Mathematics Department

Dr JR Silvester Department of Mathematics King's College London Strand, London WC2R 2LS United Kingdom Room 413A, Strand Building Tel: +44-(0)20-7848 2864 (direct) Tel: +44-(0)20-7848 2217 (general office) Fax: +44-(0)20-7848 2017 E-mail: jrs@kcl.ac.uk

Research Interests

I am interested in algebra, geometry and number theory, mostly rings and general linear groups, polynomial algebras, and questions to do with the Jacobian conjecture. However, since becoming admissions tutor I have not found enough time for serious research, so my recent publications are mostly short articles with which I hope to entertain the readers of the Mathematical Gazette.

Recent and forthcoming publications

• The Political Frogs, Maths Gazette 81, 58-63 (1997)
• Decimal deja vu, Maths Gazette 83, 453-463 (1999)
• Ceva = (Menelaus)², Maths Gazette 84, 268-271 (2000)
• Determinants of Block Matrices, Maths Gazette 84, 460-467 (2000)
• Geometry Ancient & Modern, Oxford University Press (2001)
• (with A.D.Barnard) Circle theorems and a property of the (2,3,4) triangle, Maths Gazette 85, 312-316 (2001)
• Painting by numbers (or Polyominoes revisited), Maths Gazette 86, 68-72 (2002)
• All done with mirrors, UKMT newsletter (March 2002)
• (with A.D.Barnard) Keep it simple!, Maths in School 32 No.3, 24 (2003).
• Letter to the Editor (re: how many presents did my true love send to me?), Maths Gazette 87, 343 (2003).
• (with K.R.McLean) Loops of regular 2ⁿ-gons, Maths Gazette 87, 512-513 (2003).
• Factorial Factors, Maths Gazette 88, 119-123 (2004).
• Extensions of a theorem of Van Aubel, Maths Gazette 90, 2-12 (2006).
• The r-subsequences of the Fibonacci sequence, Maths Gazette 90, 263-266 (2006).
• Feedback to a feedback by Michael de Villiers (in the November 2006 Mathematical Gazette, pp 517-8), to appear, Mathematical Gazette, July 2007.

Talks

• Talk given as one of the LMS Poplular Lectures 2002, and elsewhere:
  • Geometry Ancient & Modern. More details.
• Talk given at the Mathematical Association Conference at Norwich, Easter 2003, and also at the UKMT Teachers Meetings, Greenwich, June 2002 and Nottingham, June 2003:
  • An "Areal" View of Geometry: here are the questions (PDF) and slides (PDF) for the talk, and the Geometer's Sketchpad file to download, for those with the software to run it.
• Talk given at the Mathematical Association Conference at Norwich, Easter 2003, and also at the UKMT Teachers Meeting, Greenwich, June 2003:
  • Factorial Factors: slides (PDF). Pages 21-24 have been added since I gave the talk. There is some overlap with the above article with the same title.
• Talk given for the Saturday Advanced Mathematics class at Nottingham University, and (in various versions) also for Imperial College, Wilson's School, North London Collegiate School, UKMT at Trinity College Cambridge, Summer School at KCL, and Lord Wandsworth College (February - September 2004), St Benedict's School (June 2005):
  • All Tilings Great and Small. Here are the sets of slides 1, 2 (PDF) from the talk , plus the Geometer's Sketchpad files to download (for those with the software to run them): regular and semi-regular tilings, impossible cases, hinged tilings, Penrose tilings, Voderberg tilings, complex numbers, spiral tilings, tromino tilings. Tilings with rotational symmetry: 5-fold, 6-fold, 7-fold, 12-fold. Link to Java applet to play Penrose tiling.
• Talk given in various versions/with various titles as it evolved over the period February 2005 - March 2007, to The Invariants (Oxford), Imperial College, Wilson's School, The Archimedeans (Cambridge), The Invariants (again), King's College London, Wilson's School (again), as a Keynote Talk to the Mathematical Association's annual conference 2006, in Loughborough, and to Solihull School, and Thames Valley University:
  • Computer vs pencils and paper or Computers: can we trust them? or Pendulums, Pencils, and the Poristic Polygons of Poncelet.
    Files from the talks (Sketchpad unless it says otherwise): harmonic series (Maple), exponential series (Excel), 64 = 65, trailer, Poncelet's porism, sextics, massive Maple calculation, Greenhill point, link to ETC, pendulums, Java pendulums applet, coaxal circles, pencils, group?, Euclidean proof of Poncelet, inversion proof, elliptic curve proof, nine point circle example.
• Talk given to various teachers' meetings at King's College, at Roedean School and Exeter University; also St Benedict's School, St Olave's Grammar School, St Francis Xavier College, St Swithun's School, and Wilson's School, March 2006 - March 2007:
  • Primes and Polygons.
    Sketchpad files from the talk: ruler and compasses constructions, Mascheroni constructions, regular pentagon, regular 17-gon.
    PDF file: Notes on Mersenne and Fermat primes.
• Talk given at the Mathematical Association's annual conference at Keele, 2007:
  • A painless introduction to elliptic curves: pdf file, Sketchpad file from the talk.
• Talk given to the teachers' conference at King's College, April 2007:
  • Geometry in the classroom: pdf file, Sketchpad file from the talk. Link to Euclid's Elements online.