California State University, San Marcos

Foundations of mathematics (and related philosophical and historical issues) especially new approaches to the paradoxes ("algorithmic logic"),

Finite fields (especially sets of values of algebraic mappings),

Arithmetical algebraic geometry (and related areas of geometry and algebraic topology).

History of mathematics,

Abstract algebra,

Mathematical logic,

Geometry (including basic algebraic geometry and algebraic topology, as well as the foundations of Euclidean and hyperbolic geometry).

(with Jeff Barrett)
*Computer implication and Curry's paradox*

Journal of Philosophical Logic (2004)

(with Jeff Barrett)
*
Stability and Paradox in Algorithmic Logic*

Journal of Philosophical Logic (2006)

(with Jeff Barrett)
*Abstraction in algorithmic logic*

Journal of Philosophical Logic (2008)

(with Jeff Barrett)
*A note on the physical possibility of ordinal computation*

The British Journal for the Philosophy of Science (2010)

(with Franz Lemmermeyer)

*Counterexamples to the Hasse principle*

American Mathematical Monthly (Fall 2011).

(with Farshid Hajir and Christian Maire)

*Finitely ramified iterated extensions*

International Mathematics Research Notices (2005)

(with Mike Fried and Linda Holt)

*Davenport pairs over finite fields*

Pacific Journal of Mathematics (2004)

*An explicit sign formula for the determinant of cohomology*

Communications in Algebra (1999)

*Total relative displacement of permutations*

Journal of Combinatorial Theory, Series A (1999)

*On value sets of polynomials over a finite field*

Finite Fields and Their Applications (1998)

(with E Okamoto, G. R. Blakely)

*Algebraic properties of permutation polynomials*

IEICE Transactions on Fundamentals of
Electronics, Communications, and Computer Science (1996)

Book: *An arithmetic Riemann-Roch theorem for singular arithmetic surfaces*

Memoirs of the American Mathematical Society (1996)

(with E Okamoto, G. R. Blakely, P. F. Stiller)

*Simple permutation ciphers using permutation polynomials*

International Symposium on Information Theory and Its Applications (1994)

(with E Okamoto, G. R. Blakely, I. Borosh)

*Properties of permutation polynomials*

The 17th Symposium on Information Theory and Its Applications (1994)

*An arithmetic Riemann-Roch theorem for singular arithmetic surfaces*

Doctoral Dissertation, Harvard University (Advisor: Barry Mazur)

*Fermat's Proof*

(An essay about a proof that Fermat did manage to fit in his margin)

*Legendre's Theorem, Legrange's Descent*

(classic results on second order Diophantine equations)

Lagrange (1769),
*Sur la solution
des probl�ms ind�termin�s du second degr�*

(Select pages for now)

Linear Algebra (Modified Moore method approach)

History of Mathematics Handouts

Basic Definitions of Algebra (Reference for second semester algebra)