
Nils Bruin
Department of Mathematics
Simon Fraser University
Burnaby, BC
CANADA V5A 1S6
Tel : (778) 782 3794
Fax : (778) 782 4947
Email :
nbruin@sfu.ca


This is my personal page. Currently I am an Assistant Professor at the Department of Mathematics of Simon
Fraser University.
For various redirections, see my links page.
Previous track
September 1990  July 1995 
Undergraduate Mathematics at Leiden university, resulting in Master
Thesis Generalization
of the ABCconjecture, under supervision of Prof. Dr. R. Tijdeman. 
July 1995  July 1999 
AIO (Graduate student) at Leiden University, working
on the generalised Fermat equation, under supervision of Prof. Dr. R. Tijdeman
and Dr. F. Beukers. 
6 October 1999 
Defense of PhD thesis Chabauty Methods and covering techniques applied
to generalised Fermat equations. 
September 1999  August 2000 
NWOresearcher at Utrecht University, under supervision of Dr. F. Beukers 
July 2000  September 2000 
Clay Mathematical Institute
Liftoff Mathematician at Utrecht, Netherlands and MSRI Berkeley, CA.

September 2000  December 2000 
MSRI Postdoctoral fellow 
Januari 2001  August 2002 
PIMS Postdoctoral fellow at Simon Fraser University and University of British Columbia, Canada 
September 2002  September 2003 
Senior Research Associate with the MAGMA group within the School of
Mathematics at the University of Sydney 
October 2003  
Assistant Professor at the Department of Mathematics of Simon Fraser University 
Awards
Publications
 Nils Bruin, K. Gyory, L. Hajdu, Sz. Tengely, Arithmetic progressions consisting of unlike powers, Indag. Mathem. 17 (4), 539555
(2006). Also available as preprint.
 Nils Bruin, Julio Fernandez, Josep Gonzalez, JoanCarles Lario, Rational points on twists of X0(63). Available as Preprint at Universitat Politecnica Catalunya, 2006. Accepted for publication in Acta Arithmetica.
 Nils Bruin, E. Victor Flynn, Josep Gonzalez, Victor Rotger,
On finiteness conjectures for endomorphism algebras of abelian surfaces
, Math. Proc. Camb. Phil. Soc. (2006) 141, 383408. (or see
arXiv preprint).
See Electronic Data
for supporting electronic resources.
 Michael A. Bennett, Nils Bruin, Kalman Gyory, Lajos Hajdu,
Powers from Products of Consecutive Terms in Arithmetic Progression,
Proc. London Math. Soc. (3) 92 (2006), no. 2, 273  306.
 Nils Bruin, E. Victor Flynn, Exhibiting Sha[2] on Hyperelliptic Jacobians, Journal of Number Theory, 118 (2006), pp. 266291.

Nils Bruin and Michael Stoll,
Deciding existence of rational points on curves: an experiment, 2006. See Electronic data for additional resources.
 Nils Bruin, The primitive solutions to x^3+y^9=z^2, Journal of Number Theory,
111 (2005), pp. 179189. (Or see arXiv preprint, 2003). See Transcript of
computations for supporting electronic resources.

Nils Bruin and E. Victor Flynn,
Towers of
2covers of hyperelliptic curves, Trans. Amer. Math. Soc. 357 (2005),
43294347.
(or see
Towers of 2covers of hyperelliptic curves, Preprint
PIMS0112, 2001)
 Nils Bruin,
The arithmetic of Prym varieties in genus 3, 2004. See Electronic Data for supporting electronic resources.
 Nils Bruin and E. Victor Flynn,
Rational Divisors in Rational Divisor Classes, in
Duncan Buell (Ed.): Algorithmic
Number Theory, 6th International Symposium, ANTSVI, Lecture Notes in
Computer Science 3076 Springer 2004, pp. 132  139.
 Nils Bruin,
Visualisation of Sha[2] in Abelian Surfaces, Math. Comp. 73 (2004),
no. 247, 1459  1476. See Electronic Data for supporting
electronic resources.(or see arXiv
preprint, 2002).
 Nils Bruin, Some ternary Diophantine equations of
signature (n,n,2), to appear in Discovering Mathematics with Magma,
W. Bosma, J. Cannon (eds). (Springer), 2003. See Electronic Data
for supporting electronic resources.
 Nils Bruin,
Chabauty methods using elliptic curves. J. Reine Angew. Math. 562 (2003),
27  49.
 N.R. Bruin, Chabauty
methods and covering techniques applied to generalized Fermat
equations, CWI Tract
133, 77 pages, 2002.
 Nils Bruin and Noam D. Elkies, Trinomials ax^7+bx+c and ax^8+bx+c with
Galois Groups of Order 168 and 8*168, in
Claus Fieker, David R. Kohel (Eds.): Algorithmic Number Theory, 5th International Symposium, ANTSV, Lecture Notes in Computer Science 2369 Springer 2002,
pp. 172  188.
 Nils Bruin and E. Victor Flynn,
NCovers of hyperelliptic curves, Math. Proc. Cambridge Philos. Soc. 134 (2003), no. 3, 397405.

