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Summer Undergraduate Research Program in Mathematics
An Important Next Step
Since 1985, St. Norbert College has hosted the Regional
Pi Mu Epsilon (PME) Undergraduate Mathematics Conference under the direction of Dr. Rick Poss. PME is the National Mathematics Honorary Society and, through Summer 2009, St. Norbert has inducted 247 members. In the past few years, participants in the PME Conference have traveled to our campus from 22 midwestern colleges and universities. In November 2010, the 25th Annual PME Conference set records for the total number of students in attendance at 190, and the total number of students speakers at 37 with 5 from St Norbert.
The St. Norbert Mathematics Discipline has long supported undergraduate research in the field. Faculty have encouraged students to participate in summer research experiences at other institutions and nurtured those experiences by advising students as they presented their work at regional and national conferences. The Mathematical Association of America and PME hold a national meeting each August with St. Norbert frequently having the second most student speakers, next to Youngstown State in Ohio (resulting in a friendly competition), for at least the last decade. Since 1985, Rick Poss has accompanied St. Norbert student speakers to every summer national meeting, now part of Mathfest. No other college or university in the country has had student speakers at every one of these meetings. Our presence at this national level has been notable.
Although our students are among the most active in the country in presenting the results of their research, it is now time to take it to the next level and offer those research experiences here on our campus. In Spring 2008, led by the enthusiasm of Dr. Terry Jo Leiterman, the Mathematics Discipline became energized to provide the increasing number of interested students a quality summer research experience here at St. Norbert College.
Student Needs and Outcomes
Opportunities to develop analytical thinking abilities are essential ingredients in an undergraduate degree program. Competent training in critical thought will keep St. Norbert students competitive with their peers at other institutions. More than ever, a holistic and interdisciplinary education is important for our students' success beyond the classroom. Mathematics, with its inherent ties to the natural and social sciences, is an ideal field for students to develop collaborative critical thinking and problem solving skills.
A program in undergraduate research at St. Norbert serves our students. Its benefits lie within the tradition of the College's mission, which insists we provide an educational environment that is intellectually, spiritually, and personally challenging. The national
Council on Undergraduate Research outlines that a student research experience
- Enhances student learning through mentor relationships with faculty
- Increases retention in the sciences, technology, engineering, and mathematics (STEM) pipeline
- Increases enrollment in graduate education and provides effective career preparation
- Develops critical thinking, creativity, problem-solving, and intellectual independence
- Promotes an innovation-oriented culture.
Projects can also serve community interests, in addition to offering an engaging preparation for our students that plan to be future teachers of mathematics. St. Norbert College continually maintains a reputation as a leader with a unique tradition of academic excellence. A summer research program at the College provides an opportunity to showcase that spirit in a partnership with faculty, students, scientific inquiry, and mathematics.
Program Description
The summer research program consists of 10 weeks of full time work beginning with an application process in the spring semester open to St. Norbert mathematics students. During that time, the program is advertised in our courses, the Sigma Nu Delta math club newsletter, and on our website. It is strongly encouraged that interested students meet with a member of the Mathematics Discipline before putting together an application. In the application, students are asked to provide a list of their mathematics and other relevant course work, their research interests, and how a research experience would contribute to their professional goals. Students do not need to propose a project. Faculty members may already have projects in mind. Applications are then reviewed by the Mathematics Discipline and students are chosen by the faculty mentor. In August, the research students are expected to present their work at the national Mathfest meeting.
Alumni Support
Since Fall 2006, the Mathematics Discipline has been working hard to generate momentum and get a research program started. In Spring 2008, Terry Jo Leiterman managed to find funding ($8000) to support two St. Norbert students to do research in the summer. Both students presented their results at that summer's Mathfest. The cost per undergraduate student includes a modest stipend, on-campus housing, and FICA. Leiterman worked with the students for no pay and members of the math faculty have agreed to continue mentoring future summer research projects without compensation.
Affording students an opportunity to pursue their academic curiosities in math at St. Norbert College cannot be realized without the support of our alumni. We need an income of at least $8000 per year to ensure that future students can have the same experience. Under the direction of a faculty mentor, this amount pays two students to work full time for 10 weeks on a research project. The income minimally offsets the loss of summer employment, which is often necessary to cover our students' educational expenses during the semesters.
We are modifying the current Math Club Travel and Travel Endowment funds to include the use of these funds to support student research. The first money that is received each year will be set aside to fund student research in the current summer. After the immediate needs are met, money will then go into the endowment fund. Supporting two students each year through this program leaves us with a goal to meet of $160,000 in the endowment.
Update on the Challenge
To motivate your support, Rick Poss had set up a challenge in 2008. Rick agreed to match your gifts on a 1-for-2 basis up to $50,000 on his part. That is, for every $2 you give, he pledged $1. We are pleased to announce early support from alumni at a total of $16,000. Thank you! To give us more time to reach our endowment goal of $160,000, Rick has pledged to personally support the Summer Research Program at $8000 per year for six years, starting in 2010. The Mathematics Discipline is hoping you will also offer your support. Our target for this year is $30,000. That's a large sum of money, but there are thousands of SNC mathematics alumns out there! Each $10 you give will support one hour of student research, which includes student wages and housing.
