[ David Gruenewald ]



Research
Interests
I am a postdoctoral researcher (on the job market). In 2012 I completed a postdoc working with John Boxall in the Laboratoire de Mathématiques Nicolas Oresme at the Université de Caen, supported by the ANR's PACE project (Pairings and Advances in Cryptology for Ecash).
Before that, I spent 3 months in Department of Mathematics at Radboud Universiteit Nijmegen, supported by the DIAMANT cluster working with Wieb Bosma on complex continued fractions. Before that I was a postdoc at eRISCS at the Université d'AixMarseille (in Marseille). Before that, I was a doctoral student in the Number Theory group of the School of Mathematics and Statistics at the University of Sydney, where my supervisor was David R. Kohel.
I am interested in computing with modular forms and their associated moduli spaces.
My attention is currently focused on genus 2, where there are practical applications to hyperelliptic curve cryptography.
I was awarded my PhD in December 2009. In my thesis entitled "Explicit Algorithms for Humbert Surfaces", I find explicit practical models for moduli spaces of Abelian surfaces, in particular Humbert surfaces and Shimura curves. The equations can be found here.
In July 2008 I went to Microsoft Research for the northern summer, working as a research intern under the guidance of Kristin Lauter and Reinier Bröker. We implemented an improved version of the CRT algorithm in Magma which makes use of (3,3)isogeny relations I had previously computed.
I enjoy all computational aspects of number theory. At the beginning of 2006 I did some computations for Alf van der Poorten on width 6 Somos sequences arising from continued fraction expansions of genus 2 curves. I primarily use Magma for my computations, but have also worked with Sage and Mathematica.
Publications
Abstract: We discuss heuristic asymptotic formulae for the number of isogeny classes of pairingfriendly abelian varieties of fixed dimension g ≥ 2 over prime finite fields. In each formula, the embedding degree k ≥ 2 is fixed and the rhovalue is bounded above by a fixed real ρ_{0} > 1. The first formula involves families of ordinary abelian varieties whose endomorphism ring contains an order in a fixed CMfield K of degree g and generalizes previous work of the first author when g=1. It suggests that, when ρ_{0} < g, there are only finitely many such isogeny classes. On the other hand, there should be infinitely many such isogeny classes when ρ_{0} > g.
The second formula involves families whose endomorphism ring contains an order in a fixed totally real field K_{0}^{+}of degree g. It suggests that, when ρ_{0} > 2g/(g+2) (and in particular when ρ_{0} > 1 if g = 2), there are infinitely many isogeny classes of gdimensional abelian varieties over prime fields whose endomorphism ring contains an order of K_{0}^{+}. We also discuss the impact that polynomial families of pairingfriendly abelian varieties has on our heuristics, and review the known cases where they are expected to provide more isogeny classes than predicted by our heuristic formulae.
Abstract: Conjecturally, the only real algebraic numbers with bounded partial quotients in their regular continued fraction expansion are rationals and quadratic irrationals. We show that the corresponding statement is not true for complex algebraic numbers in a very strong sense, by constructing for every even degree d algebraic numbers of degree d that have bounded complex partial quotients in their Hurwitz continued fraction expansion. The Hurwitz expansion is the generalization of the nearest integer continued fraction expansion for complex numbers. In the case of real numbers the boundedness of regular and nearest integer partial quotients is equivalent.
Abstract:
For a complex abelian surface A with endomorphism ring isomorphic to the maximal order in a quartic CM field K, the Igusa invariants j_{1}(A), j_{2}(A), j_{3}(A) generate an unramified abelian extension of the reflex field of K. In this paper we give an explicit geometric description of the Galois action of the class group of this reflex field on j_{1}(A), j_{2}(A), j_{3}(A). Our description can be expressed by maps between various Siegel modular varieties, and we can explicitly compute the action for ideals of small norm. We use the Galois action to modify the CRT method for computing Igusa class polynomials, and our run time analysis shows that this yields a significant improvement. Furthermore, we find cycles in isogeny graphs for abelian surfaces, thereby implying that the ‘isogeny volcano’ algorithm to compute endomorphism rings of ordinary elliptic curves over finite fields does not have a straightforward generalization to computing endomorphism rings of abelian surfaces over finite fields.
