The Topology Research Group at Melbourne includes researchers working in a broad range of topological disciplines including: the interplay of geometry and the topology of manifolds in dimensions three and four, surgery theory, algebraic topology and homotopy theory.
The Topology Research Group is part of the larger Pure Mathematics Group.
A/Prof Diarmuid Crowley works in the differential topology of high dimensional manifolds and surgery theory. His specific interests span the topological residues of geometric structures such as G_2-structures and contact structures, 7-manifolds, the foundations of surgery, spaces of diffeomorphism and mapping class groups and exotic spheres and the Gromoll filtration.
A/Prof Craig Hodgson works in hyperbolic geometry, low dimensional topology and differential geometry. His research interests include computation of geometric structures using triangulations, and the study of geometric and arithmetic invariants of hyperbolic 3-manifolds. He has also developed a powerful deformation theory of hyperbolic structures using harmonic differential forms to represent cohomology classes. Recently he has been investigating relationships between new invariants of 3-manifolds arising from physics, such as the 3D-index, and classical topology and geometry, including surfaces and hyperbolic structures.
Prof Paul Norbury works in algebraic geometry and topology with particular focus on the moduli space of curves. His recent work is on cohomological field theories which organises algebraic topological invariants of the compactification of the moduli space of curves. He applies the technique of topological recursion which underlies many geometric problems in mathematical physics. He also has interest in gauge theory and the moduli space of super Riemann surfaces.
Dr Marcy Robertson works in algebraic topology and is particularly interested in using higher categories and the theory of operads to recast classical topological objects in a homotopical setting. Her recent work has focused on using higher operads to model the moduli space of genus g curves and Teichmüller space. She is also interested in using props and properads to study cohomology operations and deformation theory.
Emeritus Prof J. Hyam Rubinstein works in low dimensional topology and is especially interested in three and four-dimensional manifolds. He is also interested in studying shortest networks with applications to mining, machine learning and financial mathematics.
Dr. TriThang Tran works in algebraic topology and is interested in homological stability, configuration spaces and braid groups. He is also engaged in research in mathematics education, teaching and learning.
Researchers in Related Areas:
Dr Nora Ganter works on Moonshine, categorification and is interested in the applications of elliptic cohomology to representation theory.
Prof Christian Haesemeyer works in in the area of algebraic K-theory and motivic homotopy theory. He often uses techniques from algebraic topology and homotopy theory and has an ongoing interest in surgery theory as it relates to quadratic forms.
The Topology group runs several research and working seminars:
- Topology Seminar is a research seminar featuring a mixture of guest and local speakers that meets Mondays at 2:15 pm during the semester.
- Moduli Spaces Seminar is a research seminar that meets Wednesdays at 11:00am during the semester.
- Graduate Topology Seminar is a topics seminar for Ph.D. and Masters students.
The following is a list of selected publications by members and affiliates of the topology group. A full list of publications is available on individual webpages.
- D. Crowley, J. Nordström, The rational homotopy type of (n-1)-connected manifolds of dimension up to 5n-3, J. Topol. 13 (2020), 539–575. arXiv:1505.04184
- P. Hackney, M. Robertson, D. Yau. Modular operads and the nerve theorem. Advances in Mathematics, 370, 107206 (39pp) (2020). arXiv:1906.01143
- P. Hackney, M. Robertson, D. Yau. A graphical category for higher modular operads. Advances in Mathematics, 365, 107044 (61pp) (2020). arXiv:1906.01144
- P. Boavida de Brito, G. Horel, M. Robertson. Operads of genus zero curves and the Grothendieck-Teichmüller group. Geometry & Topology, 23, 299-346 (2019). arXiv:1612.04660
- G. Borot, P. Norbury. Loop equations for Gromov-Witten invariants of P1 SIGMA 15 (2019), 061, 29 pages. arXiv:1905.01890
- L. Chekhov, P. Norbury. Topological recursion with hard edges. Inter. Journal Math. 30, No. 3 (2019) 1950014 (29 pages). arXiv:1702.08631
- P. Dunin-Barkowski, N. Orantin, P. Norbury, A. Popolitov, S. Shadrin. Dubrovin’s superpotential as a global spectral curve. J. Inst. Math. Jussieu 18 (2019), 449-497. arXiv:1509.06954
- D. Crowley, J. Nordström. The classification of 2-connected 7-manifolds, Proc. London Math. Soc. 119 (2019), 1–54. arXiv:1406.2226
- C. Haesemeyer, C. A. Weibel, The norm residue theorem in motivic cohomology. Annals of Mathematics Studies, 200. Princeton University Press, Princeton, NJ, 2019. xiii+299 pp.
