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Research

Research Interests


● Gauge theoretical moduli spaces in various dimensions, such as Seiberg-Witten theory, moduli spaces of Higgs bundles and character varieties, gauge theory and moduli spaces on foliated spaces.

● Applications of gauge theory to 4-manifolds including group actions and families.

● Mathematical physics, particularly geometric aspects of quantum field theory and string theory including: generalised geometry, T-duality and twisted K-theory.

Publications and Preprints


Preprints

[1] Equivariant Seiberg-Witten-Floer cohomology (with P. Hekmati) (2021) arXiv:2108.06855

[2] Non-trivial smooth families of K3 surfaces (2021) arXiv:2102.06354 

[3] The alpha invariant of complete intersections (2020) arXiv:2002.06750

[4] An adjunction inequality obstruction to isotopy of embedded surfaces in 4-manifolds (2020) arXiv:2001.04006

[5] On the Bauer-Furuta and Seiberg-Witten invariants of families of 4-manifolds (with H. Konno) (2019) arXiv.1903.01649

[6] A folitated Hitchin-Kobayashi correspondence (with P. Hekmati) (2018) arXiv:1802.09699


Journal Articles

[7] Tautological classes of definite 4-manifolds, to appear in Geometry and Topology, (2020) arXiv:2008.04519

[8] A note on the Nielsen realization problem for K3 surfaces, to appear in Proceedings of the American Mathematical Society (with H. Konno) (2019) arXiv.1908.03970

[9] Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants, Algebraic and Geometric Topology 21 (2021) 317-349 arXiv.1907.03949 [published version]

[10] A gluing formula for families Seiberg-Witten invariants, Geometry and Topology 24 (2020) 1381-1456  (with H. Konno) arXiv.1812.11691 [published version]

[11] Obstructions to smooth group actions on 4-manifolds from families Seiberg-Witten theory, Advances in Mathematics 354 (2019), 106730, 32 pp. arXiv:1805.07860 [published version]

[12] Special Kähler geometry of the Hitchin system and topological recursion, Advances in Theoretical and Mathematical Physics 23 No. 8, 1981-2024 (with Z. Huang) (2017) arXiv:1707.04975 [published version]

[13] Cayley and Langlands type correspondences for orthogonal Higgs bundles, Transactions of the American Mathematical Society 371 (2019), no. 10, 7451-7492 (with L. Schaposnik) arXiv:1708.08828 [published version]

[14] On the image of the parabolic Hitchin map, The Quarterly Journal of Mathematics 69 (2018), no. 2, 681-708 (with M. Kamgarpour) arXiv.1703.09886 [published version]

[15] Complete integrability of the parahoric Hitchin system, to appear in International Mathematics Research Notices (with M. Kamgarpour and R. Varma) (2016) arXiv:1608.05454

[16] Monodromy of the SL(n) and GL(n) Hitchin fibrations, Mathematische Annalen Vol. 370 (3) 2018, pp 1681-1716 arXiv:1612.01583 [published version]

[17] Monodromy of rank 2 twisted Hitchin systems and real character varieties, Transactions of the American Mathematical Society 370 (2018), 5491-5534 (with L. Schaposnik) arXiv:1506.00372 [published version]

[18] Arithmetic of singular character varieties and their E-polynomials, Proceedings of the London Mathematical Society (3) 114 (2017), no. 2, 293-332 (with P. Hekmati) arXiv:1602.06996 [published version]

[19] Classification of the automorphism and isometry groups of Higgs bundle moduli spaces, Proceedings of the London Mathematical Society (3) 112 (2016) 827-854 arXiv:1411.2228 [published version]

[20] Moduli spaces of Contact instantons, Advances in Mathematics 294 (2016) 562-595 (with P. Hekmati) arXiv:1401.5140 [published version]

[21] Automorphisms of C* moduli spaces associated to a Riemann surface, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 12 (2016), 007, 14 pages (with I. Biswas and L. Schaposnik) arXiv:1508.06587 [published version]

[22] Transitive Courant algebroids, string structures and T-duality, Advances in Theoretical and Mathematical Physics Vol 19 No. 3, 613-672 (2015) (with P. Hekmati) arXiv:1308.5159 [published version]

