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# Dr Tim Burness

## Dr Tim Burness

BSc(Warw.), MSc(Warw.), PhD(Lond.)

Reader in Pure Mathematics
### Area of research

Group theory and representation theory
## Summary

## PhD projects

### Bases for permutation groups

### Fixed point spaces and applications

## Biography

## Teaching

## Memberships

### Organisations

### Pure Mathematics

### Research themes

## Links

## Selected publications

## Recent publications

- All details
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- Publications

BSc(Warw.), MSc(Warw.), PhD(Lond.)

Office 1.26

Fry Building,

Woodland Road,
BS8 1UG

(See a map)

+44 (0) 117 42 84902

t.burness@bristol.ac.uk

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.

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.

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 *G ^{F}*, which is the set of fixed points of a Frobenius morphism

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:

http://seis.bristol.ac.uk/~tb13602/research.html

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.

Senior Lecturer in Pure Mathematics, University of Bristol (since August 2014)

Lecturer in Pure Mathematics, University of Bristol

Lecturer in Pure Mathematics, University of Southampton

Lady Davis Postdoctoral Fellow, Hebrew University of Jerusalem

Junior Research Fellow, St John's College, University of Oxford

PhD Mathematics, Imperial College London (2005)

MSc Mathematics, University of Warwick (2001)

BSc Mathematics, University of Warwick (2000)

At Bristol, I teach the 3rd year course *Group Theory, *I co-teach a 4th year course in *Representation Theory,* and I give tutorials in Pure Mathematics to first year undergraduates.

I also supervise 3rd and 4th year undergraduate projects, and I currently have two PhD students (Elisa Covato and Scott Harper).

- Burness, T & Giudici, M, 2016, ‘Classical groups, derangements and primes’. Cambridge University Press
- Burness, TC, Ghandour, S & Testerman, DM, 2016, ‘Irreducible geometric subgroups of classical algebraic groups’.
*Memoirs of the American Mathematical Society*, vol 239. - Burness, TC, Ghandour, S, Marion, C & Testerman, DM, 2015, ‘Irreducible almost simple subgroups of classical algebraic groups’.
*Memoirs of the American Mathematical Society*, vol 236. - Burness, TC, Liebeck, MW & Shalev, A, 2013, ‘Generation and random generation: From simple groups to maximal subgroups’.
*Advances in Mathematics*, vol 248., pp. 59-95 - Burness, TC, Liebeck, MW & Shalev, A, 2009, ‘Base sizes for simple groups and a conjecture of Cameron’.
*Proceedings of the London Mathematical Society*, vol 98., pp. 116-162

- Burness, TC, Garonzi, M & Lucchini, A, 2019, ‘On the minimal dimension of a finite simple group’.
*Journal of Combinatorial Theory, Series A*. - Burness, TC & Harper, S, 2019, ‘Finite groups, 2-generation and the uniform domination number’.
*Israel Journal of Mathematics*. - Burness, T, 2019, ‘Simple groups, generation and probabilistic methods’. in:
*Groups St Andrews 2017 in Birmingham, 5th-13th August 2017: London Mathematical Society Lecture Note Series*. Cambridge University Press - Burness, TC, Liebeck, MW & Shalev, A, 2019, ‘The length and depth of algebraic groups’.
*Mathematische Zeitschrift*, vol 291., pp. 741-760 - Burness, T & Harper, S, 2019, ‘On the uniform domination number of a finite simple group’.
*Transactions of the American Mathematical Society*, vol 372., pp. 545-583

View complete publications list in the University of Bristol publications system

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