Slow Pitch Colloquium

Unlike the real numbers or integers, there are number systems where division by zero can happen. In this talk, we look at graphs of the so called zero divisors of commutative rings where each nonzero element is a vertex and two vertices are connected if and only if their product is zero. Our primary goal is the classification of when Z(R) is an ideal. In Section 2, we deal with the case of finite R with identity. Here we are able to give a complete characterization of when the zero divisors form an ideal. Also, we classify a large subset of graphs as either realizable or unrealizable as zero divisor graphs. In Section 3, we turn to general rings, possibly infinite or lacking identity, and classify when Z(R) is an ideal when the zero divisor graph is of diameter 0, 1, or 2, and give conditions that a zero divisor graph must satisfy if it has diameter 3 and Z(R) is an ideal and give an example of such a ring. In the final section, we deal with the case where the zero divisor graph is complete bipartite.

This talk gives a taste of two areas of upper division mathematics and will be accessible to undergraduates. If the language in the abstract is scary, don't worry, its not that bad.

Date	Speaker	Title and Abstract
Wednesday, April 22	Jason Hill	The Mathematics Behind Photography Something less serious this week. Photography demands two distinct and relatively disjoint skillsets: On one side it is creative and artistic, and on the other it is technical, mathematical and structured. In this talk, I'll describe some aspects of the mathematical side of photography. How can knowing about wavelengths of the visible light spectrum, base-2 calculations or linear transformations help a photographer?
Wednesday, March 18	David Grant	Things going on in our grad program As you know, the graduate committee and the departmental faculty have been working on restructuring the grad program. I'd like the chance to talk to you about what has been proposed, what has or has not been decided, and how changes would or would not affect our current students. It's also a chance to answer questions and get suggestions.
Wednesday, February 25	Richard Dodson	Constraint Satisfaction A gentle introduction to how recent applications of algebraic techniques have provided insight into the tractability of constraint satisfaction problems. In particular, we can say that specific subclasses of the general constraint satisfaction problem determined by specific algebraic operations are tractable.
Wednesday, February 18	Stephen Lewis	Representations and Combinatorics Representations of finite groups have become a popular topic, with heavy theoretical uses, and applications in chemistry, particle physics, crystallography, and many other fields. The talk will begin by drawing the connection between groups and vector spaces with representations, explore quickly some results including providing a method for classifying all representations of a given finite group, and discuss elementary character theory. If time allows, we will explore an application of set-partition combinatorics to understanding the character theory of unipotent upper triangular matrices with entries in a finite field.
Wednesday, January 21	Mike Martinez	When Do the Zero Divisors Form an Ideal, Based on the Zero Divisor Graph Unlike the real numbers or integers, there are number systems where division by zero can happen. In this talk, we look at graphs of the so called zero divisors of commutative rings where each nonzero element is a vertex and two vertices are connected if and only if their product is zero. Our primary goal is the classification of when Z(R) is an ideal. In Section 2, we deal with the case of finite R with identity. Here we are able to give a complete characterization of when the zero divisors form an ideal. Also, we classify a large subset of graphs as either realizable or unrealizable as zero divisor graphs. In Section 3, we turn to general rings, possibly infinite or lacking identity, and classify when Z(R) is an ideal when the zero divisor graph is of diameter 0, 1, or 2, and give conditions that a zero divisor graph must satisfy if it has diameter 3 and Z(R) is an ideal and give an example of such a ring. In the final section, we deal with the case where the zero divisor graph is complete bipartite. This talk gives a taste of two areas of upper division mathematics and will be accessible to undergraduates. If the language in the abstract is scary, don't worry, its not that bad.
Wednesday, January 14	Nick Pratarelli	The Primes Form an Additive Basis for the Positive Integers Every positive integer can be written as a sum of a bounded number of primes. Weird, I know. But it's true. Come see (very nearly) how. In connection with and/or to this end other topics of general interest will be discussed.
Wednesday, December 10	John Fuhrmann	Overview of Category Theory This will be an introductory talk on "abstract nonsense" also known as category theory. Topics will include categories, functors, natural transformations, monoids as categories, and categories as monoids. No previous working knowledge of category theory will be required.
Wednesday, December 3	Stephanie Lage	Quantum Mechanics
Wednesday, November 19	Ann Scarritt, John Fuhrmann, Jason Hill	Teaching Opportunities at SASC What is SASC (Student Academic Services) and why might you want to teach there as a graduate student in the Mathematics Department? The short answer: SASC provides an amazing opportunity for gaining unique teaching experience and growth in a very interactive and diverse environment. During this slow pitch, we will introduce SASC and explain what possible benefits there are with teaching in such an environment. To keep things interactive, we will discuss learning styles (something every teacher should consider) and you will discover what type of learning style you have.
Wednesday, November 12	Topaz Dent	An Introduction to the Constraint Satisfaction Problem as an Application of Universal Algebra This will be a basic talk on the constraint satisfaction problem, a problem of interest in computer science and complexity theory. We will explore examples of the constraint satisfaction problem, interpret a CSP as an algebra, and discuss the consequences of this interpretation. This talk will be accessible to undergraduates. Some knowledge of abstract algebra is helpful but not necessary.
Wednesday, November 5	Josh Sanders	Ultrafilters In this talk I will discuss filters and ultrafilters, prove a few easy general facts about them, and demonstrate applications for them in the areas of topology/analysis, algebra, and model theory (in particular, I'll sketch a proof of the coolest theorem that most people don't know... the compactness theorem.) The first 2/3 of this talk will be accessible to everyone, and the last 1/3 or so will be a continuation of Paige's talk from two weeks ago (although you'll still get the idea if you didn't go).
Wednesday, October 29	Ilia Mishev	Coxeter Groups and Hypergeometric Series Coxeter groups form invariance groups for certain linear combinations of 4_F_3(1) hypergeometric series with unit argument and help us describe 3-term relations among those.
Wednesday, October 22	Paige Cudworth	Model Theory An introduction to model theory will be given. One of the foundational theorems of model theory, the compactness theorem, will be discussed.
Wednesday, October 8	Jason B. Hill	Creating a Website for Teaching Mathematics This talk is aimed at grad students and instructors teaching at CU who want to use a website to aid communication and organization in their classes. Servers connections (Euclid) and blackboard systems (CU Learn) will be discussed. For beginners, an intro to HTML and page design will be given. For those more experienced, intros to CSS, PHP and analytics will be given. This talk is not meant to be exhaustive by any means, but should aid those of all experience levels. Supplemental resources will be made available.
Friday, September 19	Dr. Richard M. Green Dr. Markus J. Pflaum	Reflection Groups and Coxeter Groups (Green) and an Introduction to Research Areas in Topology (Pflaum) A reflection in Euclidean space is a linear transformation fixing pointwise some hyperplane through the origin, and transforming a vector orthogonal to this hyperplane to its negative. A finite reflection group is a finite group generated by reflections. A Coxeter group is a group generated by elements of order 2 subject to defining relations of a specific type. Examples of Coxeter groups include the symmetric and dihedral groups. It is an important result that the finite reflection groups are precisely the finite Coxeter groups. I will explain how the full symmetry group of the cube has the structure of a Coxeter group, and discuss some related combinatorial properties.