# 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 Summerfield is a current Pure Mathematics PhD Student at the School of Mathematics, researching algebraic groups in positive characteristics. Under the supervision of Professor Simon Goodwin, he is working towards a classification of infinitesimally isolated nilpotent orbits in the Lie algebras of algebraic groups, supported by a scholarship from the EPSRC Doctor Training Partnerships.

## Qualifications

- MMath in Mathematics, University of Oxford, 2021

## Biography

Joel Summerfield completed an Undergraduate Masters in Mathematics at Somerville College, University of Oxford, in 2021. After this he began a PhD at the University of Birmingham, which he is currently undertaking.

## Doctoral research

- PhD title
- The Classification of Infinitesimally Isolated Nilpotent Orbits
- Supervisor
- Professor Simon Goodwin
- Course
- Pure Mathematics PhD

## Research

### Research themes

- Algebraic groups
- Lie theory
- Sheets and decomposition classes
- Nilpotent orbits
- Geometric invariant theory

### Research activity

An algebraic group acts on its Lie algebra via the adjoint action, which induces an equivalence relation via the Jordan decomposition. The resulting equivalence classes are called Decomposition Classes, and separate the Lie algebra into centraliser-dimension-stable irreducible pieces. There is a well-known connection between Decomposition Classes and Sheets, which are maximal irreducible components of fixed centraliser-dimension. Most existing results in the area only cover the characteristic 0 case, or when the group satisfies the Standard Hypotheses, and this research project aims to explore what happens when these assumptions are removed.