Nils Bruin, On powers as sums of two cubes, in
Wieb Bosma (ed.),Algorithmic Number Theory 4th International Symposium,
ANTSIV Leiden,
The Netherlands, July 27, 2000 Proceedings. Springer LNCS 1838.
Associated files: prf335.g,
tor334.mpl,
prf335.mpl.

Nils Bruin, Chabauty methods and covering techniques applied to generalised
Fermat equations, PhDthesis, University of Leiden, 1999.
errata

Nils Bruin,
Chabauty methods using covers on curves of genus 2, Report no.
W9915, University of Leiden.

Nils Bruin,
Chabauty methods using elliptic curves, Report no. 199914, University
of Leiden.

Nils Bruin,
The diophantine equations x^2+y^4=+z^6 and x^2+y^8=z^3, Compositio Math. 118 (1999)
305321. Associated files

H.J. Broersma, N. Bruin, J.L. Hurink, L.E. Meester, S.S. op de Beek, J.H.
Westhuis, Throughput
of ADSL modems, in Proceedings of the 33rd European Study Group with Industry, Syllabus 46: 1126, CWI, Amsterdam (1999).

Nils Bruin,
Generalization of the ABCconjecture, Master Thesis, Leiden University,
1995
For the thesis, the stellingen (dutch) are
also available.
Online Lecture
MSRI makes its workshop lectures available via RealVideo.
During the workshop on arithmetic geometry, December 11  15, I gave a lecture on
explicit covering techniques. If you are far from MSRI, then you may prefer to contact a mirror site.
Slides and talks
Dagstuhl meeting Algorithms and Number Theory, Friday May 18, Visualising Sha[2] in Abelian surfaces, pdf slides, postscript abstract.
Workshop Computational Arithmetic Geometry, 1820 June 2003, Sydney, Prym varieties of curves of genus 3, pdf slides
Software
Algae
Algorithms for arithmetic on elliptic curves over general number fields,
based on KASH. The
main feature is an implementation of 2descent and 2isogenydescent routines
(taking care of even class numbers).
Current version: Algae 0.beta. You can download it as
ell.shar.
You may be interested in reading the README and
the (at this time very crude and perhaps intimidating) documentation.
UPDATE March 27, 2001. Bug fix. The full 2descent code did not compute properly at the infinite primes where ec.gamma is negative. Most people would probably not use this feature anyway, but it is fixed now.
WARNING. Selmer group computations also involve places at infinity.
Precision settings may affect their outcome. The curve $y^2=x^31042011$ is an
example of this phenomenon. With OrderPrec(50), you get a rank bound of 3,
where with OrderPrec(300) the bound is 4, which is the rank of the curve. This
is due to incorrect answers returned by EltCon. The command HEAP(O,
"USE_ELT_PROD_REP", 50) has been recommended by the KASH team and seems to
remedy the problem. It tells KASH to use another, safer, method for computing
EltCon in O. If the package constructs an order automatically, then it sets this
flag for that order.
NEW. A development version for computing 2 (isogeny) Selmer groups of
elliptic curves over number fields in MAGMA is now available as mAlgae.
You might also be interested in
TECC. It is another
elliptic curve package, also based on KASH. It offers functionality mostly
complimentary to ell.g. Unfortunately, the packages do not have
compatible data structures for representing elliptic curves (yet?).
I have not tested using both packages within the same session.
Denis Simon also has a
program for computing 2Selmer groups of elliptic curves over number fields,
based on PARI/GP.
Software related to x^3+y^3=z^p
Software related to my thesis
A precursor to the 2descent program mentioned above.
These come without any documentation or warranty, so you're pretty
much on your own if you want to use them. Should you do so and find them
useful, I'd be glad to hear.
Teaching
 Fall 2005: Calculus for Science students, first year undergraduate course.
(MATH151)
 Summer 2005: Elementary Number Theory, third year undergraduate course.
(MATH342)
 Fall 2004: Introduction to Algebraic Systems, third year undergraduate
course. (MATH332)
 Fall 2004: Algebraic Curves, graduate course.
 Summer 2004: Linear Algebra, fourth year undergraduate course. (MATH438)
 Spring 2004: Algebraic Number Theory, graduate course.
 Spring 2002: Advanced Real Analysis. (MATH320)
 Fall 2001: Elementary Linear Algebra. (MATH232)
Conferences and Workshops
February 4  9, 2007: BIRS workshop on Explicit Methods for Rational Points on Curves
July 9  14, 2006:
CNTA IX in Vancouver
July 5  9, 2004: PIMS Workshop Computational Arithmetic
Geometry
Various other (mathematical) subjects
Maple bug
Be aware that Maple's
simplify
command does not always test for invertibility of x when simplifying
0/x. Waterloo software is aware of this and they have acknowledged
that it is a bug and have promised to repair it in future versions. There
is an example script of this behaviour.
Short example:
simplify(0/(RootOf(x^2+1)^2+1));
evala(0/(RootOf(x^2+1)^2+1));