Queen's University

Dr. Gregory G. Smith

Professor, Department of Mathematics and Statistics
MATHEMATICS AND STATISTICS, LIFE & PHYSICAL SCIENCES

Autobiography

I am a Professor in the Department of Mathematics and Statistics at Queen's University.  My research interests include algebraic geometry, commutative algebra, and combinatorics.  I also have taught a wide range of courses to undergraduate and graduate students, including Linear Algebra, Elementary Geometry, and Computational Commutative Algebra.  I am honoured to be a recipient of an American Mathematical Society Project NExT Fellowship (2001), a Clay Mathematics Institute Liftoff Fellowship (2001), the Andre-Aisenstadt Prize (2007), and the Coxeter-James Prize (2012).

Over the next few years, I expect to take on a number of graduate students. Although I am primarily interested in PhD students, I like to have students start at the Master's level. Highly qualified applicants are encouraged to contact me directly.


 

Most Recent Project

Toric vector bundles and parliaments of polytopes

PolytopesWe introduce a collection of convex polytopes associated to a torus-equivariant vector bundle on a smooth complete toric variety. We show that the lattice points in these polytopes correspond to generators for the space of global sections and we relate edges to jets. Using the polytopes, we also exhibit toric vector bundles that are ample but not globally generated, and toric vector bundles that are ample and globally generated but not very ample.

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Other Projects

  • Sums of squares and varieties of minimal degree

    Let X be a real nondegenerate projective subvariety such that its set of real points is Zariski dense. We prove that every real quadratic form that is nonnegative on X is a sum of squares of linear forms if and only if X is a variety of minimal degree. This substantially extends Hilbert's celebrated characterization of equality between nonnegative forms and sums of squares. We obtain a complete list for the cases of equality and also a classification of the lattice polytopes Q for which every nonnegative Laurent polynomial with support contained in 2Q is a sum of squares

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