Abstracts Archives 2013-14
12/4 Samantha Reynolds, Willamette University '14
College Entrance Exam Firms, Nonprofit Efficiency, and Testing Fees
College entrance exam companies such as the College Board or the ACT claim nonprofit status. Theoretically these companies should not have high costs and considering that they aren’t profit driven, we would expect to see low testing fees. In reality this is not the case and many would claim that it stems from the inefficiency of the nonprofit. I analyzed whether high test fees could be the result of a company’s primary mission rather than inefficiency. Using the team incentive problem and the role of a budget breaker, I showed that nonprofits can induce workers to provide an effort level that minimizes costs in order to maximize net revenue. Assuming the firm has idealistic workers, the model can be extended where we still maximize net revenue without a principal playing the role of a budget breaker. The primary mission of nonprofits takes the form of a publicly valued good or service and that by maximizing revenue they can maximize the amount allocated to producing the public good. This implies that test takers may pay high fees not because the firm necessarily is inefficient but because the firm is trying to maximize how much of the public good is produced.
11/14 Professor Inga Johnson, Math Department
Topology, Homology, and Applications to Data
Topology is the subfield of mathematics that is concerned with the study of shape. Mathematicians have studied topological questions for the past 250 years. In the past few years a new interdisciplinary field has blossomed bringing together topologists, statisticians, computer scientists, engineers and others, to use topological ideas to study data sets in new and exciting ways. We will discuss one of the new topological tools that has been developed called persistence homology.
This talk will be an introduction to topology and the concept of homology. We will then use homology to a look at examples of how topological ideas can be used to give new and surprising insight towards understanding data. This talk will emphasize examples and concepts. Prerequisites will be minimal.
10/31 Jeff Schreiner-McGraw and Will Agnew-Svoboda
A unipancyclic (UPC) graph is a graph containing exactly one cycle of every possible size. Only a handful of these are known to exist, although searches have been performed through all graphs with 56 or fewer vertices. We generalized this problem by seeking to find and characterize UPC matroids. There are UPC matroids that are not graphic, so this does result in a larger family. In this talk, we will discuss the progress from the summer's research program.
10/24 Nancy Ann Neudauer, Pacific University
What is a Matroid? Investigations of asymptotic enumeration in matroids
In 1933, three Harvard junior-fellows tied together recurring themes in mathematics into what Gian Carlo Rota called one of the most important ideas of our day. They were finding independence everywhere they looked. Do you? We find that matroids are everywhere: Vector spaces are matroids; We can define matroids on a graph. Matroids are useful in situations that are modeled by both graphs and matrices. We consider how we can ask research questions about matroids, and look into results from a student's investigation.
Two matroids are commonly defined on a graph: the familiar cycle matroid and the more rarely-encountered bicircular matroid. The bases of the cycle matroid are the spanning trees of the associated graph; the bases of the bicircular matroid are all subgraphs of the graph, each of whose connected components contain exactly one cycle and (possibly) other edges. We enumerate the bases of the bicircular matroid for several classes of graphs. For a given graph, usually there are more bases of the bicircular matroid than of the cycle matroid. We ask when these numbers are the same. We also consider when there are more bases of the cycle matroid, and what this translates to in terms of the structure of the graph. No prior knowledge of matroids or graphs is needed!
10/3 Yumi Li, Math Major
Put Your Thinking CAPS On (Exploring Finite Geometry in the Card Game SET®)
Besides being a great card game, SET® serves as an excellent model for the ﬁnite geometry AG(n,3). Using the SET® cards as a visual representation, we will explore the structure of maximal caps and how we can manipulate them to discover new properties and substructures of AG(n,3). This work was done at the Research Experience for Undergraduates program at Lafayette College.
9/19 Ryan Wright, Janrain Inc.
Computing the Coming Robot Apocalypse: The math behind Artificial Intelligence and Machine Learning
Let’s face it, it’s only a matter of time before machines rise up and take over the world. From image recognition, to Netflix recommendations, to predicting the future, Machine Learning and Artificial Intelligence are at the heart of some of the coolest technology being developed today. We give a quick introduction to how these technologies work and explain why math is how we welcome our future robot overlords.