I am a number theorist with interests in arithmetic and
algebraic geometry. I am in particular interested in the study of K3 surfaces,
their Picard groups, groups of automorphisms, ample cone, counting the number
of rational points or curves with bounded height in orbits, and vector heights
and canonical vector heights. Such problems are often related to counting the
number of integer points with bounded height on certain hyperbolic
hypersurfaces, counting lattice points in hyperbolic geometries, or the study
of multi-branch linear trees and their Hausdorff dimensions. I also have
an interest in classical problems in geometry, particularly those involving
constructions with traditional but little studied tools.
Reprints or preprints are available upon request for the
following papers. For the papers that are still to appear, preprints are
available here. Some links are to the journal's website, which may require a
subscription (usually through your institute's library) or password.
Unfortunately, copyright considerations prevent me from 'publishing' my
copies. A pdf viewer (Acrobat Reader) is available, free, from
Adobe. Reviews of these papers can also be accessed through MathSciNet
(again, for those who subscribe). Links to MathSciNet for individual
papers are below, or click here
for the complete list. MathSciNet has links to most (if not all) of these
articles. Click the article link that appears with the review. Many
of these links will follow your own library's proxy.
- A Survey of Classical and
Modern Geometries: with Computer Activities, Prentice Hall, Upper
Saddle River, NJ, 2001, xiv,
for this book.
- Higher dimensional Apollonian packings, revisited, Geometria Dedicata, to appear, DOI: 10.1007/s10711-017-0280-7. Preprint.
- The Apollonian circle packing and ample cones for K3 surfaces, to appear. Preprint.
- The Neron-Tate pairing and elliptic K3 surfaces, to appear. Preprint.
- Lattice points on hyperboloids
of one sheet, New York J. Math, 20, 1253 – 1268 (2014). Paper (journal's site). MR3312058.
- Automorphisms of surfaces in a class of Wheler K3 surfaces
with Picard number 4, Rocky Mountain J.
Math., 46 (2016) no. 2, 399 - 412. Preprint (with different title). MR3529075.
- Orbits of points on certain K3
surfaces, with an appendix by Ronald van Luijk, J.
Number Theory, 131 (3), 578 � 599 (2011). Preprint
- The ample cone for a K3
surface, Canad. J. Math., 63
(3), 481 � 499 (2011). Article (journal�s site).
(pdf, 1.2MB). MR2828530.
- With David McKinnon, K3
surfaces, rational curves, and rational points, J.
Number Theory, 130 (7), 1470 � 1479 (2010). Preprint
(via arXiv). MR2645231
- With Ronald van Luijk, K3
surfaces with Picard number three and canonical heights, Math. Comp., 76 (259), 1493 --
1498 (2007). Paper
(pdf -- journal's site). MR2299785(2008e:14055).
- Fractals and eigenvalues of
the Laplacian on certain noncompact surfaces, Experiment. Math., 15
(1), 33 � 42 (2006). Paper
(journal's site). MR222933(2007a:58033).
- The Markoff-Hurwitz equations
over number fields, Rocky Mountain J.
Math., 35 (3), 695 � 712 (2005). Paper (pdf �
reprinted with permission). MR
- With Kensaku Umeda, The
asymptotic growth of integer solutions to the Rosenberger equations,
Bull. Austral. Math.
Soc., 69, 481 � 497 (2004). Paper (pdf).
- Canonical vector heights on K3
surfaces with Picard number three - an argument for non-existence, Math. Comp., 73, 2019 �
2025 (2004). Paper
journal's site). MR2005e:14058.
- Canonical vector heights on
algebraic K3 surfaces with Picard number two, Canad. Math. Bull., 46 (4),
495 � 508 (2003). Paper
(pdf � journal's site). MR 2004i:11065.
- Orbits of curves on certain K3
(2), 115 � 134 (2003). Abstract(pdf).
(via journal's site). MR 2004d:11050.
- Constructions using a compass
and twice-notched straightedge, Amer. Math. Monthly, 109 (2),
151 � 164 (2002). Paper
(pdf � reprinted with permission). Erratum(pdf).
- Rational curves with zero self
intersection on certain K3 surfaces, Fifth Conference of the
Canadian Number Theory Association, edited by K. Williams and R. Gupta, CRM
Proceedings and Lecture Notes, 19, 1 � 6 (1999). Abstract
- Rational curves on K3 surfaces
in PxPxP, Proc.
