Research Interests:

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.  

Grants:

• NSA H98230-08-1-0022, �Arithmetic on K3 surfaces and related problems, 2008 -- 2010.
• NSF DMS-0403686, "Arithmetic on K3 surfaces and related problems," 2004 -- 2008.

Publications:

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. 

Book:

• A Survey of Classical and Modern Geometries: with Computer Activities, Prentice Hall, Upper Saddle River, NJ, 2001, xiv, 370.  Website for this book.

Research Articles:

1. Higher dimensional Apollonian packings, revisited, Geometria Dedicata, to appear, DOI: 10.1007/s10711-017-0280-7. Preprint.
2. The Apollonian circle packing and ample cones for K3 surfaces, to appear. Preprint.
3. The Neron-Tate pairing and elliptic K3 surfaces, to appear. Preprint.
4. Lattice points on hyperboloids of one sheet, New York J. Math, 20, 1253 – 1268 (2014). Paper (journal's site). MR3312058.
5. 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.
6. Orbits of points on certain K3 surfaces, with an appendix by Ronald van Luijk, J. Number Theory, 131 (3), 578 � 599 (2011).  Preprint  MR2753271
7. The ample cone for a K3 surface, Canad. J. Math., 63 (3), 481 � 499 (2011).  Article (journal�s site). Preprint (pdf, 1.2MB).  MR2828530.
8. With David McKinnon, K3 surfaces, rational curves, and rational points, J. Number Theory, 130 (7), 1470 � 1479 (2010).  Preprint (via arXiv).  MR2645231
9. 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).
10. Fractals and eigenvalues of the Laplacian on certain noncompact surfaces, Experiment. Math., 15 (1), 33 � 42 (2006).  Paper (journal's site).  MR222933(2007a:58033).
11. The Markoff-Hurwitz equations over number fields, Rocky Mountain J. Math., 35 (3), 695 � 712 (2005).  Paper (pdf � reprinted with permission).  MR 2150305(2006j:11043).
12. With Kensaku Umeda, The asymptotic growth of integer solutions to the Rosenberger equations, Bull. Austral. Math. Soc., 69, 481 � 497 (2004).  Paper (pdf).  MR 2005e:11037. 
13. Canonical vector heights on K3 surfaces with Picard number three - an argument for non-existence, Math. Comp., 73, 2019 � 2025 (2004). Paper (pdf �  journal's site).  MR2005e:14058.  Erratum.
14. 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.
15. Orbits of curves on certain K3 surfaces, Compositio Math, 137 (2), 115 � 134 (2003). Abstract(pdf).  Paper (via journal's site).  MR 2004d:11050.
16. Constructions using a compass and twice-notched straightedge, Amer. Math. Monthly, 109 (2), 151 � 164 (2002).  Paper (pdf � reprinted with permission). Erratum(pdf). MR 2003d:51015.
17. 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 (pdf). MR 2000b:14068.
18. Rational curves on K3 surfaces in PxPxP, Proc. Amer. Math. Soc., 126 (3), 637 � 644 (1998). Paper (pdf � journal's site). MR 98h:14028.
19. The exponent for the Markoff-Hurwitz equations, Pacific J. Math., 182 (1), 1 � 21 (1998).  Paper (pdf � journal's site). MR 99e:11035.
20. Rational Points on K3 Surfaces in PxPxP, Math. Ann., 305, 541 � 558 (1996).  Abstract and remark (pdf). MR 97g:14020.
21. On the Unicity Conjecture for Markoff Numbers, Canad. Math. Bull., 39 (1), 3 � 9 (1996).  Abstract and remark (pdf).  MR 97d:11110.
22. Products of Consecutive Integers and The Markoff Equation, Aequationes Math, 51, 129 � 136 (1996).  Abstract (pdf). MR 97b:11037.
23. Asymptotic Growth of Markoff-Hurwitz Numbers, Compositio Math, 94, 1 � 18 (1994).  Abstract, erratum, and remark (pdf).  MR 95i:11025.
24. Integral Solutions of Markoff-Hurwitz Equations, J. Number Theory, 49, 27 � 44 (1994).  Paper (pdf � journal's site).  MR 95g:11066.
25. 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 (pdf). MR 94e:11071.
26. The ample cone and orbits of curves on K3 surfaces, to appear.   Preprint (pdf, 1.6MB).

Invited Contributions:

1. The 43rd International Mathematical Olympiad, Leader�s Report, Canad. Math. Soc. Notes, 34 (7), 12 � 13 (2002).  Report(pdf).
2. 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).
3. 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), publisher's website. 

Talks:

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.

1. 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 (pdf, 2.5Mb).

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.   
1. Patrick Hummel, Solid constructions using ellipses, The PME Journal, 11(8), 429 � 435 (2003).  Article(pdf). 
2. 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:
1. Arthur Baragar and Kensaku Umeda, The asymptotic growth of integer solutions to the Rosenberger equations, Bull. Austral. Math. Soc., 69, 481 � 497 (2004).  Paper (pdf).  MR 2005e:11037.