 Computing Humbert surfaces and applications, in Arithmetic, Geometry, Cryptography and Coding Theory 2009, Contemporary Mathematics, vol. 521, Amer. Math. Soc., Providence, RI, 2010, pp. 5969 (preprint version).
Talks
(Invited talks are highlighted in red)
 Complex numbers with bounded partial quotients: AustMS Conference, The University of Sydney, 1^{st} October 2013.
 Heuristics on pairingfriendly abelian varieties: Computational Algebra Seminar, The University of Sydney, 18^{th} April 2013.
 Computing isogeny graphs in genus 2 using CM lattices, ECC 2012, Querétaro, Mexico, 31^{st} October 2012
 Computing "isogeny graphs" using CM lattices: LACAL seminar, EPFL, 9^{th} March 2012
 Computing "isogeny graphs" using CM lattices: Workshop on Algorithms for Curves, Moduli, and Isogenies, Laboratoire d'informatique (LIX), École Polytechnique, Palaiseau, 7^{th} July 2011
 Computing "isogeny graphs" using CM lattices: Geocrypt 2011, Corsica, 22^{nd} June 2011 (pdf)
 Hyperelliptic Curves, Cryptography and Factorization Algorithms: Séminarie de Cryptographie, Université de Caen, 27^{th} January 2011
 Hyperelliptic Curves, Cryptography and Factorization Algorithms: Algemeen Wiskundecolloquium, Radboud Universiteit Nijmegen, 8^{th} December 2010 (pdf)
 Humbert Surfaces and Applications: DIAMANT Symposium, Lunteren, 27^{th} November 2010
 Explicit CM in Genus 2: Intercity Number Theory Seminar, Radboud Universiteit Nijmegen, 1^{st} October 2010
 Humbert Surfaces and Isogeny Relations: Séminaire de Cryptographie, Institut de Recherche en Mathématiques de Rennes, 15^{th} January 2010
 Humbert Surfaces and Applications: Réunion CHIC, Institut Henri Poincaré, Paris, 6^{th} October 2009 (pdf)
 An Introduction to Hyperelliptic Curves: Crypto'Puces, île de Porquerolles, 4^{th} June 2009 (pdf)
 Humbert Surfaces and Isogeny Relations: AGCT12, CIRM at Luminy, Marseille, 3^{rd} April 2009 (pdf)
 Explicit CM in Genus 2: Computational Algebra Seminar, The University of Sydney, 19^{th} March 2009 (pdf)
 Explicit CM in Genus 2: Number Theory Seminar, Institut de Mathématiques de Luminy, 9^{th} October 2008
 Computing Humbert Surfaces: AustMS Conference, La Trobe University, 25^{th} September 2007 (pdf)
 Computing Humbert Surfaces: Journées Arithmétiques, University of Edinburgh, Scotland, July 2007
 Introduction to the Weil Conjectures: Algebraic Geometry seminar, UNSW, 13^{th} June 2007
 Néron Models: Complex Multiplication lecture series, Number Theory Seminar, Sydney, semester 1, 2006
 Polarized Abelian Varieties: lecture series following Shimura’s Complex Multiplication of Abelian Varieties, Number Theory Seminar, Sydney, 15^{th} September 2005
 Humbert Surfaces: Number Theory Seminar, Sydney, 25^{th} May 2005
 The Moduli of Abelian Varieties: Number Theory Seminar, Sydney, 16^{th} March 2005
 Endomorphisms of Complex Abelian Varieties: Abelian Varieties lecture series, Number Theory Seminar, Sydney, 29^{th} October 2004
 Principally Polarized Complex Abelian Varieties and their Moduli Space: Abelian Varieties lecture series, Number Theory Seminar, Sydney, 15^{th} October 2004
 Line Bundles on Complex Tori: Abelian Varieties lecture series, Number Theory Seminar, Sydney, August  September 2004
 Ndimensional Spheres, Cubes and the Tower of Hanoi: talk given at MANSW’s (Mathematical Association of NSW) Talented Students’ Day,
 23^{rd} July, 2004 at University of Technology Sydney
 22^{nd} July, 2005 at Macquarie University
 21^{st} July, 2006 at University of New South Wales
 20^{th} July, 2007 at The University of Sydney
 Hecke Operators: third talk in the Modular Forms lecture series, Number Theory Seminar, Sydney, April 2004
