# Edward Price III - Talks

**Talks given.**

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

**The Ordinal Numbers and Transfinite Induction**Purdue University Mathematics Department Student Colloquium, September 2018

Abstract: This talk gives a brief introduction to the ordinal numbers with a focus on understanding the use of transfinite induction. We begin with an intuitive understanding of cardinal and ordinal numbers. We then briefly describe some ordinal number arithmetic and basic results to get a feel for how the ordinals work and the complexity of this number system. We then discuss transfinite induction, finishing the talk with some examples of proofs using transfinite induction.

**So you think it's easy to solve (x-2)(x-3) = 0?**Invited Talk at Rose-Hulman Institute of Technology, March 28, 2018

Abstract: When looking at the equation (x-2)(x-3) = 0, it's easy to identify the solutions as x = 2 and x = 3. But how do we know that these are the only solutions? In this talk, we will explore a part of abstract algebra known as ring theory, and study the different solutions we can find to this equation (and equations like it) in various rings. For example, we will see that if we interpret the above equation in terms of 2x2 matrices, there are infinitely many solutions in addition to the scalar 2 matrix and the scalar 3 matrix (taking 2 or 3 times the 2x2 identity matrix, respectively). Or, for example, if you're working in the integers modulo 12 (which can be thought of as doing arithmetic on a 12-hour clock), there is an additional solution x = 6. By exploring situations where we get additional solutions, we aim to come to a better understanding of why in "normal" algebra we can guarantee that x = 2 and x = 3 are the only solutions to (x-2)(x-3) = 0.

**Field Theory and Constructions Using a Straightedge and Compass**Purdue University Mathematics Department Student Colloquium, September 2017

Abstract: This talk aims to explain classical straightedge and compass constructions, and in particular, the famous problems of squaring the circle, doubling the cube, and trisecting an arbitrary angle. We will discuss how elementary field theory can be used to show that these three tasks are impossible.

**The Ordinal Numbers and Transfinite Induction**Purdue University Mathematics Department Student Colloquium, September 2015

Abstract: This talk gives a brief introduction to the ordinal numbers with a focus on understanding the use of transfinite induction. We begin with an intuitive understanding of the ordinal numbers, and then proceed to define the ordinal numbers through the von Neumann construction. We briefly describe some ordinal number arithmetic to get a feel for how the ordinals work and the complexity of this number system. We then discuss the processes of transfinite recursion and induction, finishing the talk with some examples of proofs using these processes.

A copy of my presentation is available here.

**Proving a Sentence is Independent**Purdue University Mathematics Department Student Colloquium, January 2014

Abstract: A sentence

*φ*in first-order logic is independent from a set of sentences*K*if*K*neither proves nor refutes*φ*. For example, through the work of Kurt Gödel and Paul Cohen, we know that the Axiom of Choice is independent from ZF and the Continuum Hypothesis is independent from ZFC. But how does one go about proving that a sentence is independent from a given set of sentences? In this talk, we will have a brief introduction to first-order logic and see how one can prove that a sentence is independent from a set of sentences*K*using models. In particular, we will focus on the example where*K*consists of the axioms of group theory.**Buidling Group Extensions Storing Only 2-Cocycles for Relators**Illinois Section of the Mathematical Association of America, March 2012

**Using Cayley Graphs to Build Groups**MathFest in Lexington, Kentucky, August 2011

Abstract: In mathematics, one method for defining a group is by a presentation. Every group has a presentation. A presentation is often the most compact way of describing the structure of a group; however, there are also some difficulties that arise when working with groups in this form. One of the problems is called the word problem which is an algorithmic problem of deciding whether two words represent the same element. I am studying the word problem on group extensions. To solve this problem in a group extension

*G*, we will need to know how to solve the word problem in a normal subgroup*N*and a quotient group*G/N*. This information, along with information about how these groups stick together to make*G*, is enough to solve the word problem in this case. In this talk, I will discuss some observations I have made using the Cayley Graphs for*N*and*G/N*to study the structure of the resulting group extension*G*.**Modifications of Thomae's Function**Illinois Section of the Mathematical Association of America, April 2011

Benedictine University Undergraduate Research, Scholarship, and Arts Event (Awarded Best Session Presentation), April 2011Abstract: Thomae's function has interesting properties such as continuity at all irrationals, discontinuity at all rationals, and nowhere differentiability. Can there be a function which is continuous at all rationals and discontinuous at all irrationals? We will explore these and other analytic properties of Thomae's function, including recent modifications which allow Thomae's function to be differentiable at a set of points.

**A High School Lesson on Morley's Theorem Using Cinderella**Associated Colleges of the Chicago Area Student Symposium, April 2010

Abstract: Morley's Theorem states that the three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle. In this talk, we discuss the proof of this theorem and then describe a high school geometry lesson to explore and prove this result using Cinderella.

Abstract: In this talk, we will discuss the process of doing arithmetic in a group extension. A group extension is a group which is built from two smaller groups, namely a normal subgroup and a quotient group. In addition to these two groups we also need a map, called the 2-cocycles, on each pair of elements in the quotient group. This map describes how to lift products in the quotient group to products in the group extension. In this talk we will describe how to do arithmetic in a group extension assuming the 2-cocycles are known. We will then show that in fact we only need to know the 2-cocycles for each relator in a finite presentation for the quotient group Q to be able to do arithmetic in the group extension.