Amer. Math. Soc., 126
(3), 637 � 644 (1998). Paper
(pdf � journal's site). MR 98h:14028.
- The exponent for the
Markoff-Hurwitz equations, Pacific J. Math., 182 (1),
1 � 21 (1998). Paper (pdf �
journal's site). MR 99e:11035.
- Rational Points on K3 Surfaces
in PxPxP, Math.
Ann., 305, 541 � 558 (1996). Abstract
and remark (pdf). MR 97g:14020.
- On the Unicity Conjecture for
Markoff Numbers, Canad.
Math. Bull., 39 (1), 3 � 9 (1996). Abstract
and remark (pdf). MR 97d:11110.
- Products of Consecutive
Integers and The Markoff Equation, Aequationes
Math, 51, 129 � 136 (1996). Abstract
- Asymptotic Growth of
Markoff-Hurwitz Numbers, Compositio Math,
94, 1 � 18 (1994). Abstract,
erratum, and remark (pdf). MR 95i:11025.
- Integral Solutions of
Markoff-Hurwitz Equations, J.
Number Theory, 49, 27 � 44 (1994). Paper
(pdf � journal's site). MR 95g:11066.
- The Hurwitz Equations, Number
Theory with an Emphasis on the Markoff Spectrum, edited by A.
Pollington and W. Moran, Lecture Notes in Pure and Applied Mathematics, 147,
Marcel Dekker, New York, NY, 1 � 8 (1993). Abstract
- The ample cone and orbits of
curves on K3 surfaces, to appear. Preprint
- The 43rd International
Mathematical Olympiad, Leader�s Report, Canad. Math. Soc. Notes,
(7), 12 � 13 (2002). Report(pdf).
- Snapshots of the �99 Canadian
IMO team, featured in R.E. Woodrow�s �The Olympiad Corner No. 204,�
a regular feature in Crux
Mathematicorum with Mathematical Mayhem, 26
(2), 65 � 69 (2000).
- Hurwitz equation, Encyclopaedia
of Mathematics, http://eom.springer.de,
Springer-Verlag, 2002. Originally in Encyclopaedia of Mathematics
on CD-ROM, Kluwer Academic Publishers, 1997. Article local site (pdf),
I have transitioned to the use of Prosper to produce talks,
which can be easily posted (the output is pdf). Please keep in mind that
many details, caveats, and clarifications are presented orally, so if something
in these slides are of interest, please consult the appropriate preprints or
reprints for exact statements. Update 2013: I have been hesitant
to post any more talks, since so much is lost without the oral
presentation. I have left the following on, since it covers much of my
research from 2003 to the present.
- The ample cone and orbits of
curves and rational points on K3 surfaces, Analytic
Methods for Diophantine Equations, Banff International Research Station,
May 16th, 2006, Banff, Canada. Slides
Articles by my students:
- While Patrick Hummel was a high school student in Las
Vegas, he asked me for a project. I gave him one. I
didn't hear from him for quite some time, so I thought that maybe he
didn't think it was all that interesting. He contacted me again when
he had solved it. I suggested he write it up, and gave him a very
quick tutorial on LaTeX. Again, he didn't come back until after he
had written up his results. I helped him with the diagrams and
suggested he submit it to the Pi Mu Epsilon journal. It won a 2003
Richard V. Andree Award, given for those articles judged to be among the
best-written articles in the Journal for the year. Patrick's second
article was written under the guidance of Dinakar Ramakrishnan while he
was on a Caltech Summer Undergraduate Research Fellowship. Patrick
did graduated from CalTech, earned a Ph.D. in Economics from Stanford, and
last I heard, works for Yahoo.
- Patrick Hummel, Solid
constructions using ellipses, The PME Journal, 11(8),
429 � 435 (2003). Article(pdf).
- Patrick Hummel, On
consecutive quadratic non-residues: a conjecture of Issai Schur,
J. Number Theory, 103 (2003), 257 � 266.
- Kensaku Umeda earned his masters degree under my
supervision. His results are written up in the following paper:
- Arthur Baragar and Kensaku Umeda, The asymptotic growth of integer solutions to the Rosenberger
Austral. Math. Soc., 69, 481 � 497 (2004). Paper (pdf).