Back to homepage

Description of Research

My research centres around using Morse theory to study the topology of quotients of singular spaces, in particular those that arise from gauge-theoretical constructions (such as moduli spaces of Higgs bundles and Nakajima quiver varieties). The equations defining these spaces originate in physics as special cases of the Yang-Mills equations, and there exist deep connections between these moduli spaces and many other areas of mathematics, such as low-dimensional topology, geometric representation theory and the geometric Langlands program. In addition, these equations and their moduli spaces of solutions also have applications to control theory and computer network reliability.

Current Projects


  1. G.Wilkin, Morse theory for the space of Higgs bundles, Communications in Analysis and Geometry, Vol. 16, No. 2 (2008), pp283-332. (arxiv, pdf, publication)
  2. G.Daskalopoulos, J.Weitsman, R.Wentworth, G.Wilkin, Morse theory and hyperkähler Kirwan surjectivity for Higgs bundles, Journal of Differential Geometry, Vol. 87, No. 1 (2011), pp81-116. (arxiv, pdf, publication)
  3. M.Harada, G.Wilkin, Morse theory of the moment map for representations of quivers, Geometriae Dedicata, Vol. 150, No. 1 (2011), pp307-353. (arxiv, pdf, publication)
  4. G.Daskalopoulos, R.Wentworth, G.Wilkin, Cohomology of SL(2,C) character varieties and the action of the Torelli group, Asian Journal of Mathematics, Vol. 14, No. 3 (2010), pp359-384. (arxiv, pdf, publication)
  5. G.Wilkin, Homotopy groups of moduli spaces of stable quiver representations, International Journal of Mathematics Vol. 21, No. 9 (2010), pp1219-1238. (arxiv, pdf, publication)
  6. I.Biswas, G.Wilkin, Morse theory on the space of Higgs G-bundles, Geometriae Dedicata Vol. 149, No. 1 (2010), pp189-203. (arxiv, pdf, publication)
  7. R.Wentworth, G.Wilkin, Morse theory and stable pairs, Variational Problems in Differential Geometry, pp142-181, London Math. Soc. Lecture Note Ser., 394, Cambridge Univ. Press, Cambridge, 2012. (arxiv, pdf)
  8. S.Bradlow, G.Wilkin, Morse theory, Higgs fields and Yang-Mills-Higgs functionals, J. Fixed Point Theory Appl Vol. 11, No. 1 (2012), pp1-41. (arxiv, pdf, publication)
  9. R.Wentworth, G.Wilkin, Cohomology of U(2,1) representation varieties of surface groups, Proc. London Math. Soc. Vol. 106, No. 2 (2013), pp445-476. (arxiv, pdf, publication)
  10. G.Wilkin, Moment map flows and the Hecke correspondence for quivers, Advances in Mathematics, Vol. 320 (2017), pp730-794. (arxiv, pdf, publication)
  11. N.Ho, G.Wilkin, S.Wu, Higgs bundles on a nonorientable manifold, Communications in Analysis and Geometry, Vol. 26, No. 4 (2018), pp857-886. (arxiv, pdf, publication)
  12. I.Biswas, G.Wilkin, Anti-holomorphic isometries of hyperkähler manifolds and branes, Journal of Geometry and Physics Vol. 88 (2015), pp52-55. (arxiv, pdf, publication)
  13. G.Wilkin, The reverse Yang-Mills-Higgs flow in a neighbourhood of a critical point, Journal of Differential Geometry Vol. 115, No. 1 (2020), pp111-174. (arxiv, pdf, publication)
  14. G.Daskalopoulos, C.Mese, G.Wilkin, Higgs bundles over cell complexes and representations of finitely presented groups, Pacific Journal of Mathematics, Vol. 296-1 (2018), pp31-55. (arxiv, pdf, publication)
  15. N.Ho, G.Wilkin, S.Wu, Conditions of smoothness of moduli spaces of flat connections and of representation varieties, Mathematische Zeitschrift, Vol. 293, No. 1-2 (2019), pp1-23. (arxiv, pdf, publication)
  16. G.Wilkin, Equivariant Morse theory for the norm-square of a moment map on a variety, International Mathematics Research Notices, Vol. 2019, Issue 15 (August 2019), pp4730-4763. (arxiv, pdf, publication)
  17. S. Kim, G.Wilkin, Analytic convergence of harmonic metrics for parabolic Higgs bundles, Journal of Geometry and Physics, Vol. 127 (2018), pp55-67. (arxiv, pdf, publication)
  18. M. Pflaum, G. Wilkin, Equivariant control data and neighbourhood deformation retractions, Methods and Applications of Analysis (special issue in memory of John Mather), Vol. 26, No. 1 (2019), pp13-36. (arxiv, pdf, publication)
  19. V. Mathai, G. Wilkin, Fractional quantum numbers via complex orbifolds, Letters in Mathematical Physics, Vol. 109, No. 11 (2019), pp2473-2484. (arxiv, pdf, publication)
  20. O. Garcia-Prada, G. Wilkin, Action of the mapping class group on character varieties and Higgs bundles, Documenta Mathematica, Vol. 25 (2020), pp841-868. (arxiv, pdf, publication)
  21. V. Mathai, G. Wilkin, Fractional quantum numbers, complex orbifolds and noncommutative geometry, Journal of Physics A: Mathematical and Theoretical (published online June 2021). (arxiv, pdf, publication)
  22. G. Wilkin, Local behaviour of the gradient flow of an analytic function near the unstable set of a critical point. To appear in Rocky Mountain Journal of Mathematics (arxiv, pdf, publication)
  23. G. Wilkin, Convolution for quiver varieties via cup product on a Morse complex. (arxiv, pdf)

Edited volumes

  1. The geometry, topology and physics of moduli spaces of Higgs bundles, Lecture Notes Series, Institute for Mathematical Sciences, NUS, World Scientific Publishing. publication

Last Updated 12 September, 2023