Syzygies and Hilbert Functions

Examines the invariant of Castelnuovo-Mumford regularity, blowup algebras, and bigraded rings. This work outlines the status of two conjectures: the lex-plus-power (LPP) conjecture and the multiplicity conjecture. It considers minimal free resolutions of integral subschemes and of equidimensional Cohen-Macaulay subschemes of small degree.