Research
My research is in arithmetic algebraic geometry, an area at the intersection of number theory and algebraic geometry. I am also interested in computational aspects of number theory, algebraic geometry, representation theory, and other algebraic topics.
Preprints (in preparation)
- Cohomological vanishing on Siegel modular varieties and applications to lifting Siegel modular forms (with S. Mullane)
- A theta operator for Siegel modular forms (with M. Flander)
- Analytic evaluation of Hecke eigenvalues (with O. Colman and N. Ryan)
Preprints (submitted)
- HLinear: Exact dense linear algebra in Haskell (with M. Westerholt-Raum)
- Theta operators on Siegel modular forms and Galois representations (with A. McAndrew)
Publications
- Distinguishing newforms (with S. Chow), International Journal of Number Theory 11, no. 3 (2015), 893-908. (arXiv copy) Sage source code and data
- Distinguishing eigenforms modulo a prime ideal (with S. Chow), Functiones et Approximatio Commentarii Mathematici 51, no. 2 (2014), 363-377. (arXiv copy)
- Hecke eigenvalues of Siegel modular forms of ''different'' weights (with R. Sayer), Journal of Number Theory 143, no. 10 (2014), 125-141. (arXiv copy)
- Computing level one Hecke eigensystems (mod p) (with C. Citro), LMS Journal of Computation and Mathematics 16 (2013), 246-270. (arXiv copy)
- Computations of vector-valued Siegel modular forms (with N. Ryan and D. Sulon), Journal of Number Theory 133, no. 11 (2013), 3921-3940. (arXiv copy)
- Experimental evidence for Maeda's conjecture on modular forms (with A. McAndrew), Tbilisi Mathematical Journal 5, no. 2 (2012), 55-69, special issue on symbolic computation using Sage. (arXiv copy)
- Distinguishing Hecke eigenforms, International Journal of Number Theory 7, no. 5 (2011), 1247-1253. (arXiv copy)
- Enumerating Galois representations in Sage (with C. Citro), in K. Fukuda et al (Eds), Proceedings of the International Congress on Mathematical Software 2010, Lecture Notes in Computer Science 6327 (2010), 256-259, Springer, Heidelberg. (arXiv copy)
- All Siegel Hecke eigensystems (mod p) are cuspidal, Mathematical Research Letters 13, no. 5-6 (2006), 813-823. (arXiv copy)
- Upper bound on the number of systems of Hecke eigenvalues for Siegel modular forms (mod p), International Mathematics Research Notices, no. 55 (2004), 2983-2987. (arXiv copy)
- Hecke eigenvalues of Siegel modular forms (mod p) and of algebraic modular forms, Journal of Number Theory 106, no. 2 (2004), 345-384. (arXiv copy)
Erdős number 3: Erdős-Pomerance-Ryan-Ghitza
Other
- Review of Lectures on N_X(p) by Jean-Pierre Serre, Gazette of the Australian Mathematical Society 41, no. 5 (November 2014), 305-308.
- The abc conjecture, as easy as 1, 2, 3… or not, The Conversation, 26 November 2012
- An elementary introduction to Siegel modular forms (survey), September 2004
Students (alphabetical)
- Dwi Setya Wibawanti (MSc): (quasi-)modular forms
Past students (inverse chronological)
- Angus McAndrew (MPhil 2015): Galois representations and theta operators for Siegel modular forms
- James Withers (MPhil): explicit computation of Hecke module structure on spaces of modular forms. Currently in the PhD program at Boston University.
- Rob Sayer (MSc 2014): the paramodular conjecture. Currently in the PhD program at Boston University.
- Max Flander (PhD 2014): operators for Siegel modular forms (mod p). Currently Associate Search Data Analyst at SEEK.
- Angus McAndrew (MSc 2013): modular forms and Maeda’s conjecture.
- Leon Di Stefano (MSc 2013): geometry of Gaussian graphical models. Currently Clinical Research Assistant at the Academic Child Psychiatry Unit, Royal Children’s Hospital.
- Mitchell Brunton (Vacation scholar 2013): multiple zeta values. Currently in the BSc program in mathematics at the University of Melbourne.
- Sam Chow (MPhil 2012): distinguishing modular eigenforms (mod p). Currently in the PhD program in mathematics at the University of Bristol (since September 2012).
- Angus McAndrew (Vacation scholar 2012): comparison of integer relation algorithms.
- Matt Collins (Hons 2011): representations of algebraic Heisenberg groups
- Scott Mullane (MSc 2011): Vanishing sheaf cohomology. Currently in the PhD program in mathematics at Boston College (since September 2011).
