Commercial applications of maths
Mathematics has many applications across the whole range of intellectual endeavour; by way of illustration, we give here a rather eclectic sample of uses and users of mathematics that we happen to have heard about. Recent years have seen an explosion in the use of advanced mathematics in finance. Probability and statistical theory has long been used by actuaries to calculate what premiums should be charged for insurance; more recently the same mathematical ideas were used by Credit Suisse Financial Products in their Credit Risk+ software, which helps banks to assess the risks associated with making commercial loans. Advanced mathematics is also used in the pricing and management of financial instruments known as derivatives. The home page of author and consultant Paul Wilmott is an entertaining place to learn more about this area.
Another area of mathematics which has seen a lot of recent activity is the use of number theory in computer communication software. On the one hand, when sending images across the internet, one wants to try to speed things up by compressing the data as much as possible. On the other hand, if noise in the phone lines introduces errors in the transmission, one wants to be able to detect this and either correct it or ask for a retransmission of part of the data. All this can be achieved by various data encoding schemes, some of which involve the mathematics of fractals, and others the theory of matrices over finite fields. Another important problem arises when you want to buy something offered for sale over the internet and pay for it with a credit card. Typically you won't have spoken to the company before so you won't have agreed any kind of secret code with them. For safety, you need to assume that any eavesdropper will see and record absolutely all data exchanged between you and the company. You want to be able to send the company your credit card number, but you don't want to reveal it to any eavesdropper. This seems impossible, and will probably seem more and more impossible the more you think about it. Nonetheless, it is in fact possible; the techniques used (Diffie-Hellman key exchange and the Rivest-Shamir-Adleman algorithm) are both mathematically beautiful and of the utmost commercial importance. You can read more about these ideas at the RSA Data Security website.
Computer modelling of physical processes is another huge area of application of mathematics. For example, PowerGen's Power Technology Centre (just south of Nottingham) employs a number of mathematicians to do computational modelling of the flow of fluids and gasses in power stations. The Met. Office employs many mathematicians to do theoretical and numerical modelling both of day-to-day weather changes and of long-term issues such as global warming and the depletion of the ozone layer.
The Cambridge-based company D-Cubed sells software that does algebraic analysis of geometric problems that arise in computer-aided design, leading to faster and more accurate solutions than those obtained by more traditional numerical (as opposed to algebraic) approaches. Their web pages have some nice illustrations of the issues involved. A number of companies carry out research in pure mathematics in the hope of generating commercially valuable spin-offs in the long term. For example, Microsoft Research's theory group does extensive work in statistical physics, probability theory, combinatorics, geometry and topology, theoretical computer science, and algorithms. The group includes the eminent mathematician Michael Freedman, recipient of a Fields Medal in 1986.