# Home page

## Members

Diarmuid Crowley Thorsten Hertl Craig Hodgson

Scott Mullane Paul Norbury Marcy Roberston TriThang Tran

## Affiliated Members

Nora Ganter Christian Haesemeyer Yaping Yang Gufang Zhao

### Topology is part of the Pure Mathematics Research Group

The Topology Research Group at Melbourne has 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.

### Members:

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.

Thorsten Hertl works in differential geometry and algebraic topology, with a focus on spaces of geometric structures and their moduli; e.g. spaces positive scalar curvature metrics or spaces of Riemannian metrics with exceptional holonomy. In this work he uses tools from combinatorial topology, stable and unstable homotopy theory, Riemannian geometry, and Index theory. He is currently investigating approaches to constructing non-trivial elements in higher homotopy groups of G_2 moduli spaces, as well as invariants for detecting such elements.

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.

Dr Scott Mullane works in the area of moduli spaces, most frequently moduli spaces related to Riemann surfaces including the moduli space of curves, strata of differentials, and Hurwitz spaces. He is interested in how the different perspectives from topology, geometry, dynamics, and algebraic geometry relate and inform each other on these spaces.

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.

A/Prof 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:

A/Prof 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.

Dr Gufang Zhao works on oriented cohomology theories and applications in representation theory and enumerative geometry. He is also interested in structures of moduli of sheaves on Calabi-Yau 3-folds.

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 and preprints by members and affiliates of the topology group (ordered alphabetically by surname). Full publication lists are available on individuals webpages.

### Diarmuid Crowley

**D. Crowley**, Cs. Nagy, The smooth classification of 4-dimensional complete intersections. To appear in Geom. Topol.. arXiv:2003.09216**D. Crowley**, T. Schick, W. Steimle, The derivative map for diffeomorphisms of discs: An example, Geom. Topol. 27 (2023), 3699–3713. arXiv20.12.13634- A. Conway,
**D.****Crowley**, M. Powell, Infinite homotopy stable class for 4-manifolds with boundary, Pacific J. Math. 325 (2023), 209–237. arXiv:2210.00927 **D.****Crowley**, J. Bowden, Contact open books with flexible pages, Bull. Lond. Math. Soc. 55 (2023), 1302–1313. arXiv:2106.06253**D. Crowley**, J. Nordström, Exotic G_{2}-manifolds, Math. Ann. 381 (2021), 29–74. arXiv:1411.0656**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

**Thorsten**** Hertl**

**T. Hertl**, Moduli spaces of positive curvature metrics in dimension four and beyond. arXiv:2310.14115**T. Hertl**, Line bundle twists for unitary bordisms are ghosts. arXiv:2202.09919

### Paul Norbury

**P. Norbury.**A new cohomology class on the moduli space of curves.*Geom. Top.***27**(2023), 2695-2761.- A. Giacchetto, D. Lewanski,
**P. Norbury.**An intersection-theoretic proof of the Harer-Zagier formula.*Algebraic Geometry***10**(2023), 130-147. - W. Chaimanowong, M. Swaddle, M. Tavakol,
**P. Norbury.**Airy structures and deformations of curves in surfaces.*J. London Math. Soc.***109**(2023). - M. Kazarian,
**P. Norbury.**Polynomial relations among kappa classes on the moduli space of curves. To appear in IMRN. **P. Norbury.**A family of finite measures on the moduli space of curves and super volumes. arXiv:2312.14558- L. Anagnostou, S. Mullane,
**P. Norbury.**Weil-Petersson volumes, stability conditions and wall-crossing. arXiv:2310.13281 **P. Norbury.**Enumerative geometry via the moduli space of super Riemann surfaces. arXiv:2005.04378

**Marcy Robertson**

- 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 - P. Hackney,
**M. Robertson**, D. Yau. Infinity Properads and Infinity Wheeled Properads. Springer Lecture Notes in Mathematics, 358 pp, 2015. arXiv:1410.6716

**TriThang Tran**

- A. Kupers, J. Miller,
**T. Tran**. Homological stability for symmetric complements. Trans. AMS, 368, (2015), 7745-7762. arXiv:1410.5497

**Nora Ganter**

**N. Ganter.***Categorical Tori,*SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), Paper No. 014, 18 pp. arXiv:1406.7046**N, Ganter**.*Inner products of 2-representations*. Adv. Math. 285 (2015), 301–351. arXiv:1110.1711**N. Ganter**.*The elliptic Weyl character formula.**Compos. Math.*150 (2014), no. 7, 1196–1234. arXiv:1206.0528**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**N, Ganter.***Orbifold Genera, Product formulas and Power Operations*. Adv. Math. 205 (2006), no. 1, 84–133. arXiv:math/0407021