- D. Crowley, T. Schick, W. Steimle. Harmonic spinors and metrics of positive scalar curvature via the Gromoll filtration and Toda brackets, J. Topol. 11 (2018), 1076–1098. arXiv1612.04660
- G, Cortiñas, C. Haesemeyer, M. E. Walker, C. A. Weibel. The K-theory of toric schemes over regular rings of mixed characteristic. Singularities, algebraic geometry, commutative algebra and related topics, 455-479, Springer, Cham (2018). arXiv:1703.07881
- N. Do, P. Norbury. Topological recursion on the Bessel curve. Comm. Number Theory and Physics 12 (2018), 53-73. arXiv:1608.02781
- N. Ganter. Categorical Tori, SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), Paper No. 014, 18 pp. arXiv:1406.7046
- J. Bowden, D. Crowley, A. Stipsicz. The topology of Stein fillable manifolds in high dimensions II, Geom. Topol. 19 (2015), 2995–3030. arXiv:1409.7504
- N, Ganter. Inner products of 2-representations. Adv. Math. 285 (2015), 301–351. arXiv:1110.1711
- A. Kupers, J. Miller, T. Tran. Homological stability for symmetric complements. Transactions of the American Mathematical Society, 368, 7745-7762, (2015). arXiv:1410.5497
- P. Hackney, M. Robertson, D. Yau. Infinity Properads and Infinity Wheeled Properads. Springer Lecture Notes in Mathematics, 358 pp, 2015. arXiv:1410.6716
- N. Ganter. The elliptic Weyl character formula. Compos. Math. 150 (2014), no. 7, 1196–1234. arXiv:1206.0528
- G. Cortiñas, C. Haesemeyer, M. E Walker, C.A Weibel. Bass’ NK groups and cdh-fibrant Hochschild homology. Invent. Math. 181 (2010), no. 2, 421–448. arXiv:0802.1928
- N. Ganter. Hecke operators in equivarian elliptic cohomology and generalized Moonshine. Groups and symmetries, 173–209, CRM Proc. Lecture Notes, 47, Amer. Math. Soc., Providence, RI, 2009. arXiv:0706.2898
- G. Cortiñas, C. Haesemeyer, M. Schlichting, C. Weibel. Cyclic homology, cdh-cohomology and negative K-theory. Ann. of Math. (2) 167 (2008), no. 2, 549–573. arXiv:math/0502255
- N, Ganter. Orbifold Genera, Product formulas and Power Operations. Adv. Math. 205 (2006), no. 1, 84–133. arXiv:math/0407021
Here is a chronological list of visitors to the topology group and affiliates.
- Alessandro Giacchetto, visited Paul Norbury from MPIM, Bonn, February and March 2020.
- Prof. Martin Markl, visited Marcy Robertson from the Czech Academy of Sciences, December 2019.
- Prof. Charles Wiebel, visited Christian Haesemeyer, August 2019.
- Prof Søren Galatius, visited Diarmuid Crowley from the University of Copenhagen, School Visiting Scholar, Feburary and March 2019.
- Prof Jim Davis, visited Diarmuid Crowley from Indiana University, February 2019.
- Dr Huijun Yang, visited Diarmuid Crowley from Haifeng University for a one year post-doc supported by the China Scholarship Council 2018.