A monoid M is a set with a binary operation * which satisfies the usual group axioms except for the existence of inverses. The factorization of integers in the normal sense takes place in the multiplicative monoid of positive integers. As is well known, such factorizations into prime (or irreducible) elements is unique up to the order of appearance of the prime factors. Such is not the case in various other types of monoids (such as numerical monoids or block monoids). The project for summer of 2014 is meant to build on the work of many of my other past REU groups in analyzing monoids where factorization into irreducibles is not unique. These investigations will include the study of many combinatorial constants associated to non-unique factorizations, such as elasticity, delta sets, the omega function, and the catenary degree.
Textbooks play a central role in the mathematics classroom. The content and its presentation of mathematics within textbooks influence what is taught, how it is taught, and ultimately, what is learned. Additionally, mathematics textbooks are influenced by many factors, including recommendations from professional organizations, national and state standards, and high-stakes assessments. For example, textbooks for prospective teachers may present content addressed by teacher certification standards documents or exams.
Recently, I completed an examination of the statistics content of textbooks for prospective elementary teachers in the U.S. This summer, we will conduct three investigations related to this topic. First, we will examine the statistics content of U.S. mathematics textbooks intended for students in grades 1 to 5. Second, we will examine and compare the recommendations and standards for preparing teachers to teach statistics in the U.S. and other nations. Finally, we will compare our findings from these two studies with the statistics content of textbooks for prospective elementary teachers.
The notion of a convex set is a simple one. Nonetheless, arguments based on properties of convex sets abound in mathematics, especially in the branch of analysis known as functional analysis. Classical functional analysis can be regarded as the study of vector spaces of continuous functions, and the use of convexity in their study has been very fruitful. However, modern functional analysis has recently focused on the study of various vector spaces of matrices and linear transformations, where it has been discovered that a newer more general notion of a convex set plays an important role. This new notion of convexity is known as `matrix convexity,' and while some major pieces of its theory are now in place, there still remains a lot which has not been worked out.
Our project will be to generalize facts about convex set to facts about matrix convex sets. Students working on this project will learn some of the rudiments and language of modern functional analysis, but will primarily be using linear algebra to establish results.