Frits Beukers

Name : Frits Beukers
Position : Full Professor
Dept. : Mathematics
 Utrecht University
the Netherlands
Room : MI 520
EMail : f.beukers at uu.nl
Phone : +31-30-2531419 (work)

Onderwijs/Education

Ringen en Galoistheorie, een niveau 2 bachelor cursus te geven in Blok 3 en 4, 2012.
Modular Forms, Master Mathematical Sciences, met Sander Dahmen, te geven in Blok 3 en 4, 2012
Algebraische meetkunde, getaltheorie en de mobiele telefoon, slides van de voordracht en hints voor inspiratie.
Lineaire Algebra B, een niveau 1 bachelor cursus in Blok 4, 2012.

Seminars (local and intercity)

Utrecht Seminar on Algebra and Number Theory
Intercity Seminar Number Theory

Recent scientifice preprints/papers

F.Beukers, Consequences of Apery's work on irrationality,
Rencontres Arithmetiques de Caen, zeta(3) irrationnel: les retombees, 1995
D.Zagier, F.Beukers, Lower bounds of heights of points on hypersurfaces
Acta Arithmetica 79(1997), 103-111
F. Beukers Ternary form equations.
J.Number Theory 54 (1995), 113-133
H.P.Schlickewei, F.Beukers, The equation x+y=1 in finitely generated groups
Acta Arithmetica 78(1996), 189-199
F. Beukers, On a sequence of polynomials.
J.pure and applied Algebra 117,118(1997), 97-103
F. Beukers Integral points on cubic surfaces
CRM Proceedings and Lecture Notes 19 (1999), 25-33
F. Beukers The diophantine equation Ax^p+By^q=Cz^r
Duke Math. J. 91 (1998), 61-88
R.Cushman, F.Beukers Zeeman's monotonicity conjecture
Journal of Differential Equations 143(1998), 191-200
T.N.Shorey, R.Tijdeman, F.Beukers Irreducibility of polynomials and arithmetic progressions with equal products of terms
Proc. Conference in honour of A.Schinzel (eds: Gyory, Iwaniec, Urbanowicz) W.de Gruyter 1999, 11-26
J.Sanders, J.P.Wang, F.Beukers One symmetry does not imply integrability
extension of paper which appeared in J.of Differential Equations 146 (1998), 251-260.
F.Beukers On Dwork's accessory parameter problem, Math.Z. 241 (2002), 425-444.
F.Beukers The maximal differential ideal is generated by its invariants, Indag. Math. 11 (2000), 13-18.
C.Smyth, F.Beukers Cyclotomic points on curves (16 pages), Number Theory for the Milennium I, Proceedings of Milennial Conference, Urbana-Champaign 2000, Volume 1, p67-86, A.K.Peters, 2002
R.Cushman, F.Beukers, The complex geometry of the spherical pendulum, Celestial Mechanics, Contemporary Mathematics 292(2002), 47-70. This version is expanded and corrected from the original.
A. van der Waall, F.Beukers Lame equations with algebraic solutions (19 pages), J.Differential Equations 197(2004), 1-25.
F.Beukers, A refined version of the Siegel-Shidlovskii theorem (9 pages, see also href 0405549), Annals of Mathematics 163(2006), 369-379.
With H.Montanus: A compilation of all extremal, semi-stable elliptic fibrations of K3-surfaces and the associated paper titled Explicit calculation of elliptic fibrations of K3-surfaces and their Belyi-maps. Number theory and polynomials, 33--51, London Math. Soc. Lecture Note Ser., 352, Cambridge Univ. Press, Cambridge, 2008.
F.Beukers, Irrationality of p-adic L-values, Acta Mathematica Sinica (English Series) 24, 663-686.
F.Beukers, Unitary monodromy of Lam'e differential operators, Regul. Chaotic Dyn. 12 (2007), no. 6, 630--641.

