Emeritus professor of Mathematics at the University of Georgia.
My research interests are in Capacity Theory, Arithmetic Geometry, Arithmetic Dynamics, Arithmetic aspects of Mathematical Logic, Algebraic Number Theory, Algorithms, and Computational Mathematics.
I have graduated eight doctoral students and four masters students:
· (Doctoral): Steve Donnelly, Milton Nash, Charles Pooh, Daeshik Park, Zubeyir Cinkir, Nathan Walters, John Doyle, and Kenneth Jacobs.
· (Masters): Pongsak Ajjimarangsee, Chandra French, Mee Seong Im, Renee Canfield.
Here is my current Vita.
I have written four research monographs:
· Capacity theory on algebraic curves, Lecture Notes in Mathematics 1378, Springer-Verlag, New York (1989), 437 pp.
· Existence of the sectional capacity (with C. F. Lau and R. Varley), Memoires of the American Mathematical Society, vol 145, no. 690 (2000), 130 pp.
· Potential Theory and Dynamics on the Berkovich Projective Line (with M. Baker), volume 159 in the AMS Surveys and Monographs Series (2010), 428 pp. (The file is identical to the published text; with the permission of the AMS, it is being made available online a year after its publication.)
· Capacity Theory with Local Rationality: The Strong Fekete-Szego theorem on Curves, volume 193 in the AMS Surveys and Monographs Series (2013), 437 pp. (the link is to an early draft of the book, not to the final one published)
Papers concerning primality testing:
· On distinguishing prime numbers from composites (with L. Adleman and C. Pomerance), Annals of Mathematics 117 (1983), 173-206.
· Recent advances in primality testing, Notices of the AMS (August 1983), 475-477.
Papers concerning logic and decision procedures:
· Undecidability and defininability in the theory of global fields, Trans. AMS 262 (1981), 195-217.
· Arithmetic over the ring of all algebraic integers, J. Reine Angew. Math 368 (1986), 127-133.
Papers concerning arithmetic capacity theory:
· Capacity theory on curves and canonical heights, Sem. Group d'etude d'analyse ultrametrique, 12e annee, 1984/85, no. 22
· The capacity pairing (with T. Chinburg), J. Reine Angew. Math. 434 (1993), 1-44.
· On the relation between Cantors capacity and Chinburg's sectional capacity, Duke Mathematical Journal 70 (1993), 517-574.
· Arithmetic capacities on PP^N (with C.F. Lau), Math. Zeit. 215 (1994), 533-560.
· An intersection pairing for curves with analytic contributions from nonarchimedean places , in: Canadian Mathematical Society Conference Proceedings 15 (1995), American Mathematical Society, 324-327.
· A Fekete-Szegö theorem with splitting conditions I, Acta Arithmetica 93 (2000), 99-116.
· A Fekete-Szegö theorem with splitting conditions II, Acta Arithmetica 103 (2002), 347-410.
· Capacity theory and arithmetic intersection theory (with T. Chinburg and C.F. Lau), Duke Mathematical Journal 117 (2003), 229-285.
Papers involving applications of arithmetic capacity to classical potential theory:
· A Robin formula for the Fekete-Leja transfinite diameter, Math. Annalen 337 (2007), 329-338.
· Transfinite diameter and the resultant (with L. DeMarco), J. Reine Angew. Math 611 (2007), 145-161.
Papers in arithmetic geometry:
· A formula for the grössencharacter of a parametrized elliptic curve, J. Number Theory 17 (1983), 389-402.
· On the grössencharacter of an abelian variety in a parametrized family, Trans. A.M.S. 276 (1983), 213-233.
· The well-adjusted models theorem over Dedekind Rings (with T. Chinburg), in: Arithmetic Algebraic Geometry, van der Geer, Oort, and Steenbring, eds.; Birkhauser, Boston (1991), 3-24.
· On Bilu's equdistribution theorem, in : Contemporary Mathematics 23 (1999), T. Branson, ed. American Mathematical Society, 159-166.
· A finiteness property of torsion points (with M. Baker and S. Ih), Algebra and Number Theory 2 (2008), 217-248.
Papers concerning arithmetic dynamics:
· Harmonic analysis on metrized graphs (with M. Baker), Canadian Journal of Mathematics 59 (2007), 225-275.
· Equidistribution of small points on curves, rational dynamics, and potential theory(with M. Baker), Annales d l'Institut Fourier 56 (2006), 625-688.
· Configuration of the Crucial Set for a Quadratic Rational Map (with J. Doyle and K. Jacobs), Research in Number Theory 2 (2016), 1-16.
· The Minimal Resultant Locus, Acta Arithmetica 169 (2015), 251-289.
· The Geometry of the Minimal Resultant Locus, to appear in Algebra and Number Theory
Papers in algebraic number theory:
· Zeros of p-adic exponential polynomials II (with A.J. van der Poorten), J. London Math. Soc 36 (1987), 1-15.
· Remarks on generalized power sums (with A.J. van der Poorten), Bull. Austral. Math. Soc. 36 (1987), 311-329.
· A note on the Hadamard kth root of a rational function (with A.J. van der Poorten), J. Austral. Math. Soc. 43 (1987), 324-327.
· Notes on van der Poortens proof of the Hadamard quotient theorem I, Sem. Th. Numbers Paris (1986-1987), Progress in Mathematics 75, 349-382. Birkhauser, Boston.
· Notes on van der Poortens proof of the Hadamard quotient theorem II, Sem. Th. Numbers Paris (1986-1987), Progress in Mathematics 75, 383-409. Birkhauser, Boston.
Papers involving numerical computations:
· Primes in arithmetic progressions (with O. Ramaré), Math. Comp. 65 (1996), 397-425.