Ronald M. van Luijk

Theses

• On perfect Cuboids, my undergraduate thesis.
• Rational points on K3 surfaces, my graduate thesis.

Papers

• Nontrivial elements of Sha explained through K3 surfaces and accompanying files, Math. Comp. 78 (2009), 441-483. (joint work with Adam Logan, also available at arXiv:0706.0541).
• Non-Euclidean Pythagorean triples, a problem of Euler, and rational points on K3 surfaces and in PDF, Mathematical Intelligencer, Vol. 30, No. 4 (2008), 4--10 (joint work with Robin Hartshorne, also vailable at arXiv:math.NT/0606700).
• The diameter of the circumcircle of a Heron triangle and in PDF, Elemente der Mathematik, Vol. 63, Issue 3 (2008), p. 118-121.
• K3 surfaces with Picard number one and infinitely many rational points and in PDF, Algebra and Number Theory, Vol. 1, No. 1 (2007), 1-15.
• K3 surfaces with Picard number three and canonical vector heights and in PDF, Mathematics of Computation, Volume 76 (2007), 1493-1498 (joint work with Arthur Baragar, different version available at arXiv:math.AG/0602166 ).
• An elliptic K3 surface associated to Heron triangles and in PDF, Journal of Number Theory, Volume 123 (2007), 92-119, also available at arXiv:math.AG/0411606).
• A K3 surface associated to certain integral matrices with integral eigenvalues and in PDF Canadian Mathematical Bulletin, Volume 49 (4), 2006, 560-577, also available at arXiv:math.AG/0411600).
• Quartic K3 surfaces without nontrivial automorphisms and in PDF, Mathematical Research Letters (MRL), Volume 13 (2006), Issue 3, 423-439, also available at arXiv:math.AG/0511542).
• A linear algebra exercise, with F. Beukers and R. Vidunas, Nieuw Archief voor Wiskunde (NAW), Juni 2002, 139-140.

• Cubics points on cubic curves and the Brauer-Manin obstruction for K3 surfaces and accompanying files, also available at arXiv:0708.2752.
• Density of rational points on diagonal quartic surfaces (joint with A. Logan and D. McKinnon).
• On character varieties on two-bridge knot groups (joint with M. Macasieb and K. Petersen)
• Two-coverings of Jacobians of curves of genus two (joint with E.V. Flynn and D. Testa)
• Lines on Fermat surfaces and accompanying files, also available at arXiv:0812.2377 (joint with M. Schütt and T. Shioda)

Slides

• An elliptic K3 surface associated to Heron triangles (Colloquium talk at Queen's University)
• K3 surfaces with Picard number one and infinitely many rational points (number theory seminar at Queen's University)
• Explicit computations on the Manin conjectures (workshop "Explicit methods in number theory" at Oberwolfach)
• Toward an explicit 2-descent on the Jacobian of a generic curve of genus 2 (AMS meeting San Antonio, Jan. 2006)
• Nontrivial Sha for curves of genus 2 arising from K3 surfaces (UC San Diego, May 2006)
• Manin conjectures for K3 surfaces (Banff, Canada, May 2006)
• Introduction Number Theory in Cryptography (Bogota, Colombia, September 2006)
• Cubic points on cubic curves and the Brauer-Manin obstruction on K3 surfaces (Bristol, UK, July 2007)

Pictures

• A K3 surface with infinitely many rational points (The cover of my thesis)

Various

• Wiskunde Olympiade with Jan van de Craats and Thijs Notenboom, Nieuw Archief voor Wiskunde, December 2000, 448-450 (Dutch).
• Hex, dots and boxes with Sander van Rijnswou, Nieuw Archief voor Wiskunde, December 2001, 358-361 (Dutch).
• IMO95 puzzles, Crux Mathematicorum, No. 2, Vol. 22, March 1996, 70.