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Dr Tim Burness

Group theory and representation theory

My main area of research is in group theory. I am interested in simple groups, both finite and algebraic, with a particular focus on subgroup structure, conjugacy classes and representation theory.

I am also interested in permutation groups and related combinatorics, and in the application of probabilistic and computational methods.

PhD projects

Bases for permutation groups

If G is a permutation group on a set S then a subset B of S is a base for G if the pointwise stabiliser of B in G is trivial. The base size of G, denoted b(G), is the smallest size of a base for G. Bases have been widely studied since the early days of group theory in the nineteenth century, and they are used extensively in computational group theory. There are many possible projects in this area:

  • Investigate b(G) when G is a finite almost simple primitive group. For example, it is known that b(G) ≤ 7 if G is “non-standard”, and the proof uses probabilistic methods. It would be interesting to develop these methods to determine the exact base size for all non-standard groups. This will involve a detailed study of the subgroup structure and conjugacy classes in the almost simple groups of Lie type.
  • Study the finite primitive groups G with the extremal property b(G) = 2. For example, if G = V:H ≤ AGL(V) is an affine group, then b(G) = 2 if and only if the irreducible subgroup H ≤ GL(V) has a regular orbit on V, and determining the possibilities for H and V is a well-studied problem in representation theory.
  • Investigate bases and related base-measures for interesting families of infinite permutation groups.

Fixed point spaces and applications

In the study of group actions, there are many interesting problems concerning the fixed point sets of elements or subgroups. For example, if G is an algebraic group acting on an algebraic variety X then the set of fixed points of g ∈ G is a subvariety and we can study its dimension as we vary X and the element g. Further, we can use bounds on the dimension of these fixed point sets to estimate the proportion of fixed points of elements in a corresponding action of the finite group GF, which is the set of fixed points of a Frobenius morphism F of G. This interplay between algebraic and finite groups is a common theme in my research.

The case where G is a simple algebraic group is particularly interesting. Indeed, fixed point ratios for finite simple groups have been applied in a wide range of problems in recent years, e.g. base sizes, generation problems for finite groups, and the study of monodromy groups of compact connected Riemann surfaces.

An overview of my recent research activities (with references) can be found here:

If you are interested in any of the above projects, or if you would like to know more about my research, then please feel free to contact me by email.