Matthew Weaver - Talks
Talks given.Below is a list of talks I have given, starting with the most recent.
- The Equations Defining Rees Algebras of Ideals in Hypersurface Rings
AMS Spring Sectional Meeting, March 2022 (upcoming)
Abstract: The defining equations of Rees algebras provide a natural pathway to study the so-called blowup algebras. However, these equations are often elusive and so far have only been classified for Rees algebras of specific classes of ideals in polynomial rings. It is an interesting question as to what the equations defining Rees algebras of ideals in other rings are, hence we consider ideals of hypersurface rings. In this talk, we consider perfect ideals of grade two and perfect Gorenstein ideals of grade three in these rings and their Rees algebras. By introducing the modified Jacobian dual matrix, we are able to formulate recursive algorithms which produce minimal generating sets of the defining ideals of these Rees algebras. We also investigate the Cohen-Macaulayness of these rings.
- Polynomial Relations and the Rees Algebra
Purdue University Graduate Research Day, November 2021
Abstract: Given an ideal of a commutative ring, is it possible to determine all of the polynomial relations among a generating set? The answer is uncertain in general and a complete classification of these equations is known only in a few instances. In this talk we'll describe how some of these polynomial relations can be formulated and how they relate to Rees algebras.
- The Equations Defining Rees Algebras of Ideals of Hypersurface Rings
Commutative and Homological Algebra Market Presentations (CHAMP) Seminar, October 2021
Abstract: The defining equations of Rees algebras provide a natural pathway to study the blowup algebras. However, a minimal generating set of the defining ideal is rarely understood outside of a few classes of ideals. Moreover, most of these results only pertain to ideals of polynomial rings. With this, it is an interesting question as to what the defining equations of the Rees algebra are for an ideal outside of this setting. In this talk we consider ideals of codimension two of hypersurface rings and the equations defining their Rees algebras. By introducing the modified Jacobian dual, we apply a recursive algorithm with this matrix and produce a minimal generating set of the defining ideal.
- Rees Algebras of Codimension Three Gorenstein Ideals of Hypersurface Rings and their Defining Equations
Notre Dame Algebraic Geometry and Commutative Algebra Seminar, September 2021
Abstract: One of the most natural ways to study the Rees algebra of an ideal is through its defining ideal and its generators, the defining equations. Unfortunately determining such a minimal generating set is difficult in general and results are only known for Rees algebras of specific classes of ideals. In particular, the Rees algebra of a perfect Gorenstein ideal of codimension three has been studied extensively in recent years, but only when such an ideal belongs to a polynomial ring. In this talk we extend some of these results to the situation of the Rees algebra of such an ideal of a hypersurface ring and explore the defining equations. By introducing the modified Jacobian dual and a recursive algorithm of gcd-iterations we produce a minimal generating set of the defining ideal and determine the Cohen-Macaulayness of the Rees algebra.
- Polynomial Relations and Defining Ideals of Rees Algebras
Purdue University Mathematics Department Student Colloquium, September 2021
Abstract: Given an ideal of a Noetherian ring, is it possible to determine all of the polynomial relations among a set of generators? Surely you could write some of them out, but how could you ever know you've found them all? In this talk, we'll explore some examples of ideals where this is possible and will relate our findings to Rees algebras. In particular, the defining ideal of the Rees algebra encodes all of these polynomial relations and this algebra has deep implications to both commutative algebra and algebraic geometry.
- The Equations Defining Rees Algebras of Codimension Three Gorenstein Ideals of Hypersurface Rings
Purdue University Mathematics Department Commutative Algebra Seminar, September 2021
Abstract: The procurement of the defining equations of Rees algebras of ideals is a natural way to study these algebras and encodes the polynomial relations of any generating set of an ideal. The Rees algebra of codimension three Gorenstein ideals has been studied extensively in recent years, but we intend to focus on and generalize a classic result in the study of the blow-up algebras. In the mid-1990s, Morey gave a classification of the defining ideal of the Rees algebra of such an ideal of a polynomial ring in a particular setting by showing only one nontrivial equation exists and then described it as the gcd of the minors of a Jacobian dual matrix. In this talk we consider the defining ideal of the Rees algebra of a codimension three Gorenstein ideal of a hypersurface ring by introducing the modified Jacobian dual and provide a recursive algorithm of gcd matrix iterations to obtain a complete generating set of the defining ideal.
- Rees Algebras of Ideals and Modules of Hypersurface Rings and their Defining Equations
Purdue University Mathematics Department Commutative Algebra Seminar, October 2020
Abstract: The Rees algebra of an ideal has been a longstanding construction providing an algebraic viewpoint into the notion of blowing up an algebraic variety along a subvariety. Viewing this algebra as an epimorphic image of a polynomial ring, we study the kernel of this map and refer to it as the defining ideal. In the mid-nineties Morey and Ulrich gave a complete description of the defining ideal of the Rees algebra of a linearly presented perfect ideal of grade two in a polynomial ring and proved it was of the expected form. It is an interesting question as to what the defining ideal looks like if any of their assumptions are weakened or altered and how far it strays from the expected form. In this talk we attempt to do just that and study the defining ideal of Rees algebras of linearly presented ideals and modules of hypersurface rings.
- Defining Equations of Rees Algebras
Purdue University Graduate Research Day, November 2019
Abstract: We define and explore some properties of the Rees algebra and its defining equations and their implications in both commutative algebra and algebraic geometry. The Rees algebra gives a way to study the blow-up of affine schemes from geometry in a more algebraic setting. When viewed in this light, it is possible to produce to the defining equations which cut out the blow-up as a quasi-projective variety. This has deep implications in resolutions of singularities for the geometers and bridges connections to other algebraic structures for the algebraists. This is an expository talk and is meant to be accessible to any graduate student with introductory experience in algebra or geometry.
- Defining Ideals of Rees Algebras
Commutative Algebra Student Seminar, April 2019
Abstract: We begin by viewing the Rees algebra in a new light as the epimorphic image of a polynomial ring. The natural question is then what the kernel of this map is, as then one can characterize the Rees algebra as a factor ring. This kernel is what is referred to as the defining ideal of the Rees algebra and its generators are often called the defining equations. In this talk we discuss the structure of the defining ideal and known defining equations of certain classes of ideals of low codimension.
- Macaulay2: An Introduction
Commutative Algebra Student Seminar, January 2019
Abstract: The aim of this talk is introduce the listener to Macaulay2, a computer algebra program. We begin by discussing basic syntax and operations and then proceed to writing expressions such as for loops, while loops, and conditional statements. From there we discuss how to write simple functions to accomplish tasks and computations within commutative algebra.