Dr Lewis Topley PhD

Dr Lewis Topley

School of Mathematics
UKRI Future Leaders Fellow and Birmingham Fellow
Lecturer

Contact details

Address
University of Birmingham
Edgbaston
Birmingham
B15 2TT
UK

Lewis Topley is a Lecturer in the School of Mathematics. He also holds a UKRI Future Leaders Fellowship and a Birmingham Fellowship.

His research focuses on Representation Theory and Lie theory. Broadly speaking the goal of these subjects is to understand the symmetries of geometric objects by expressing them as matrices of linear transformations. The field encompasses an exciting array of methods, exhibiting strong interactions with theoretical physics and algebraic geometry.

Qualifications

  • PhD in Mathematical Sciences, University of Manchester 2013
  • MSc in Pure Mathematics, Imperial College London 2009
  • BSc in Mathematics, University of Edinburgh 2008

Biography

After completing his PhD at the University of Manchester in 2013, Lewis took postdocs at the University of East Anglia, the University of Padova and the University of Kent.

His research at UEA was supported by the doctoral prize fellowship, awarded each year to a promising doctoral graduate from any discipline.

At the University of Padova, Lewis held a Marie Sklodowska-Curie fellowship. He used the research funding there to travel widely, starting collaborations at the University of Oregon and the University of Birmingham. These collaborations led to important progress in the modular theory of Yangians and finite W-algebra. The synthesis of these two theories is one of Lewis’ most fruitful topics of current research. 

In 2016 Lewis returned to the UK to take up a postdoc at the University of Kent with Stéphane Launois. There he developed his research interests in Poisson geometry, deformation theory and quantum groups.

In 2019 he condensed his research objectives into a UKRI Future Leaders fellowship proposal, which he was awarded. This grant will run until January 2024, with a possible 3 year extension. Funding will be available to hire a postdoctoral research associate working in the field of geometric representation theory and W-algebras. On top of this Lewis is actively recruiting PhD students.

Lewis joined the University of Birmingham in August 2020. As well as carrying his UKRI fellowship status he has been awarded a Birmingham Fellowship for the initial five year period of his contract.

Teaching

Lewis currently supervises one PhD student as well as organising weekly reading groups for PhD students, joint with the University of Warwick.

Postgraduate supervision

Lewis offers supervision on a range of PhD topics related to representation theory and Lie theory. Interested candidates should write directly to Lewis to discuss potential projects.

Research

Overview of research interests:

  • Ordinary and modular representation theory of Lie algebras
  • Finite W-algebras
  • Poisson algebras and Poisson varieties
  • Yangians and quantum groups
  • Symplectic representation theory

Detailed summary

Lewis’s research focuses on various quantum algebras arising in Lie theory. The basic motivation for all of his research is to better understand the representations of various families of Lie algebras, both over the complex numbers and over fields of positive characteristic. The most interesting examples are the Lie algebras of reductive groups which exhibit an extremely elegant theory of representations, with many mysteries still to uncover. However the Cartan type Lie algebras, affine Lie algebras and all of the associated centralisers provide interesting testing ground for breaking out of the reductive setting.

One of the most interesting modern tools for reductive groups is the theory of finite W-algebras. These arise via quantum Hamiltonian reduction, and should be viewed as non-commutative analogues of the transverse Poisson structures to symplectic leaves, first defined and studied by Weinstein. Their representation theory involves itself in that of Lie algebras via Skryabin’s equivalence, relating these new representations to the classical category of Whittacker modules. In the ordinary setting one obtains an elegant approach to understanding the primitive ideals of enveloping algebras, perfectly suited to the discussion of associated varieties and Joseph’s theory of Goldie ranks. In the modular setting one obtains a Morita equivalence with the reduced enveloping algebra which control the representation theory of the original Lie algebras.

Together with Jon Brundan, Lewis carried out the very first study of Yangians in the modular setting. These have been studied extensively by physicists and have now shown themselves to be a key tool in the modular representation theory of general linear Lie algebras. Consequently many new avenues of research have been opened up.

In recent years a new theme has been emerging in representation theory. The enveloping algebras of Lie algebras of reductive groups, as well as their finite W-algebras, have been identified as the universal quantizations of certain conic symplectic singularities. This allows us to view all of these theories through a single lens, and ties the representation theory to the theory of Poisson deformations of symplectic varieties. In a sense this relegates Lie theory to a single facet of a much larger program of research, and this broader perspective is already brining new tools to bear on classical problems. This new wave has been christened symplectic representation theory, and will serve as a starting point for many of Lewis’s future investigations.

Publications

Recent publications

Article

Goodwin, S & Topley, L 2021, 'Restricted shifted Yangians and restricted finite W-algebras', Transactions of the American Mathematical Society, vol. 8, pp. 190–228. https://doi.org/10.1090/btran/63

Goodwin, S & Topley, L 2019, 'Minimal dimensional representations of reduced enveloping algebras for gln', Compositio Mathematica, vol. 155, no. 8, pp. 1594-1617. https://doi.org/10.1112/S0010437X19007474

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