Mathematical Sciences: Group Theory and Combinatorics

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This research is concerned with understanding the so-called Monster Group. There are three directions which the principal investigator will consider. One is to consider the relations between the group-characters of the Monster and certain classical modular forms. The second is the constructions of the group in 198684-dimension space. the third is the representation of the group in terms of hyperbolic reflections. This research is in the general area of finite group theory. It is concerned with studying a particular group called the Monster Group. This group is not well understood but has a very rich structure. At present calculations in the group are difficult, but whenever they have been performed successfully, wonderful and simple patterns have emerged. There are deep connections with other branches of mathematics and it seems quite likely that there are equally fundamental connections with theoretical physics.