanding \"new\" universe.
Scientific Computing 科学计算
Every science (and, alas, pseudoscience) is mathematized. All have their computational problems. One may wish to simulate Saturn's rings, a satellite trajectory, an electronic circuit, the vibration of an ice-laden power transmission line, the fuel consumption of a proposed design of automobile engine, an economic subsystem, the weather, the galaxy. (Or maybe just the Zodiac.) Constructing a mathematical model of your system is one thing (see the modelling section but once your mathematical equations are constructed, you have to solve them. That is, in order to produce tables of output, or more useful graphs and pictures, you have to be able to convert your problem into a sequence of good old arithmetic operations. This is scientific computing.
Once you have a scheme for doing so, you have to convert your scheme into something a computer can work with; this process is programming. Writing computer games is also programming, as is writing the web browser you're using to read this, but computers and programming were invented to solve scientific problems, and the Web and all its glory are incidental to the original purpose.
Chaos and Nonlinear Dynamics 混沌和非线性动力学
This is a \"hot\" topic, or at least it has been in the last ten years. We consider the reliability of scientific computing methods for chaotic dynamical systems, and applications of chaotic dynamics to assess the reliability of scientific computing methods to predict (or falsely predict) global warming. We look at ways of predicting when chaotic behaviour can occur (such as in oscillation of structures) so it can be avoided, and of applying chaotic behaviour to chemical and molecular systems when it produces useful results (such as in mixing of materials).
But what is \"chaos\"? In the previous century, linear mathematical models were used to predict the dynamical behaviour of objects; these linear models had solutions that one could write down explicitly, like exp(-0.1*t)*sin(t). You tell me the time t, and I can tell you what exp(-0.1*t)*sin(t) is---the answer is perfectly predictable, and if you only give me three places of accuracy in t then I can give you three places of accuracy in the answer (in the absolute error sense).
But for chaotic systems, the situation is very different. There are no formulas that anyone could write down for the solution (usually); instead we have computer programs such as
x = input
for i=1:100,
x = 3.9*x*(1-x)
end;
output x
for which incredibly tiny changes in the input produce drastic, chaotic, changes in the output. Though the system is perfectly deterministic, not random at all, the fact that our knowledge of the input is imperfect destroys the predictability of the outcome. |
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