Date of Completion

5-4-2016

Embargo Period

5-4-2016

Major Advisor

Jerzy Weyman

Associate Advisor

Ralf Schiffler

Associate Advisor

Kyu-Hwan Lee

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

In this thesis we investigate an important singularity invariant, Bernstein-Sato polynomials, also called b-functions. Together with the classical notion of the b-function (of one variable) of a single polynomial, we also consider more general ones, such as the b-function of several variables (of several polynomials) and Bernstein-Sato polynomials (of one variable) of arbitrary varieties. We propose several techniques for their computation in various equivariant settings. The main applications will belong to the quiver setting, namely (semi-)invariant polynomials (resp. nullcones, orbit closures) of quivers.

We give a formula relating b-functions of semi-invariants corresponding to each other under castling transforms (or reflection functors). This, in particular, allows the computation of the b-functions for all Dynkin quivers, and also extended Dynkin quivers with prehomogeneous dimension vectors.

We give another computational technique using slices, that is efficient for semi-invariants with "small weights". Among other uses of slices, we give a way of finding locally semi-simple representations and an easy rule for the determination of the canonical decomposition for type D quivers.

We compute the Bernstein-Sato polynomial of the ideal generated by maximal minors (which is a type A_2 orbit closure) and sub-maximal Pfaffians. We settle the Strong Monodromy Conjecture in these cases.

Using our computational results, we give various results on the geometry of orbit closures of quivers. In particular, we prove that codimension 1 orbit closures of Dynkin quivers have rational singularities. We prove the same result for extended Dynkin quivers for dimension vectors that are not "too small". We give results on the reduced property of the nullcone of quivers and establish a connection between b-functions of several variables and the Bernstein-Sato polynomial of varieties. With these we give some criteria for rational singularities in higher codimensions.

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