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Group Theory

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Group theory is the study of algebraic structures called groups, which arise naturally throughout mathematics. Groups encode the symmetries in a vast range of mathematical and physical systems, and group theory provides a powerful and unified language for studying these symmetries. Research in group theory at Bristol is focussed on finite and algebraic groups, and geometric group theory.

Permutation groups have been studied since the early days of group theory in the 19th century. In the last thirty years, the subject has been revolutionised by the Classification of Finite Simple Groups, and this has led to many interesting problems, and the development of powerful new techniques to solve them.

Problems concerning the subgroup structure, conjugacy classes and representation theory of simple groups are a focus of much research in this area. The interplay between finite simple groups of Lie type and simple algebraic groups is another common theme. Here the main tools come from Lie theory, representation theory, algebraic geometry and combinatorics. In addition, probabilistic and computational techniques play an important role.

In Geometric Group Theory, finitely generated infinite groups are studied using tools from geometry and topology. The subject has developed rapidly over the last thirty years, particularly inspired by ideas of M. Gromov, although it has roots going back to work of Dehn and others in the early 20th century.

The groups studied and tools used come from many different areas of mathematics, particularly low-dimensional topology, algebraic topology, combinatorial group theory, quasiconformal analysis and differential geometry.

The subject is often motivated by a number of rich examples such as Artin groups, Coxeter groups, mapping class groups, 3-manifold groups, lattices in Lie groups, nilpotent groups, polycyclic groups, solvable groups, and many others. A particular interest here are hyperbolic and relatively hyperbolic groups.

Group theory is an essential tool across mathematics. At Bristol, this is reflected in the links with other research groups in the school, including Representation theory, Ergodic theory and dynamical systems, and Combinatorics. There is also a very active Algebra and Geometry seminar.