 Introduction to Modular Forms: talk given to the Sydney University Mathematics Society, October 2003
 Introduction to Elliptic Curves: talk given to the Sydney University Mathematics Society, May 2003
Conference and Workshop participation
 Australian Mathematical Society Annual Meeting, The University of Sydney, 30^{th } September – 3^{rd} October 2013
 ECC 2012, Querétaro, Mexico, 28^{th} – 31^{st} October 2012
 ECC 2011, INRIA, Nancy, France, 19^{th} – 21^{st} September 2011
 Geocrypt 2011, near Bastia, Corsica, 20^{th} – 24^{th} June 2011
 AGCT13, CIRM Luminy, Marseille, 14^{th } – 18^{th } April 2011
 DIAMANT Symposium, Lunteren, 26^{th} – 27^{th} November 2010
 ANTSIX , INRIA Nancy, France, 19^{th} – 23^{rd} July 2010
 Réunion CHIC, Institut Henri Poincaré, Paris, 5^{th} – 6^{th} October 2009
 Crypto'Puces, Île de Porquerolles, 2^{nd} – 6^{th}June 2009
 AGCT12, CIRM Luminy, Marseille, 30^{th }March – 3^{rd} April 2009
 ESF Workshop on Codes, Cryptography and Coding Theory, Institut de Mathématiques de Luminy, 25^{th }– 29^{th} March 2009
 Australian Mathematical Society Annual Meeting, La Trobe University, 25^{th }– 28^{th} September 2007
 Journées Arithmétiques, University of Edinburgh, Scotland, 2^{nd }– 6^{th} July 2007
 Australian Mathematical Society Annual Meeting, Macquarie University, 25^{th }– 29^{th} September 2006
 Computing with Modular forms, MSRI, Berkeley, 31^{st} July – 11^{th} August 2006
 Explicit Arithmetic Geometry, Institut Henri Poincaré, Paris, 6^{th} – 10^{th} December 2004
Contact Details
Email: 
davidg at maths . usyd . edu . au 
Teaching
(All MATHxxxx tutorials mentioned below correspond to lecture courses given at the University of Sydney)
 2016
 Tutor at National Mathematics Summer School, Australian National University, Canberra, 3^{rd} – 16^{th} January
 MATH1002 Linear Algebra tutorials
 2015
 MATH1002 Linear Algebra tutorials
 MATH1003 Integral Calculus and Modelling tutorials
 MATH1014 Introduction to Linear Algebra tutorials
 2014
 2013
 Tutor for 2Unit maths bridging course, Sydney University, 11^{th} – 26^{th} February
 MATH1901 Differential Calculus (Advanced) tutorials
 MATH1002 Linear Algebra tutorials
 MATH1903 Integral Calculus and Modelling (Advanced) tutorials
 2009
 Tutor at National Mathematics Summer School, Australian National University, Canberra, 4^{th} – 17^{th} January
 Tutor for 2Unit maths bridging course, Sydney University, 9^{th} – 24^{th} February
 2008
 Tutor at National Mathematics Summer School, Australian National University, Canberra, 6^{th} – 19^{th} January
 Tutor for 2Unit maths bridging course, Sydney University, 11^{th} – 26^{th} February
 MATH1011 Life Sciences Calculus tutorials
 2007
 Tutor at National Mathematics Summer School, Australian National University, Canberra, 7^{th} – 20^{th} January
 Tutor for 2Unit maths bridging course, Sydney University, 12^{th} – 27^{th} February
 MATH1011 Life Sciences Calculus tutorials
 2006
 Tutor at National Mathematics Summer School, Australian National University, Canberra, 8^{th} – 21^{st} January
 Tutor for 2Unit maths bridging course, Sydney University, 13^{th} – 28^{th} February
 MATH1002 Linear Algebra tutorials
 2005
 Tutor at National Mathematics Summer School, Australian National University, Canberra, 2^{nd} – 15^{th} January
 Tutor for 2Unit maths bridging course, Sydney University, 14^{th} February  1^{st} March
 MATH1901 Differential Calculus (Advanced) tutorials
 MATH1012 Life Sciences Algebra tutorials
 2004
 MATH1001 Differential Calculus tutorials
 MATH1003 Integral Calculus tutorials
Does this web page look familiar? I borrowed the template from Ben Smith (I’ll return it or the favour one day!)
Last modified on: 5th February 2016