# Matthew Weaver - Talks

**Talks given.**

Below is a list of talks I have given, starting with the most recent.

**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**Purdue University Mathematics Department Student Commutative Algebra 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**Purdue University Mathematics Department Student Commutative Algebra 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.