Seminar

Grobner bases for twisted commutative algebras

Speaker: Steven Sam

3:45 PM

Recent work of Draisma-Kuttler, Snowden, and Church-Ellenberg-Farb in commutative algebra, algebraic geometry, and homological algebra, seeks to understand stability results (say, of some invariant of a sequence of algebraic objects) as the finite generation of some other algebraic object.

Examples include the Delta-modules in the study of free resolutions of Segre embeddings and FI-modules in the study of cohomology of configuration spaces. This finite generation often reduces to establishing the Noetherian property: subobjects of finitely generated objects are again finitely generated. I will discuss the situation of modules over twisted commutative algebras, which realize some of these topics as special cases. I will introduce a Grobner basis theory and show how it proves these Noetherian results. The talk will be mostly combinatorial and I will suggest some open problems. This is based on joint work with Andrew Snowden.

Smoothing of limit linear series on metrized complexes of algebraic curves.

Speaker: Madhusudan Manjunath

5:00 PM

The theory of limit linear series on curves of compact type (reducible curves whose dual graph is a tree) was introduced by Eisenbud and Harris in

1986 and this theory has several applications to algebraic curves. This theory has recently been generalized  to objects called ``metrized complexes of curves" by Amini and Baker. A metrized complex of curves is essentially a metric graph with algebraic curves plugged into the vertices of this metric graph. Eisenbud and Harris showed that any limit $g^1_d$ on a curve of compact type can be smoothed. We study the question of smoothing a limit $g^1_d$ on a metrized complex.  We provide an effective characterization of a smoothable limit $g^1_d$ on a metrized complex and the talk will include examples demonstrating this characterization.  This is ongoing work with Matthew Baker and Luo Ye.