My most recent research projects are centred around the cohomology of sheaves on small categories. In particular I've been working with sheaves on lattices arising naturally from hyperplane arrangements. Much of this is part of ongoing work with Brent Everitt, and arose from my interest in knot homology theory and in particular Khovanov homology.

Before that I was working in quantum topology looking at homotopy quantum field theories with a side interest in gerbes. Even further back I was interested in the rich algebraic structure found in the homology of certain infinite loop spaces associated to cohomology theories and their relationship to cohomology operations.

2208.01367 | Mario Gutiérrez, Hugo Parlier, Paul Turner, Quadratis Puzzles, Proceedings of Bridges 2022: Mathematics, Art, Music, Architecture, Culture, pages 49–56. |

1908.04500 | B. Everitt and P. Turner, Sheaf homology of hyperplane arrangements, Boolean covers and exterior powers, Mathematische Zeitschrift, volume 302, pages 1451–1475 (2022). |

1902.00399 | B. Everitt and P. Turner, Deletion-restriction for sheaf homology of geometric lattices, Advances in Mathematics, 402, 25 June 2022, 24 p., 108354. |

1803.10079 | P. Turner, Khovanov homology and diagonalisable Frobenius algebras, J. Knot Theory and Its Ramifications, Vol. 29, No. 01, 1950095 (2020). |

1409.6442 | P. Turner, A hitchhiker's guide to Khovanov homology. In Lectures on Quantum Topology in dimension three, T. Le, C. Lescop, R. Lipshitz, P. turner, Panoramas et Synthèses, No 48 (2016), Société Mathématiques de France. |

1406.3476 | B. Everitt and P. Turner, Cellular cohomology of posets with local coefficients. Journal of Algebra, vol 439 (2016) 134-158. |

1202.1856 | B. Everitt, R. Lipshitz, S. Sarkar and P. Turner, Khovanov homotopy type and the Dold-Thom functor. Homology, Homotopy and Applications, vol 18(2) (2016), 177-181. |

1112.3460 | B. Everitt and P. Turner, The homotopy theory of Khovanov homology. AGT 14 (2014) 2747-2781. |

1001.5379 | P. Turner and E. Wagner, The homology of digraphs as a generalisation of Hochschild homology. Journal of Algebra and Its Applications 11 (2) (2012), 1250031(13 pages) |

0808.1686 | B. Everitt and P.R. Turner, Bundles of coloured posets and a Leray-Serre spectral sequence. Trans. AMS. 334 (6) (2012), 3137 --3158 |

0711.0103 | B. Everitt and P.R. Turner, Homology of coloured posets: a generalisation of Khovanov's cube construction. J. Alg. 322 (2009), 429 -- 448 |

0606464 | P.R. Turner, Five Lectures on Khovanov homology. J. Knot Theory and its Ramifications, vol 26, Issue 3, 1741009 (2017) (41 pages). |

0606369 | P.R. Turner, A spectral sequence for Khovanov homology with an application to (3,q)-torus links. Algebr. Geom. Topol. 8 (2008), 869 --884 |

0502445 | M. Mackaay and P.R. Turner, Bar-Natan's Khovanov homology for coloured links. Pac. J. Math., 229 (2) (2007), 429 -- 446 |

0509692 | M. Mackaay, P.R. Turner and P. Vaz, A remark on Rasmussen's invariant for knots. J. Knot Theory and its Ramifications, 16 (3) (2007), 333 -- 344. |

0506229 | V. Turaev and P.R. Turner, Unoriented topological quantum field theory and link homology. Alg. Geom. Top. 6(2006), 1069 -- 1093. |

0411225 | P.R. Turner, Calculating Bar-Natan's characteristic two Khovanov homology. J. Knot Theory and its Ramifications, 15 (10) (2006) 1335-1356. |

0410179 | S.K. Hansen, J. Slingerland and P.R. Turner, Abelian homotopy Dijkgraaf-Witten theory. Adv. Th. Math. Phys. 9 (2005), 321 -- 353. |

0408333 | P.R. Turner and M. Brightwell, Relative differential characters. Comm. Anal. Geom. 14(2) (2006), 269 --282. |

0212128 | P.R. Turner, A functorial approach to differential characters. Algebr. Geom. Topol. 4, (2004), 81--93. |

0201116 | U. Bunke, P. Turner and S. Willerton, Gerbes and homotopy quantum field theories, Algebr. Geom. Topol. 4, (2004), 407--437. |

0202066 | M. Brightwell, P. Turner and S. Willerton, Homotopy quantum field theories and related ideas, Proceedings of the Tenth Oporto Meeting on Geometry, Topology and Physics (2001). Int. J. Modern Phys. A, 18 (2003), October, suppl., 115--122. |

0111084 | M. Brightwell and P. Turner, Modular functors in homotopy quantum field theory and tortile structures. J. Pure and Applied Algebra, 185 (2003), no. 1-3, 43--71. |

9910026 | M. Brightwell and P. Turner, Representations of the homotopy surface category of a simply connected space, J. Knot Theory and its Ramifications, vol. 9, No. 7 (2000) 855-864. |

J-Y. Butowiez and P.R. Turner, Unstable multiplicative cohomology operations, Quart. J. Math., 4 (2000), 437-449. | |

J.R. Hunton and P.R. Turner, The homology of spaces representing exact pairs of homotopy functors, Topology Vol. 38, No. 3 (1999), 621-634 . | |

J.R. Hunton and P.R. Turner, Coalgebraic algebra, Journal of Pure and Applied Algebra 129 (1998) 297-313. | |

P.R. Turner, Dickson coinvariants and the homology of QS^0, Math. Zeitschrift 224 (1997), 209-228 . | |

P.J. Eccles, P.R. Turner and W.S. Wilson, On the Hopf ring for the sphere , Math. Zeitschrift 224 (1997), 229-233. | |

N.P. Strickland and P.R. Turner, Rational Morava E-theory and DS^0, Topology Vol. 36, No. 1 (1997), 137-151. | |

P.R. Turner, Unstable BP-operations and formal group laws, Journal of Pure and Applied Algebra 110 (1996) 91-100. | |

T. Kashiwabara, N.P. Strickland and P.R. Turner, The Morava K-theory Hopf Ring for BP, Algebraic Topology: New trends in localization and periodicity, ed. C. Broto et al., Birkhauser 1996. | |

V.Gorbunov, N. Ray and P.R. Turner, On the Hopf ring for a symplectic oriented spectrum , American Journal of Mathematics 117 (1995), 1063-1088. | |

P.R. Turner, Dyer-Lashof operations in the Hopf ring for complex cobordism, Proc. Cam. Phil. Soc.114 (1993) 453-460. |