Mr Joel Summerfield MMath

Mr Joel Summerfield

School of Mathematics
PhD Student

Contact details

School of Mathematics
Watson Building
University of Birmingham
B15 2TT

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.


  • MMath in Mathematics, University of Oxford, 2021

Doctoral research

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


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.