- Peter Humphries (Vacation scholar 2009): Computing dimensions of spaces of modular forms of weight one over the complex numbers. Currently in the PhD program in mathematics at Princeton University (since September 2012).
Seminars, conferences
- September 2016: Number theory down under 4 at Newcastle
- December 2014: Number theory session at ANZMC8
- 2012-present: Number theory seminar
- 2009: p-compact groups
- 2009: Conformal field theory
Talks (relatively recent)
2015
- Analytic evaluation of Hecke eigenvalues, AustMS conference, Adelaide, 1 October 2015
- Differential operators on modular forms, Number Theory Seminar, University of New South Wales, 24 June 2015.
- Lifting modular forms: geometry in aid of arithmetic, Algebraic geometry and number theory seminar, University of Gothenburg, 22 January 2015.
2014
- Differential operators and Galois representations, Topology and Representation Theory at Kioloa, 12 November 2014.
- Irreducibility of Hecke polynomials in higher levels, Number Theory Down Under, Newcastle, 24 October 2014.
- (Mis)Adventures with open source in mathematics, Open Science Workshop, Melbourne, 19 July 2014.
- Distinguishing Hecke eigenforms and irreducibility of Hecke polynomials, Explicit Theory of Automorphic Forms, Tongji University, Shanghai, 24 March 2014.
- Hecke polynomials and distinguishability of eigenforms of higher level, Graduate Student Number Theory Seminar, University of Illinois at Urbana-Champaign, 23 January 2014.
2013
- Harder-type congruences between modular forms, Number Theory Down Under, Newcastle, 5 October 2013.
- Vanishing on the Siegel modular threefold, AustMS conference, Sydney, 3 October 2013.
- The algebraic structure of Harder-type congruences, AustMS conference, Sydney, 2 October 2013.
- A theta operator for Siegel modular forms (mod p), Number Theory Seminar, University of Sheffield, 6 August 2013.
- Comparing Siegel eigenforms of different weights, Heilbronn Seminar, University of Bristol, 31 July 2013.
- Lifting Siegel modular forms (mod p) to characteristic zero, Number Theory Seminar, University of Sheffield, 18 July 2013.
- How can you tell apart two modular forms?, Pure Mathematics Seminar, University of Queensland, 2 April 2013.
2012
- On the number of level one systems of Hecke eigenvalues (mod p), AustMS conference, University of Ballarat, 26 September 2012.
- Deligne’s proof of “Weil implies Ramanujan-Petersson”, Part I, working seminar on the Weil conjectures, University of Melbourne, 7 June 2012.
- Motives, Algebra-Geometry-Topology Seminar, University of Melbourne, 27 April 2012.
- Congruences and geometry, Distinguished Visiting Professor Seminar, Bucknell University, 9 February 2012.
- Integer relation algorithms, Distinguished Visiting Professor Seminar, Bucknell University, 7 February 2012.
2011
- 30 years of LLL, Melbourne University Mathematics and Statistics Society, 28 October 2011.
- Lifting modular forms, AustMS conference, University of Wollongong, 27 September 2011.
- Computing with modular forms (mod p), Universität Heidelberg, 15 July 2011.
- Computing Galois representations mod p, Workshop on explicit theory of automorphic forms, applications and computations, CIRM, Luminy, 12 May 2011.
2010
- Distinguishing Hecke eigenforms, Special session in algebra and number theory, AustMS conference, University of Queensland, Brisbane, 28 September 2010.
2009
- Doing algebra with Sage, 27th Victorian Algebra Conference, University of Melbourne, 5 November 2009.
- Enumerating Galois representations (mod p), Special session in computational algebra, First PRIMA Congress, University of New South Wales, Sydney, 10 July 2009.
- Lifting modular forms from positive characteristic, Australian National University, Canberra, 15 April 2009.
- Computing with modular forms (mod p), Monash University, Melbourne, 12 May 2009.
- The double life of primes, Melbourne University Mathematics and Statistics Society, 20 March 2009.
- Algebraic geometry in Sage, Sage Days 14, Mathematical Sciences Research Institute, Berkeley, 10 March 2009.
- Some aspects of Siegel modular forms mod p, Universität Siegen, 13 January 2009.
2008
- Computing level 1 Hecke eigensystems (mod p), Special session in number theory, 7th Australia-New Zealand Mathematics Convention, University of Canterbury, Christchurch, NZ, 8-12 December 2008.
- Modular forms in arithmetic geometry, Algebra, Geometry, and Topology seminar, University of Melbourne, 20 October 2008.
- A differential operator for modular forms in positive characteristic, 26th Victorian Algebra Conference, RMIT, Melbourne, 3 October 2008.
- Geometry of modular forms mod p and Galois representations, University of Melbourne, 15 January 2008.