Algebra Seminar
Algebra Seminar
Algebra Seminar
Title: The Birational Geometry of K-Moduli Spaces
Abstract: K-stability is a rapidly developing theory that allows one to construct moduli spaces for Fano varieties. In all known examples, K-moduli spaces are uniruled, so their Kodaira dimensions are negative infinity. In this talk we will describe components of K-Moduli spaces which are birational to M_g, in particular they have maximal Kodaira dimension when g is sufficiently large. This component parameterizes certain moduli spaces of vector bundles on smooth curves, and the main difficulty is to show that these moduli spaces are K-stable. To establish this we require good understanding of their toric degenerations.
Algebra Seminar
Title: Computing Free Resolutions of OI-Modules
Abstract: Free resolutions are a powerful tool in commutative and homological algebra. Much of the structure of a module can be encoded in a free resolution. For example, in the case of graded modules, free resolutions can be used to study Betti numbers and Hilbert functions. Certain homological constructions such as the Ext and Tor functors can be computed with free resolutions as well. In this talk we show how to compute free resolutions in the case of OI-modules over a Noetherian polynomial OI-algebra, where OI denotes the category whose objects are totally ordered finite sets and whose morphisms are strictly increasing functions.
Algebra Seminar
Title: Computing the Algebra of Conformal Blocks for sl_4
Abstract: Conformal blocks are finite-dimensional vector spaces that arise from the WZNW model of conformal field theory. These have applications in algebraic geometry, particularly in describing the moduli of principal bundles and the moduli of curves. In this talk, we will discuss recent progress on computing a presentation of the algebra of conformal blocks for sl_4. We also describe equations, the tropical variety, and a large family of toric degenerations for the case of a cone with genus 0 and 3 marked points.
Algebra Seminar
Title: Mori Dream Spaces: What, Where, and Why You Would Care
Abstract: If you've asked me in the past two years about my research, I've likely launched into an explanation of Mori dream spaces, their properties, and/or my implementation of an algorithm in Macaulay2. However, my hallway explanation likely left you with even more questions: What are these objects? Where can I find them? Why would I care? Well, you're in luck! In this talk, I plan to answer all of these questions, give plenty of examples, and list some questions motivating future work.
Algebra Seminar
Algebra Seminar
Title: The Gonality of Rook Graphs
Abstract:
The gonality game is a type of chip-firing game which is played on graphs which has interesting applications to algebraic geometry. The objective of this game is to ensure that no vertex on our graph has a negative number of chips after one is stolen. In this talk we will learn optimal strategies for winning on 2-dimensional rook graphs.
Algebra Seminar
Abstract: A linear series is a subspace of the space of functions on an algebraic curve. In this talk, we will discuss tropical analogues of linear series, with focus on a few key examples. Time permitting, we will close with a number of open questions about these newly defined objects.