Dr Simon Willerton
School of Mathematical and Physical Sciences
Senior Lecturer in Pure Mathematics
+44 114 222 3823
Full contact details
School of Mathematical and Physical Sciences
J19
Hicks Building
Hounsfield Road
Sheffield
S3 7RH
- Profile
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Dr. Willerton obtained his PhD from the University of Edinburgh in 1997, where he also held a temporary lectureship. He held a Royal Society Research Fellowship in Melbourne and an EU Marie Curie Fellowship in Strasbourg before returning to the UK as a Research Associate at Heriot-Watt University. After a Visiting Professorship at the University of California, San Diego, he joined Sheffield University as a lecturer in 2002.Since then he has held visiting positions at UC San Diego, UC Riverside and the Autonomous University of Barcelona.
He was awarded a Senate Award for Excellence in Teaching and Learning in 2009 for his work with Eugenia Cheng on using YouTube in teaching mathematics and was promoted to Senior Lecturer in 2010.
- Research interests
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Dr. Willerton is interested in various ideas in low-dimensional topology coming from quantum physics, and in their relationship to geometry and algebraic topology.
In particular, methods from quantum field theory give rise to new invariants of knots and three-manifolds -- these are the so-called quantum and Vassiliev (or finite-type) invariants. A large part of the motivation for Dr. Willerton's work is to understand these invariants from a topological or geometric point of view. For instance, the Kontsevich integral is a construction which takes a knot and gives back a sort of Feynman diagram expansion: this embodies a rich algebraic structure that is reminiscent of certain objects from algebraic topology, but it is not clear at the moment how to relate these.
Well-studied examples of quantum invariants arise when one fixes a Lie group. Motivated in part by trying to understand the Kontsevich integral, Dr. Willerton has considered (with collaborators in San Diego and Oxford) the less well-studied invariants which arise when one fixes a hyper-Kahler manifold. This work has revealed unexpected algebraic structures in the derived category of coherent sheaves on a complex manifold.
The theory of gerbes is a related interest of Dr. Willerton. Gerbes can be thought of as the next step beyond line bundles. Ideas from this area feed into K-theory, string theory and the quantum invariants mentioned above.
In recent times Dr Willerton has been interested in the connections between metric spaces and category theory. This has lead in particular to him studying measures of biodiversity.
- Publications
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Show: Featured publications All publications
Featured publications
Journal articles
- The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials. Discrete Analysis, 2020(5). View this article in WRRO
- Categorifying the magnitude of a graph. Homology, Homotopy and Applications, 19(2), 31-60. View this article in WRRO
Preprints
- The Legendre-Fenchel transform from a category theoretic perspective. View this article in WRRO
All publications
Journal articles
- The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials. Discrete Analysis, 2020(5). View this article in WRRO
- Categorifying the magnitude of a graph. Homology, Homotopy and Applications, 19(2), 31-60. View this article in WRRO
- On the magnitude of spheres, surfaces and other homogeneous spaces. Geometriae Dedicata, 168(1), 291-310.
- Tight spans, Isbell completions and semi-tropical modules. Theory and Applications of Categories, 28, 696-732.
- On the asymptotic magnitude of subsets of Euclidean space. Geometriae Dedicata, 164(1), 287-310.
- On the rozansky-witten weight systems. Algebraic and Geometric Topology, 10(3), 1455-1519.
- A DIAGRAMMATIC APPROACH TO HOPF MONADS. ARAB J SCI ENG, 33(2C), 561-585.
- The twisted Drinfeld double of a finite group via gerbes and finite groupoids. ALGEBR GEOM TOPOL, 8(3), 1419-1457.
- On the first two Vassiliev invariants. EXP MATH, 11(2), 289-296.
- Free groups and finite-type invariants of pure braids. MATH PROC CAMBRIDGE, 132, 117-130.
- A combinatorial half-integration from weight system to Vassiliev knot invariant. J KNOT THEOR RAMIF, 7(4), 519-526.
- Vassiliev invariants and the Hopf algebra of chord diagrams. MATH PROC CAMBRIDGE, 119, 55-65.
- MATHEMATICS - A TOPOLOGICAL TIE-IN. NATURE, 368(6467), 103-104.
- Spread: a measure of the size of metric spaces.
- The Mukai pairing, I: a categorical approach. New York Journal of Mathematics, 16, 61-98.
- Gerbes and homotopy quantum field theories. Algebr. Geom. Topol., 4, 407-437.
- An almost-integral universal Vassiliev invariant of knots. Algebr. Geom. Topol., 2, 649-664.
- Vassiliev invariants as polynomials. Banach Center Publications, 42(1), 457-463.
Conference proceedings papers
- Homotopy quantum field theories and related ideas. INTERNATIONAL JOURNAL OF MODERN PHYSICS A, Vol. 18 (pp 115-122)
- THE KONTSEVICH INTEGRAL AND ALGEBRAIC STRUCTURES ON THE SPACE OF DIAGRAMS. Knots in Hellas '98
Working papers
- Heuristic and computer calculations for the magnitude of metric spaces. View this article in WRRO
- Problems on invariants of knots and 3-manifolds. Geom. Topol. Monogr., 4, 377-572.
- The Kontsevich integral and algebraic structures on the space of
diagrams.
Preprints
- On the magnitude of odd balls via potential functions.
- The Legendre-Fenchel transform from a category theoretic perspective. View this article in WRRO
- The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials. Discrete Analysis, 2020(5). View this article in WRRO
- Research group
- Teaching activities
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MAS115 Mathematical Investigation Skills MAS336 Differential Geometry