Research

"Discovery is the priviledge of the child." (Grothndieck)

Research Interests 

Geometric representation theory, more specifically, the geometric Langlands program and arithmetic of character varieties

Paper in preparation 

21. Counting points on character varieties (joint with my students Gyeonghyeon Nam, Bailey Whitbread, and Stefano Giannini) 

ArXiv Preprints

20.  Connections whose differential Galois group is reductive of maximal rank (joint with Dan Sage)

19. Arithmetic geometry of character varieties with regular monodromy, I (joint with PhD student Nam and colleague Anna Puskas, submitted). 

Published Papers

I try to publish in journals which are open access, cheap, and associated with societies. You can see how successful I have been!

18. Character stacks are PORC count (joint with my Master's student Nick Bridger, Journal of the Australian Mathematical Society, 2022)

17.  Hypergeometric sheaves for classical groups via geometric Langlands (joint with Daxin Xu and Lingfei Yi, Transactions of the American Mathematical Society, 2022)

16.  Airy sheaf for reductive groups (joint with Konstantin Jakob and Lingfei Yi,  Proceedings of London Mathematical Society, 2022) - video

15. Geometric Langlands for Hypergeometric Sheaves (joint with Lingfei Yi,  Transactions of the American Mathematical Society, 2021)

14. Rigid connections on the projective line via the Bruhat--Tits building (joint with Dan Sage, Proceedings of the London Mathematical Society, 2020)

13. On a geometric analog of a conjecture of Gross and Reeder (joint with Dan Sage, American Journal of Maths, 2019)

12. On the image of the parabolic Hitchin map (joint with David Baraglia,  Quarterly Journal of Math, 2017)

11. Complete integrability of the parabolic Hitchin map (joint with David Baraglia and Rohith Varma, International Mathematical Research Notices, 2017)

10. Jordan decomposition for formal G-connections (joint with MPhil student Samuel Weatherhog, Graduate Journal of Maths 2017)

9. Conductors in local geometric Langlands correspondence (Canadian Journal of Mathematics, Vol 69, no. 1, 2017, 107--126)  

8. Preservation of depths in local geometric Langlands program (joint with Tsao-Hsien Chen, Transactions of the American Mathematical Society (2017), 1345--1364)

7. Compatibility of the Feigin-Frenkel Isomorphism and the Harish-Chandra Isomorphism for Jet algebras (Transactions of the American Mathematical Society (2016), 2019--2038)

6. Ramified Satake Isomorphisms for strongly parabolic characters (Joint with Travis Schedler, 2014, Documenta Mathematica) 

5. Geometrization of principal series representation of reductive groups (Joint with Travis Schedler, 2013, Annals of Institute Fourier)

4. Geometrization of continuous characters of Z_p^* (Joint with Clifton Cunningham, Pacific Journal of Mathematics, 2013)

3. Compatible intertwiners for representations of nilpotent groups (Joint with Teru Thomas, Representation Theory, 2011) 

2. Maximal representation dimension for finite p-groups (Joint with Zinovy Reichstein and Shane Cernele, Journal of Group Theory, 2011) 

1.Stacky abelianization of algebraic groups (PhD Thesis, Transformation Groups,  2009) 

Incomplete Preprints- Not intended for publication

4. Langlands correspondence for a hypergeometric connection on the projective line (superseded by the paper with Lingfei Yi on hypergeometrics) 

3. Stabilisers of eigenvectors of finite reflection groups (Results were generalised by in MPhil thesis of Sinead Wilson)

2. Universal property of K-theory and Homology (joint with Justin Noel) - See this paper where they give a definitive account of the universal property of K-theory. 

1. Weil representation over finite field (Master's Thesis 2005) 

Non-mathematical publications

5. Show me the money: what does your registration fee cover?  

4. Interview with Maud El-Hachem: the 2021 winner of the AustMs Maryam Mirzakhani Award  (Gazette of the Australian Mathematical Society, 2020)

3. Interview with Hoa Bui: 2020 winner of the AustMs Maryam Mirzakhani Award  (Gazette of the Australian Mathematical Society, 2020)

2. Interview with Fatemeh Ansarizadeh: 2019 winner of the AustMs Maryam Mirzakhani Award  (Gazette of the Australian Mathematical Society, 2020)

1. Meeting the Australian Parliament: a mathematician's experience (Gazette of the Australian Mathematical Society, 2020)