Ilya Smirnov: Two questions about monomial ideals

Abstract:
Monomial ideals often allow to bring combinatorial insight in commutative algebra. I will provide necessary definitions and further background information for two questions. The first question comes from my joint work with Craig Huneke and Javid Validashti on an asymptotic refinement of Lech's inequality. The original inequality can be reduced to monomial ideals and a simple geometric picture, while our refinement seems to be more subtle. Thesecond question is due to Craig Huneke who asked what monomial ideals are strongly Golod. If time permits, I will survey what is known about strongly Golod ideals.