Invited Paper Session Abstracts - Undergraduate Research Projects in the Mathematical Sciences

Saturday, August 6, 1:00 p.m. - 3:20 p.m., Harrison

Organizers:
Pamela E. Harris, Williams College
Alicia Prieto Langarica, Youngstown State University

The undergraduate mathematics curriculum continues to evolve from expository classes to students working on original research projects. This curriculum change has created a need in the mathematical community for more REU programs and for faculty to develop accessible research projects for students at their respective institutions. For new faculty, developing such projects may be challenging, as their particular expertise might require extensive background and is hence not suitable for an undergraduate audience. Fortunately, faculty programs, such as The Center for Undergraduate Research in Mathematics (CURM) Mini-Grants, Preparation for Industrial Careers in Mathematical Sciences (PIC Math) Preparing Undergraduates Through Mentoring toward PhD’s (PUMP), Project NExT (New Experiences in Teaching), and Research Experience for Undergraduate Faculty (REUF), have an impressive track record of helping faculty develop and promote accessible research projects at the undergraduate level.

In this session, past CURM, PIC Math, PUMP, Project NExT, and REUF faculty participants present the mathematical results of their student’s original mathematical research and share tools they learned and used to develop these projects. In addition, speakers will provide early-career faculty with information related the respective faculty programs and open problems that are accessible to undergraduate students. Mathematical topics are broad and include number theory, graph theory, applications of PDEs, and industrial mathematics.

Counting Dessins

​1:00 p.m. - 1:20 p.m.
Naiomi Cameron, Lewis & Clark University

In this talk, I will discuss an undergraduate research agenda which originated at a 2012 AIM REUF Workshop. The objective of the research is to contribute to the efforts to better understand the Galois orbits of dessins through combinatorial, topological, geometric and/or arithmetical points of view. I will describe the motivation and outcomes of two undergraduate research experiences that have resulted from this agenda. In the first, an eight-week experience over the summer of 2014, the research group explicitly computed Belyi maps for trees with passports of size 1 or 2. In the second, a semester long experience over the spring of 2016, the research group tackled the problem of enumerating dessins whose passport is given by a triple of partitions.

PIC Math: A Course for Undergraduate Students to Do Research on Actual Problems from Industry

1:30 p.m. - 1:50 p.m.
Michael Dorff, Brigham Young University

PIC Math is a new program to prepare undergraduate students in the mathematics and statistics to succeed in careers in business, industry, and government (BIG). Funded by a $2 million NSF grant, this program strives to (a) increase awareness among faculty and students about non-academic career options, (b) pro- vide undergraduate research experience using problems from industry, and (c) pre- pare students for industrial careers. The program includes a 3-day faculty summer training workshop, a spring semester course in which students learn skills and work on research problems from industry, and an end-of-program research conference at which the students present. For the semester course, we have developed a set of educational and informative videos and prepared materials for the course such as sample syllabi, set of sample research problems from industry, sample student solutions to industrial research problems, and sample videos of student presenting their research. During the 2015-2016 academic year we have math departments at 50 different institutions with over 500 students participating.

Constructing Solutions to Truncated Moment Problems and Applications to PDE; a PUMP Undergraduate Research Group

2:00 p.m. - 2:20 p.m.
Cynthia Flores, California State University, Channel Islands

This presentation outlines an investigation of the synergy between truncated moment problems and their wide applications in the theory of partial differential equations (PDEs). For instance, some PDEs display a special time evolution of the moments up to a finite order, such as in the case of the Benjamin-Ono equation, a model for long internal gravity waves. As part of this project, undergraduate student participants study how to derive useful measures with prescribed moment properties. In this talk, we discuss the motivation of the project, the undergraduate student involvement, as well as the mathematical results.

Undergraduate Research in Pebbling

2:30 p.m. - 2:50 p.m.
Aparna Higgins, University of Dayton

Since it was introduced in 1989, the topic of pebbling on graphs has been a source of interesting and accessible problems for researchers, including undergraduates. Given a connected graph \(G\) and a distribution of non-negative integers on its vertices, a pebbling move on \(G\) is defined as the removal of two pebbles from one vertex, followed by the placement of one of those pebbles on an adjacent vertex. The pebbling number \(f(G)\) of a graph is the minimum number of pebbles needed such that, given any distribution of \(f(G)\) pebbles on \(G\), one pebble can be placed on any specified but arbitrary vertex through a sequence of pebbling moves. The talk will describe results by undergraduates on pebbling and on some of its variations. We will also show, via the example of pebbling, some general ideas on how to create problems that may be tractable for engaging undergraduates in research at one’s own institution.

Research Collaborations in the Public Sector

3:00 p.m. - 3:20 p.m.
Thomas Wakefield, Youngstown State University

As part of the PIC Math (Preparing for Industrial Careers in Mathematics) program offered by the MAA, Youngstown State offered students the opportunity to work on research problems originating from the Youngstown Police Department and other agencies under my direction. Although originally planned for one semester, the students’ work on the YPD project extended throughout the summer and into the next academic year. We will discuss the project, address the benefits and challenges of such partnerships, and offer tips for finding such partnerships.