Periodic Point Bifurcations - with Pictures! by Dan Wolf and Melissa Larson and supervised by Dr. Jody Sorensen
Buy? Sell? What are the Options? by Jennifer Geis and supervised by Dr. Tracy Bibelnieks
Bulgarian exchange: where does it end? by Tim Bancroft and supervised by Dr. Su Dorée
Wave modeling using parallel processing by Dan Wolf and Kailash Thapa and supervised by Dr. Nick Colt
Numerical analysis on computer wave simulations by Kevin Sanft supervised by Dr. Nick Coult

Periodic Point Bifurcations - with Pictures!

In the 2005-2006 school year, Augsburg senior math majors Dan Wolf and Melissa Larson worked with Dr. Jody Sorensen on a research project in the field of Dynamical Systems. Dan and Missy had no background in this field, so they started by working quickly through a textbook on the subject. Dynamical Systems involves iterating functions: you start with an x, apply a function f and then take that output and put it back into the function over and over again. The goal is to be able to predict what happens in the long term. Dynamical Systems has practical applications ranging from economic modeling to population dynamics to weather prediction.

Dan and Missy’s work focused on the bifurcation diagram, an example of which is shown below. The example is for the function f(x) = x2+c, and has the parameter c on the horizontal axis and the variable x on the vertical axis. The picture shows the periodic points for various c values - these are the points that are locked into a repeating pattern under iteration. The colors vary based on how those periodic points effect nearby points - whether they are attracting or repelling.

In looking at many examples of bifurcation diagram, Dan and Missy and Jody noticed certain common traits. For example, near places where the diagram branches, the periodic points always alternate between attracting and repelling. They were able to prove (under very general conditions) that this alternating behavior must occur. Surprisingly, the proofs ended up using the Intermediate Value Theorem, a topic covered early on in every Calculus I class.

Dan and Missy presented their work in several ways. Missy took an all-expenses-paid trip to San Antonio, Texas in January 2006 to participate in the Undergraduate Poster Session at the Joint Mathematics Meetings. Later that spring Dan and Missy presented their work at the Pi Mu Epsilon Conference at St. John’s, and then at an Augsburg math department colloquium entitled “Periodic Point Bifurcations - with Pictures!” Dr. Sorensen is working on a paper that will incorporate Dan and Missy’s work.
A Bifurcation Diagram
Missy at the Joint Mathematics Meetings at San Antonio, TX, January 2006.
Dr. Sorensen introduces Dan Wolf and Missy Larson at the Pi Mu Epsilon Conference at St. John’s University, MN
Buy? Sell? What are the Options?

Jennifer Geis is a senior Mathematics and Actuarial Science major at Augsburg College who recently researched questions in financial mathematics.
Jennifer Geis presents her research poster at the AMS/MAA Joint meetings in San Antonio, Texas, January 2006.

Over the past summer, she had the opportunity to go to North Carolina State University for a Research Experience for Undergraduates (REU). There she learned about financial mathematics and about correctly pricing stock options. Options are the right, but not the obligation, to buy or sell a stock for a certain price at a certain time in the future. Her research consisted of improving upon a least-squares Monte Carlo simulation technique for pricing American put options by implementing and testing different control variates.

In November 2005 Jennifer presented her work "Option Pricing Made Cents" at Augsburg's Mathematics Colloquium. In January, Jenn took her work on the road a the Joint Mathematics Meetings in San Antonio, Texas on January 14, 2006. At the meetings she presented a poster “Optimization Least Squares Approach for Valuation of American Put Options”. See some of Jennifer's work by viewing her Powerpoint presentation from the mathematics colloquium. To view her powerpoint presentation click here.

Jennifer is currently working under the supervision of Dr. Tracy Bibelnieks on researching data-mining techniques in conjunction with a local business.

Tim Bancroft presents his research poster at the AMS/MAA Joint meetings in Phoenix, Arizona, January 2004.

Bulgarian exchange: where does it end? 

That’s the question Tim Bancroft, a senior mathematics major at Augsburg, has studied under the direction of research advisor Prof. Su Dorée. Tim grew up in the Twin Cities and came to Augsburg because his older brother and sister had. After graduating from Augsburg Tim plans to work on his master’s in statistics at Iowa State.

