Finitely presented groups at The City College of New York, Spring 2009 conference

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How can you tell whether a knot is really knotted or whether it is possible to find a formula for the solutions of certain equations or how to make senseof the nature of the elementary particles of physics or why there are only 17 really different wall paper patterns? These and many other such questions depend on an algebraic structure called a group. Groups arise in many parts of mathematics and in science. They provide, in particular, the means to understand and exploitsymmetry. This makes it possible for them to helpin our understanding of the nature of the space we live in, the structure of crystals and various molecules and in the design of new cryptographic protocols. Indeed the security of information transmitted electronically over the internet depends crucially on a particular kind of group called a finite group. This group provides a wayto encode messages using the product of two very large primenumbers. The encoding is the easy part of the process.The hard part is the decoding and this is where group theoryis used. The present proposal deals with the current state ofknowledge of certain groups, called finitely presented groups.These finitely presented groups hold in many instances the keysto solving important problems in mathematics and scienceand the primary objective of this conference is for a numberof very prominent mathematicians to map out future directions of study.