### Christian Haesemeyer

**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.- 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 - 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

### Yaping Yang

- M. Rapčák, Y. Soibelman, Y. Yang,
**G. Zhao**, Cohomological Hall algebras, vertex algebras and instantons, Commun. Math. Phys. 376, (2020), 1803–1873. arXiv:1810.10402 **Y. Yang**, G. Zhao, The cohomological Hall algebras of a preprojective algebra with symmetrizer, Algebr. Represent. Theory 26 (2023), 1067–1085. arXiv:1911.02689- Ivan Mirković,
**Y. Yang**, Gufang Zhao, Loop Grassmannians of quivers and affine quantum groups A Conference Celebrating the Birthdays of Sasha Beilinson and Victor Ginzburg, Springer 2022. arXiv:1810.10095.

### Gufang Zhao

- M. Rapčák, Y. Soibelman, Y. Yang,
**G. Zhao**,*Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds*. To appear in Commun. Number Theory Phys. arXiv:2007.13365 **G. Zhao**and C. Zhong,*Elliptic affine Hecke algebras and their representations*. Adv. Math. 395 (2022) 108077. arXiv:1507.01245.- C. Su,
**G. Zhao,**C. Zhong,*Wall-crossings and a categorification of K-theory stable bases of the Springer resolution*, Compo. Math. 157 (2021), 2341-2376. arXiv:1904.03769 - M. Levine, Y. Yang,
**G. Zhao**,*Algebraic elliptic cohomology theory and flops 2: $SL$-cobordism**.*Adv. Math. 384 (2021) 107726. arXiv:1610.00396

Here is a chronological list of visitors to the topology group and affiliates.

- Dr Hao Guo visited Diarmuid Crowley from Tsinghua University, August-September 2024
- Dr Jonathan Bowden visited Diarmuid Crowley from Regensburg University, August-September 2024
- Prof Charles Wiebel visited Christian Haesemeyer, July 2024
- A/Prof Chris Rogers visited Marcy Robertson from UNR, September-October 2023
- Dr Iva Halcheva visited Marcy Robertson from Northeastern University, July-August 2023
- Prof Victor Turchin visited Marcy Robertson and Diarmuid Crowley from Kansas State University, May 2023
- Dr Luciana Basualdo Bonatto visited Marcy Robertson from the Max Plank Institute, February 2023

- Dr Csaba Nagy visited Diarmuid Crowley from the University of Glasgow, December 2022
- 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 Henan University for a one year post-doc supported by the China Scholarship Council 2018

The conference Geometry and topology in low dimensions celebrates the work of Craig Hodgson, following his recent retirement.

#### 2022

Diarmuid Crowley was awarded a 2022 ARC Discover Project – Topology in seven dimensions – with Johannes Nordström from the University of Bath.

Marcy Robertson was awarded an 2021 ARC Future Fellowship which aims to develop the theory of infinity (modular and cyclic) operads and to use homotopical algebraic structures to study variations on Teichmüller space.

#### 2021

Diarmuid Crowley is speaking via zoom in the Princeton Topology Seminar on November 11, 2021

Nora Ganter is speaking at the Princeton Topology Seminar on May 2, 2021.

Christian Haesemeyer and Marcy Robertson are invited participants in the 2021 CRM Intensive Research Program “IRP Higher Homotopical Structures.”

#### 2020

Marcy Robertson is speaking at “Higher Structures in Prague” a conference in honour of Martin Markl’s 60th birthday November 9th-13th, 2020.

Marcy Robertson is speaking at the MIT Topology Seminar on October 26, 2020.

Marcy Robertson is an organiser of an online conference “Operad Pop-Up’‘ to take place August 11, 2020.

Marcy Robertson is an Organiser and Research Professor at the 2020 MSRI Program “Higher Categories and Categorification.”

Christian Haesemeyer is a program participant at the Newton Institute for Mathematical Sciences in the 2020 program “K-Theory, Algebraic Cycles and Motivic Homotopy Theory.”

Nora Ganter, Yaping Yang, and Gufang Zhao co-organized with Daniel Berwick Evans and Theo Johnson-Freyd the workshop on elliptic cohomology and physics 25-28 May 2020.

#### 2019

Marcy Robertson gave the 2019 Rubinov Oration at Federation University about the use of hyperbolic geometry in the work of MC Escher.