"One of the most exciting parts of my research project was being able to travel and present my work," Tim explains. Through funding from Augsburg, Tim traveled with Professor Dorée to a national mathematics conference in Phoenix in January. In addition to enjoying the warm, sunny weather, Tim presented a poster on his work and attended numerous talks about mathematics. "There were a lot of talks," Tim explains, "but the sessions on ‘Mathematics and Sports’ were my favorite."

Tim’s research project grew out of a puzzle from Discrete Mathematical Structures class (MAT 271). In his two–player version you and a friend each begin with coins arranged in piles. At each turn you trade coins according to rule: remove the top coin from each pile, possibly eliminating piles, and give that collected pile of coins to the other player. The game continues until you begin to repeat previously encountered arrangements.

Which arrangements are reached again? How long are the ultimate cycles of arrangements? Do all starting arrangements end in the same cycle? What does the state graph look like?

The single player version, called Bulgarian solitaire, was made popular in 1983 by Martin Gardner in his Scientific American column and is a somewhat distant relative of the two-player African pebble games Mancala. The study of such games generally falls under what’s known as Recreational Mathematics but, more specifically, it lies within Combinatorics, the study of the arrangements of objects. Although the rules are easy to describe, the mathematical structures of the game are intricate. In 1985 Akin and Davis published an article in The American Mathematical Monthly that characterized the cycle structure and cyclic elements of the solitaire version. Tim’s work generalizes their results to the new, two-player version.


Mathematics students at Augsburg have many opportunities to participate in research projects. In addition to independent study, mathematics majors have the opportunity to propose Departmental Honors projects, and even to work as research assistants on longer-term faculty research projects.

Dan (l) and Kailash (r) had the opportunity to work with Augsburg's new 6-CPU beowulf cluster. This small cluster of three computers, each with two processors, can perform moderate to large scale numerical simulations very quickly. Dan and Kailash spent time learning how to design algorithms for parallel computation, and wrote computer software to do numerical modeling of waves using the cluster.

Wave modeling using parallel processing

Dan Wolf and Kailash Thapa, both Mathematics and Computer Science double majors, spent summer 2003 working on a research project in collaboration with Dr. Nick Coult. The research was conducted as part of a three year NSF-funded project to study mathematical models of waves.

Dan hails from Lacrosse, WI, and said he greatly enjoys doing research of this kind. "It's very challenging, because you don't always know whether you're on the right path, but it's very exciting too to be learning so much and to be working on the cutting edge," he said of his summer work.

Kailash comes from farther afield - Kathmandu, Nepal, to be exact. Like Dan, he enjoys the challenge of the work and the close interaction with faculty. "We're working with Dr. Coult on a daily basis, identifying issues, solving problems, running numerical experiments...with the computer resources we have available, we can do quite a lot, but we also use a lot of mathematics. I think it's great," said Kailash.

Mathematics students at Augsburg have many opportunities to participate in research projects. In addition to independent study, mathematics majors have the opportunity to propose Departmental Honors projects, and even to work as research assistants on longer-term faculty research projects.

Kailash and Dan made extensive use of calculus, differential equations, linear algebra, numerical anaylsis, and a variety of computer programming languages -- all of which they learned through courses at Augsburg.

Numerical analysis on computer wave simulations

Kevin Sanft, Mathematics and Computer Science double major, spent the 2001-2002 academic year working on a research project in collaboration with Dr. Nick Coult.

Kevin wrote:
"My project is primarily a numerical analysis problem. I am studying the wave equation and how numerical solutions to this equation can be used in computer simulations. I am also looking at the error involved in my model and exploring the relationship between the accuracy of the model and the computation time required."

When he wasn't working on numerical wave modeling, Kevin hit the links with the Augsburg golf team. He also likes spending time outdoors and fishing. He graduated in May 2002 and is now working for as a data analyst for the Mayo Clinic in Rochester, MN.


Kevin works at one of Augsburg's Sun Ultrasparc workstations.


Snapshot of a two-dimensional circular wave propagating through an inhomogeneity.


In addition to computer modeling and simulation, Kevin has to do some old-fashioned chalk-board work.