I'm interested in the interactions between geometry, algebra and theoretical physics. More specifically, I work on derived categories of coherent sheaves and their various generalizations.
A longer summary of my research, written for the general public, is here.
Publications and preprints (arXiv)
(xviii) Equivariant Fukaya categories at singular values (with Yankı Lekili)
(xvii) Serre functors of residual categories via hybrid models (with Federico Barbacovi)
Bull. London Math. Soc. 55 (2023), no. 5, 2505-2514.
(xvi) Line fields on punctured surfaces and twisted derived categories
(xv) Discriminants and semi-orthogonal decompositions (with Alex Kite)
Comm. Math. Phys. 390 (2022), 907-931
(xiv) A non-commutative Bertini theorem (with Jørgen Rennemo and Michel Van den Bergh)
J. Noncommutative Geometry 13 (2019), no. 2, 609-616.
(xiii) Hori-mological projective duality (with Jørgen Rennemo)
Duke Math. J. 168 (2019), no. 11, 2127-2205.
Some supporting calculations are in this addendum.
(xii) All autoequivalences are spherical twists
Int. Math. Res. Not. 2018 (2018), no. 10, 3137-3154.
(xi) A new 5-fold flop and derived equivalence
Bull. London Math. Soc 48 (2016), no. 3, 533-538.
(x) Quintic threefolds and Fano elevenfolds (with Richard Thomas)
J. Reine Angew. Math. (Crelle). (2018), 743, 245-259.
(ix) K-theoretic and categorical properties of toric Deligne-Mumford stacks (with Tom Coates, Hiroshi Iritani and Yunfeng Jiang)
Pure Appl. Math. Q. 11 (2015) no. 2, 239-266.
(viii) The Pfaffian-Grassmannian equivalence revisited (with Nick Addington and Will Donovan)
Alg. Geom. 2 (2015), no. 3, 332-364.
Here's a video of a lecture I gave at the Newton Institute on this topic.
(vii) Mixed braid group actions from deformations of surface singularities (with Will Donovan)
Comm. Math. Phys. 335 (2015), no. 1, 497-543.
(vi) D-brane probes, branched double covers, and non-commutative resolutions (with Nick Addington and Eric Sharpe)
Adv. Theor. Math. Phys. 18 (2014), no. 6, 1369-1436.
(v) Window shifts, flop equivalences and Grassmannian twists (with Will Donovan)
Compositio Math. 150 (2014), no. 6, 942-978.
(iv) The closed state space of affine Landau-Ginzburg B-models
J. Noncommutative Geometry. 7 (2013), no. 3, 857-883.
(iii) Equivalences between GIT quotients of Landau-Ginzburg B-models
Comm. Math. Phys. 304 (2011), no. 2, 411-432.
(ii) Gauge theory in higher dimensions, II (with Simon Donaldson)
Surveys in Differential Geometry 16 (2011), 1-14.
The A-infinity deformation theory of a point and the derived categories of local Calabi-Yaus
J. Algebra 320 (2008), no. 8, 3232-3268.
This is essentially my PhD thesis, my advisor was
The thesis version has an extra appendix on A-infinity algebras with some pretty pictures.
These are the complete lecture notes for a 4th-year undergraduate/MSc level course on differential geometry, which I taught at Imperial College for three years. They cover the foundations of manifolds, tangent vectors, and differential forms, up to Stokes' Theorem. Here are the exercises.
Group Representation Theory
Complete notes for a 3rd-year undergraduate course on representation theory, which I also taught for three years at Imperial. They cover the basics of representations of finite groups over the complex numbers (Maschke's Theorem, Schur's Lemma, character tables) and finish with the classification of semi-simple algebras. Here are the exercises.
Federico Barbacovi (graduated 2023)
Thesis: Flop-flop autoequivalences and
compositions of spherical twists.
Bradley Doyle (graduated 2021)
Thesis: Homological Projective Duality for the Plücker embedding of the Grassmannian.
Alex Kite (graduated 2019)
Thesis: Fundamental group actions on derived categories, parts of which became paper (xv) above and also arXiv:1711.08940.
If you're interested in doing a PhD with me you should apply to the LSGNT. Applications are normally open in November/December each year.
The universal closed state space of an open TFT
These are some short notes on a particular result in 2-dimensional topological field theory.
The 7 Colour Theorem
I gave a talk at the KCL Maths School and one of the students made these beautiful notes.
The Incredible Predictive Power of String Theory
A pop-maths article I wrote for the De Morgan newsletter.
LSGNT Topics in Geometry
The Geometry of Gauged Linear Sigma Models
A workshop I co-organised in June 2022 at the University of Birmingham.