Gregory G. Smith: Nonnegative sections and sums of squares

A polynomial with real coefficients is nonnegative if it takes on only nonnegative values. For example, any sum of squares is obviously nonnegative.  For a homogeneous polynomial with respect to the standard grading, Hilbert famously characterized when the converse statement hold, i.e. when every nonnegative homogeneous polynomial is a sum of squares.  In this talk, we will examine this converse for line bundles on a totally-real projective subvariety.  In particular, by working with multigraded polynomial rings, we will provide many new examples in which every nonnegative homogeneous polynomial is a sum of squares.