# Mr Joel Summerfield MMath

## In 'Staff and Students'

Back to 'School of Mathematics'School of Mathematics

PhD Student

## Contact details

- Address
- School of Mathematics

Watson Building

University of Birmingham

Edgbaston

Birmingham

B15 2TT

UK

Joel is a PhD student in the Algebra group, under the supervision of Professor Simon Goodwin. He is currently working on classifying infinitesimally isolated nilpotent orbits in projective general linear algebras, and hopes to generalise these results to the Lie algebras of arbitrary connected reductive linear algebraic groups.

## Qualifications

- MMath in Mathematics, University of Oxford, 2021

## Doctoral research

- PhD title
- The Classification of Infinitesimally Isolated Nilpotent Orbits without the Standard Hypotheses
- Supervisors
- Professor Simon Goodwin and Dr Matthew Westaway
- Course
- Pure Mathematics PhD

## Research

### Research themes

- Representation Theory
- Linear Algebraic Groups
- Nilpotent Orbits
- The Standard Hypotheses
- Projective General Linear Algebras
- Decomposition Classes

### Research activity

The Lie algebra of a connected reductive linear algebraic group can be broken into quasi-affine varieties depending on centraliser dimension within the Lie algebra. A nilpotent orbit is infinitesimally isolated if it coincides with an irreducible component of such a quasi-affine variety. The classification of infinitesimally isolated nilpotent orbits is known in the case where the underlying group satisfies the Standard Hypotheses. Joel's current work centres around extending the classification to cases where the Standard Hypotheses no longer hold, such as in certain projective general linear groups.