Time: 9h, Wednesday, March 4, 2020

Location: Room 611-612,  Building A6, Institute of Mathematics

Abstract: The strong Lefschetz property for an Artinian graded algebra A over a field K simply says that for a general linear form L, the multiplication ×L^s: [A]_i ---> [A]_(i+s) has maximal rank in every degree i and every positive integer s. In particular, if the multiplication by a general linear form ×L : [A]_i ---> [A]_(i+1) has maximal rank in every degree i, then A is said to have the weak Lefschetz property. At first glance this might seem to be a simple problem of linear algebra. However, determining which graded Artinian K-algebras have the weak and/or strong Lefschetz property is notoriously difficult and it is strongly connected to many topics in algebraic geometry, commutative algebra and combinatorics. Some of these connections are quite surprising and still not completely understood, and much work remains to be done. In this talk, I will give an overview of known results on the weak and strong Lefschetz properties, with an emphasis on the approaches and tools that have been used. I also discuss open problems.