Recent lectures, semi-scientific publications, course notes

F.Beukers Vakantiecursus 1999: P=NP? (17 pages, in Dutch)
F.Beukers A rational approach to Pi (17 pages), Notes of a lecture held on the occasion of Pi-day on July 5, 2000 in Leiden. Nieuw Archief voor Wiskunde 2000, issue 4.
W.Reinboud, F.Beukers, Snellius versneld (5 pages, in English), Nieuw Archief voor Wiskunde 3(2002), 60-63.
F.Beukers Experimentele Getaltheorie (Vakantiecursus 2001, 19 pages in Dutch)
F.Beukers The Riemann zetafunction and its relatives" Transparencies from a lecture in the Basic Notions series of the Math Colloquium at Utrecht on 13 september 2001.
Oratie gehouden op 22 Oktober 2001.
With R.M. van Luijk, R.Vidunas, A linear algebra problem, Nieuw Archief voor Wiskunde 3 (2002), 139-140.
Gauss hypergeometric functions, Notes from an MRI course given in 1993. Note: MRI stands for MATHEMATICAL RESEARCH INSTITUTE, a Dutch research school in which several universities participate. Each year, advanced and internationally oriented courses are offered.
Hypergeometric functions of one variable Notes from MRI springschool 1999 held in Groningen (organisers M.van der Put, J.Top)
Periods Slides from a lecture held in Luminy (May 11, 2002) about the algebraic independence of periods as expounded in the paper Periods by M.Kontsevich and D.Zagier.
Exploring E-functions, Slides of a lecture held on June 18, 2004, Waterloo, on the occasion of W.D.Brownawell's 60th birthday.
The diophantine equation Ax^p+By^q=Cz^r. Lectures held at Institut Henri Poincare, September 2004.
The limits of reason Studium generale voordracht, 6 april 2005, naar aanleiding van Hoofdstuk 10 van Peter Atkins' boek Galileo's finger. In pdf, in Dutch.
Introduction to the ABC-conjecture, lecture held on September 9, on the occasion of the "ABC-dag" in Leiden. For the notes of the other lectures see the Program
Waar zijn de reele getallen?, Demonstration and pictures given during a lecture before the Nationale Wiskundedagen 2008 on the nature of the real numbers. In particular is R=?Q. The complete presentation can also be downloaded (beware, about 30Mb).
Recurrent sequences coming from Shimura curves, Lecture given on June 3, 2010 in Banff at the occasion of Cam Stewart's 60th birthday.
F.Beukers Diophantische vergelijkingen: een onmogelijke uitdaging (CWI Vakantiecursus 2010, in Dutch)

Getaltheorie voor beginners

This is an introduction (written in Dutch) addressed to newcomers in Number Theory. Complete title:

Frits Beukers,
Getaltheorie voor beginners
Epsilon Uitgaven, Utrecht 1999
ISBN 90-5041-049-9

Apart from an elementary introduction a good number of chapters is devoted to recent developments in elementary number theory. You can find the pdf-file of the Preface and table of contents here. Of course it is a lot of fun to perform experiments on a computer. For this reason I collected some useful software.

Pi

This is an introduction (again in Dutch, about 50 pages) for high school students into the secrets of PI. It is issue 6 of the ZEBRA-series published jointly by Epsilon Uitgaven and the Nederlandse Vereniging van Wiskundeleraren. ISBN-number is 90-5401-062-6. You can find the pdf-file of the introduction and table of contents here. As a result of repeated requests you can find here the answers to the exercises in pdf-format.

Research

My research interests are

Diophantine equations
Hypergeometric functions
Number Theory in general (see Number Theory Web for my colleagues)

Mathematical applets

Plot phase portraits of ODE's
Richard Dawkins' biomorphs
Rubik's cube
2D-projections of a number of 4D regular n-cells
Traveling salesman problem using simulated annealing.
Knapsack problem, an NP-complete problem
Money changing problem, an applet illustrating a workshop at the Nationale Wiskunde Dagen 2001, held at Noordwijkerhout.
Collatz problem or the 3n+1 problem. An applet illustrating this infamous problem. It was written as illustration for an article in Pythagoras-magazine, a mathematical magazine for high school students and other interested people.
Kali, wallpaper design based on the 17 discrete euclidean symmetry groups of the plane.
Proof of Pythagoras' Theorem
Chaotic motion under Lorenz' dynamical system.