Commutative algebra and algebraic geometry form a deeply interwoven field that investigates the structure of polynomial rings, their ideals, and the geometric objects defined by these algebraic sets.

Several fields of mathematics have developed in total isolation, using their own 'undecipherable' coded languages. Mathematicians now present 'big algebras,' a two-way mathematical 'dictionary' ...

Algebraic geometry is a branch of mathematics which, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of polynomials.

Algebraic geometry; commutative algebra; homological algebra; algebraic K-theory. My research has been mainly in algebraic geometry, with an abiding interest in the study of algebraic cycles, ...

Transactions of the American Mathematical Society, Vol. 359, No. 2 (Feb., 2007), pp. 827-857 (31 pages) We discuss some of the basic ideas of Galois theory for commutative S-algebras originally ...

Tropical algebraic geometry is a young but rapidly developing field that translates intricate algebraic and geometric problems into combinatorial ones. In this setting, algebraic varieties (that is, ...

This project builds on my work with Andrew Berget (in "The external activity complex of a pair of matroids"). That work resolved a conjecture from 2005 by Speyer, on the positivity of a certain ...