[23] Cyclic Higgs bundles and the affine Toda equations, Geometriae Dedicata Vol. 174 (2015), pp 25-42 arXiv:1011.6421 [published version]

[24] A Fourier Mukai approach to the K-theory of compact Lie groups, Advances in Mathematics 269 (2015), pp. 335-345 (with P. Hekmati) arXiv:1406.3993 [published version]

[25] Real structures on moduli spaces of Higgs bundles, Advances in Theoretical and Mathematical Physics Vol 20 No. 3, 525-551 (2016) (with L. Schaposnik) arXiv:1309.1195 [published version]

[26] Higgs bundles and (A,B,A)-branes, Communications in Mathematical Physics 331 (2014), no. 3, 1271-1300 (with L. Schaposnik) arXiv:1305.4638 [published version]

[27] A coboundary morphism for the Grothendieck spectral sequence, Applied Categorical Structures 22 pp. 269-288 (2014) arXiv:1112:6295 [published version]

[28] Topological T-duality for general circle bundles, Pure and Applied Mathematics Quaterly Vol. 10, no. 3, pp. 367-438 (2014) arXiv:1105:0290 [published version]

[29] Variation of Hodge structure for generalized complex manifolds, Differential Geometry and its Applications 36 (2014), pp. 98-133 arXiv:1205.0240 [published version]

[30] Topological T-duality for torus bundles with monodromy, Reviews in Mathematical Physics 27, No. 3 (2015), 1550008 arXiv:1201.1731 [published version]

[31] Conformal Courant algebroids and orientifold T-duality, International Journal of Geometric Methods in Modern Physics Vol 10, no. 2 (2013), 1250084 arXiv:1109.0875 [published version]

[312] Leibniz algebroids, twistings and exceptional generalized geometry, Journal of Geometry and Physics 62 (2012), pp. 903-934 arXiv:1101.0856 [published version]

[33] Moduli of Coassociative Submanifolds and Semi-Flat G2-manifolds, Journal of Geometry and Physics 60 (2010), pp. 1903-1918 arXiv:0902.2135 [published version]

Book Chapters

[34] Introduction to Generalized Geometry and T-duality, in Open Problems and Surveys of Contemporary Mathematics SMM 6, pp. 45-97 (2013)

This book chapter is an introduction to generalized geometry, T-duality and twisted K-theory. In writing this paper one of the main objectives was to explain how these topics fit together as part of a bigger picture: these topics are aspects of the geometry underlying string theory.

Conference Papers

[35] Brauer group of moduli of Higgs bundles and connections, to appear in Hitchin 70 proceedings, Oxford University Press (with I. Biswas and L. Schaposnik) (2017) arXiv:1609.00454

[36] Topological T-duality with monodromy, String-Math 2011, Proceedings of Symposia in Pure Mathematics, Eds. J. Block, J. Distler, R. Donagi, E. Sharpe, (2012), vol 85, pp. 293-302 [link to AMS bookstore] [String-Math 2011 webpage]

This is a summary of my research on topological T-duality for torus bundles with monodromy.

DPhil Thesis

[37] G2 Geometry and Integrable Systems, arXiv:1002.1767

My thesis studied some appearances of the exceptional Lie group G2 in geometry and integrable systems. In more detail, my thesis consists of three topics:

1. We study the Hitchin component in the space of representations of the fundamental group of a Riemann surface into a split real simple Lie group in the rank 2 case. We prove that such representations are described by a conformal structure and class of Higgs bundle we call cyclic and we show cyclic Higgs bundles correspond to a form of the affine Toda equations. In each case we relate cyclic Higgs bundles to geometric structures on the surface.

2. We elucidate the geometry of generic 2-plane distributions in 5 dimensions, relating it to a parabolic geometry associated to the split real form of G2 and a conformal geometry with holonomy in G2. We prove the distribution is the bundle of maximal isotropics corresponding to the annihilator of a spinor satisfying the twistor-spinor equation.

3. We study the moduli space of coassociative submanifolds of a G2-manifold with an aim towards understanding coassociative fibrations. We consider coassociative fibrations where the fibres are orbits of a torus action of isomorphisms and prove a local equivalence to minimal 3-manifolds in R^{3,3} with positive induced metric.