# Duncan Laurie

Research interests

I am currently a PhD student at the Mathematical Institute, University of Oxford, supervised by Professor Kevin McGerty and working in the Algebra research group.

My research is in geometric representation theory, more specifically quantum groups, quantum affine algebras, crystal bases, quantum toroidal algebras, quiver varieties, and related areas.

Contact

Email: firstname [dot] lastname [at] maths [dot] ox [dot] ac [dot] uk

There is also a contact form on my Mathematical Institute profile.

Publications and preprints

Young wall realizations of level 1 irreducible highest weight and Fock space crystals of quantum affine algebras in type E

Duncan Laurie (7 Nov 2023)

Abstract: We construct Young wall models for the crystal bases of level 1 irreducible highest weight representations and Fock space representations of quantum affine algebras in types E_6^(1), E_7^(1) and E_8^(1). In each case, Young walls consist of coloured blocks stacked inside the relevant Young wall pattern which satisfy a certain combinatorial condition. Moreover the crystal structure is described entirely in terms of adding and removing blocks.

Automorphisms of quantum toroidal algebras from an action of the extended double affine braid group

Duncan Laurie (13 Apr 2023)

Abstract: We construct an action of the extended double affine braid group B̈ on the quantum toroidal algebra Uq(gtor) in all untwisted types. In the simply laced cases, using this action and certain involutions of B̈ we obtain automorphisms and anti-automorphisms of Uq(gtor) which exchange the horizontal and vertical subalgebras. Moreover, they switch the central elements C and k0^a0...kn^an up to inverse. This generalises existing results in type A due to Miki, and can be viewed as the analogue, for these quantum toroidal algebras, of the duality for double affine braid groups which Cherednik used to realise the difference Fourier transform in his celebrated proof of the Macdonald evaluation conjectures.

Other writings

Quiver varieties, quantum groups and crystal bases

Thesis for the 'Transfer of Status' milestone of my PhD, December 2021

Quotients in algebraic geometry

Mini-project for the Oxford Master's course 'C3.4 Algebraic Geometry', December 2020

Representations and characters of GLn(Fq)

Master's dissertation, April 2019

Talks

Conferences

(November 2023) "Quantum toroidal algebras: braid group actions and automorphisms" - Algebraic and Combinatorial Methods in Representation Theory ICTS program at the Tata Institute of Fundamental Research in Bengaluru, India

(July 2023) "Quantum toroidal algebras: braid group actions, automorphisms, and representations" - Workshop in Noncommutative Algebra and Representation Theory at the University of Kent, invited by Prof. Stéphane Launois

(April 2023) "Automorphisms of quantum toroidal algebras from an action of the extended double affine braid group" - RepNet Spring School in Representation Theory at the University of Kent

Seminars

(Upcoming, December 2023) "Quantum toroidal algebras and their representation theory" - Algebra & Combinatorics Seminar, Indian Institute of Science Bangalore, invited by Prof. Vyjayanthi Chari

(October 2023) "Quantum toroidal algebras: braid group actions, automorphisms, and representation theory" - Paris Algebra Seminar, invited by Prof. David Hernandez

(March 2023) "Quantum toroidal algebras and extended double affine braid groups" - Junior Algebra and Representation Theory Seminar, University of Oxford

(November 2021) "An introduction to quivers, quiver representations and quiver varieties" - Junior Algebra and Representation Theory Seminar, University of Oxford

(June 2021) "Representations and characters of general linear groups over finite fields" - Junior Algebra and Representation Theory Seminar, University of Oxford

Reading groups

(June 2023) "Borel and parabolic subgroups" - Linear Algebraic Groups reading group

(February 2022) "Descent theory, torsors and principal bundles" - Stacks reading group

Conferences

Past conferences

(November 2023) Algebraic and Combinatorial Methods in Representation Theory ICTS program at the Tata Institute of Fundamental Research in Bengaluru, India

(August 2023) LMS-Bath Symposium on Geometric and Categorical Representation theory at the University of Bath

(August 2023) LMS-Bath Summer School on Geometric and Categorical Representation theory at the University of Bath

(July 2023) LMS Research School – Algebraic Groups and Their Representations at the University of Birmingham

(July 2023) Workshop in Noncommutative Algebra and Representation Theory at the University of Kent

(April 2023) RepNet Spring School in Representation Theory at the University of Kent

(March 2023) Categories, clusters, and completions master class at Aarhus University

(January 2023) Quivers, Clusters, Moduli and Stability at the University of Oxford

(June 2022) Mini-workshop on Quiver Varieties and Related Topics at the University of Oxford

(June 2022) Representation Theory and Flag or Quiver Varieties Summer School at the University of Paris

Teaching

2022-23:

Tutor in Representation Theory for the Williams-Exeter Programme at Oxford (WEPO), which also included designing the course and writing and marking the exam

Tutor for the 3rd year course 'Introduction to Representation Theory' (two sets of classes)

2021-22:

Revision class tutor for the 3rd year course 'Introduction to Representation Theory' (two sets of classes)

Tutor for the 2nd year course 'Group Theory' at Somerville College Oxford

Teaching assistant for the 3rd year course 'Commutative Algebra'

Tutor for the 3rd year course 'Introduction to Representation Theory'

2020-21:

Teaching assistant for the Master's course 'Probabilistic Combinatorics'

Teaching assistant for the 3rd year course 'Introduction to Representation Theory'