Permutations are fundamental objects in different areas of mathematics. In combinatorics, they are usually considered to be linear arrangements, with patterns to avoid and statistics to count. In algebra, we think of them as bijections from a set to itself, giving us the symmetric group. The different viewpoints are closely related, however, and there is some interesting mathematics bridging the descriptions. In this talk we will survey some of these connections, including Joyal's elegant proof of Cayley's tree formula and a polytope called the `permutohedron'. This talk should be accessible to a general audience, including undergraduate mathematics majors.

The Hausdorff metric provides a measure of distance between compact sets in any complete metric space. The metric is important for its applications in fractal geometry, image matching, visual recognition by robots, and computer-aided surgery. In the summer of 2011, Jon Verwys, Mychael Sanchez (from New Mexico State University) and Steve Schlicker considered the problem of defining a trigonometry in the space of compact sets of n-dimensional real space as part of the GVSU Research Experiences for Undergraduates (REU) program. The first step in the process was to define and understand an appropriate measure of orthogonality of compact sets. We will present the results of this investigation and how orthogonality might be used to define trigonometry.

Numerous efforts in math education have attempted to convince instructors to shift their instructional practices from lecture to alternative student-centered practices, but these have not been successful on a large scale, despite professional development that focuses on increasing awareness and improving instructor attitudes about student-centered instructional practices. This may be due to a knowledge-attitude-practice gap (KAP Gap), which exists when knowledge and favorable attitude do not lead toward adoption of a practice. This study uses a quantitative approach (author-created electronic survey, response rate 21.2%) to measure knowledge, attitudes, and instructional practices of Michigan community college mathematics faculty, with the purpose of identifying the existence of a possible KAP Gap and the factors that might be influencing its existence.
The analysis includes a breakdown about how community college math faculty acquire their knowledge of instructional practices and their level of participation in a variety of formal and non-formal professional development activities. General faculty attitudes about teaching and the teaching environment are measured using survey instruments developed by Trigwell and Prosser (2004, 2008). Attitudes towards three instructional practices (collaborative learning, inquiry-based learning, and the lecture method) are examined in depth, especially with regard to the influence of the environment, the enabling characteristics of students, and the time requirements for using the method. Finally, instructors are asked to report about their level of use of each of the three practices (allowing the use of more than one practice) using a scale developed by Henderson & Dancy (2009).
This study is one of the first to directly identify a KAP Gap for instructional practices and to explore the variables that influence the instructional practices of college math instructors. The results suggest that knowledge plus a favorable instructor attitude is not enough to predict an instructor's use of a student-centered instructional practice (although an unfavorable attitude will predict non-use). This study also illuminated significant difference between adjunct and full-time faculty in the level of professional engagement, breadth of teaching experiences, and use of student-centered instructional practices.

For a very long time, one of the most interesting and most challenging mathematics problems was the Four Color Problem. This problem came from a young British mathematician. All of the information we have about what this young man was probably thinking has been written by others. But what if he had something else in mind? We explore some thoughts he might have had and describe some research that could have resulted from this.

Efficient energy conversion is a fundamental problem of the 21 century. A key step in converting photonic and chemical energy to useful electric potential is the generation of electrostatic double layers at catalyst interfaces. To do so efficiently requires interpercolating structures with nanoscale morphology, which are often generated by phase-separation of functionalized polymer-solvent mixtures. We will present an overview of energy methods which guide phase separation, and introduce a novel class of energies which incorporate electrostatic and solvation effects via higher order curvatures that drive interfacial generation.

After an introduction to plane fields and contact structures, we will see a concrete example of an overtwisted disk (obstruction to tightness), how to encode a contact structure via twists on a surface, and go through part of the process of distinguishing which structures are tight by counting special regions in a planar graph related to the given contact structure. (This is a joint work with Firat Arikan.)

In geometry, the definition of a convex set depends completely on points and lines (i.e. between any two points in the set, the resulting line is also in the set). One way to generalize this to other contexts is to look for structures that include something that corresponds to points and something that corresponds to lines. Directed graphs are a perfect environment in which to work with convexity. Here the "points" are the vertices, and the "lines" are directed paths. In this talk, I will discuss some work I have done with Randy Westhoff and Marty Wolf of Bemidji State University on convexity in multipartite tournaments.

Part 1: Engaging students in discussion result in better comprehension of the material and overall enhance learning. I will show how peer instruction with the use of Bluetooth technology (BT) promotes students' participation and comprehension.

Part 2: We will consider Lie groups in dimension two and three and will focus on the solutions of Killing's equations. A striking result is that several of the three-dimensional Lie groups turn out to be spaces of constant curvature.

Students: Reynaldo Herrera and Bryan Kimball
Topics: Counting Tilings (Reynaldo Herrera) and Rankings in Digraphs (Bryan Kimball) [Math 496 class projects]
Faculty Advisor: Brian Drake

Student: David Ridder
Topic: Geometric Cryptography [Math 399 project]
Faculty Advisor: Ed Aboufadel