Mr Joel Summerfield MMath

Contact details

Email: jns228@student.bham.ac.uk

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

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.