Alexander Borisov's Research Page


I have broad research interests. My primary interests lie in algebraic geometry and number theory, which often lead me to other topics. I feel that the most beautiful mathematics often lies at the crossroads of different subjects. I think that the most important mathematical ideas are not technical, even though they often need to be backed by some heavy technique. Here is the link to my vita. My publications are listed below, roughly classified by the subject matter.

Publications and Preprints:

(Some of the texts available here are not the final versions and/or differ slightly from the published papers.)

Applications of Harmonic Analysis to Number Theory:

Positive positive-definite functions and measures on locally compact abelian groups, preprint.

Convolution structures and arithmetic cohomology, Compositio Math., 136 (2003), no. 3, 237--254.

Iterations of Polynomial Maps, with Applications to Group Theory:

Polynomial maps over finite fields and residual finiteness of mapping tori of group endomorphisms, preprint (2003) (with M. Sapir). Submitted to Inventiones Math.

Toric Varieties and Lattice Polytopes:

Quotient singularities, integer ratios of factorials and the Riemann Hypothesis, preprint.

Convex lattice polytopes and cones with few lattice points inside, from a birational geometry viewpoint, preprint.

On classification of toric singularities, Algebraic Geometry. 9. J. Math. Sci. (New York) 94 (1999), no. 1, 1111--1113.

Minimal discrepancies of toric singularities, Manuscripta Math. 92 ( 1997 ), no. 1, 33 -- 45.

Singular toric Fano varieties, Russian Acad. Sci. Sb. Math., 75 (1993), no. 1, 277--283 (with L. Borisov).

Three-dimensional toric Fano varieties with terminal singularities, preprint in Russian (with L. Borisov).

Fano Varieties:

Boundedness of Fano threefolds with log-terminal singularities of given index, J. Math. Sci. Univ. Tokyo, 8 (2001), no. 2, 329--342.

Boundedness theorem for Fano log-threefolds, J. Algebraic Geom. 5 (1996), no. 1, 119--133.

Rational Polynomials:

On some polynomials allegedly related to the abc conjecture, Acta Arithmetica, 84 (1998), no. 2, 109--128.

Classes of polynomials having only one non-cyclotomic irreducible factor, Acta Arithmetica, 90 (1999), no. 2, 121--153 (with M. Filaseta, T.-Y. Lam, and O. Trifonov)

On a Question of Craven and a Theorem of Belyi, Proceedings of AMS, to appear.