Make a Gift
We need your support. Please consider offering a gift. There are several ways to do so.
1. Make a gift by mail.
Please send your gift of support to:
Terry Jo Leiterman
St. Norbert College
100 Grant Street
DePere, WI 54115.
2. Make a gift online.
Offer support online at the
College Advancement website.
Apply your gift to this program by typing "Math Research."
3. Make a pledge.
Fill out the
Gift Intention Form and mail it to Terry Jo Leiterman.
Research Projects
Summer 2008: Mathematical Ecology
Modeling Diatom Growth in Trout Lake
by
Corey Vorland
(left) and
Stephanie Schauer
(center) under the direction of
Dr. Terry Jo Leiterman
(right).
Aulacoseira is a freshwater diatom whose abundance and colony size has been measured at varying depths in Trout Lake in Northern Wisconsin. Its population growth patterns are influenced by temperature, light availability, and nutrients. In this project, the vertical distribution of Aulacoseira is investigated through a mathematical model which incorporates natural characteristics of the lake as well as effects of the diatom's buoyancy. Predicted outcomes are compared to measured observations of growth by Dr. David Poister, an environmental chemist at St. Norbert College.
Stephanie Schauer began this research project at the end of her sophomore year. She is a native of New Franken, WI. Stephanie is a mathematics major with secondary education emphasis. She gave her first research lecture in November 2007 at the PME Conference on work she completed through the freshman fellows program at St. Norbert College. Stephanie was a student in the Fall 2007 MATH 489 course which built the popular
Square Wheeled Bicycle.
Corey Vorland also began this research project at the end of his sophomore year. He is a native of Neenah, WI. Corey is a double mathematics and computer science major with a minor in history. He gave his first research lecture at Mathfest held in August 2008 at Madison, WI on the results of the Summer 2008 project.
Summer 2009: Mathematical Ecology
Modeling Diatom Growth in Trout Lake, Part 2
by
Corey Vorland (left) and
Stephanie Schauer (right) under the direction of
Dr. Terry Jo Leiterman.
Aulacoseira's growth is determined by a complex, interconnected relationship between mixing and light availability in the lake. Mixing, generated by turbulent convection, alters the location of Aulacoseira within the depth of the lake, consequently altering its ability to obtain light for growth. Aulacoseira's abundance and colony size have been measured at varying depths in Trout Lake in Northern Wisconsin. In previous work, we built a mathematical model which accounted for growth and sinking of the diatom. However, sinking was only qualitatively included. In this work, the model more rigorously incorporates sedimentation. The diatom's sinking velocity, which is not well understood in the biological community, is formulated by exploiting the low Reynold's number nature of the system in addition to diatom's cylindrical shape and its influence on drag.
Stephanie Schauer graduated from SNC in Spring 2009 as a mathematics major with secondary education emphasis. She was an active member of the Sigma Nu Delta Math Club and gave nine presentations on four research topics while a student at the College. Stephanie is currently teaching at Big Foot Union High School in Walworth, Wisconsin.
Corey Vorland graduated from SNC in Spring 2009 as a mathematics and computer science double major. He is currently studying mathematics in the PhD program at South Dakota State University.
Summer 2010: Counterfeit Coins and Egyptians Fractions
Variations of the Counterfeit Coin Problem
by Kayla Pope under the direction of
Dr. John Frohliger.
The original counterfeit coin problem involves using a balance scale to determine a single counterfeit from among a collection of coins. We generalized this problem to the case in which the number of counterfeit coins is unknown. We expressed possible solutions and weighings as vectors and used dot products to characterize the results of weighings. A weighing scheme is a tree diagram in which the nodes correspond to weighing vectors and the edges leaving a node are determined by the results of that weighing (and the preceding weighings). Our most unexpected result involved the size of a minimal weighing tree. We speculated that, with the right choice of vectors for the nodes, a successful tree could be generated as long as the number of leaves exceeded the number of possible solution vectors. To our surprise, we were able to prove that this is not always the case.
Kayla Pope is from Oshkosh, WI, and began this project the summer after her first year at SNC. She is majoring in mathematics while minoring in computer science and plans to finish her degree requirements in three years. Kayla is a member of the SNC Women's Soccer team. In Fall 2010, she was among the team's leading scorers.
Balanced Sequences and Egyptians Fractions
by Haoqi Chen (left) under the direction of
Dr. Teena Carroll (right).
Balanced sequences, sequences whose first k terms have the same sum and product, were built. The first balanced sequence found was related to the well known Sylvester sequence, whose reciprocals sum to one. Investigating this connection leads to the ancient idea of an Egyptian fraction representation of a rational number a/b. Enumeration problems were a primary focus of this work, including finding how many Egyptian Fraction representations of one there are which use only even denominators. Using the denominators present in these representations, we found a connection to the so called Pythagorean spiral sequence, which encodes an infinite number of Pythagorean triples. Through this exploration we discovered a way of generating sequences of this type which partitions the integers into equivalence classes of Pythagorean spiral sequences using the recursion relationship which defines Sylvester's sequence.
Haoqi Chen is a mathematics major with interests in physics and computer science. He is an international student from China who plans to graduate in 2012 then pursue a graduate degree, possibly in Industrial and Systems Engineering. He is an international student from China who plans to graduate in 2012. This research began as a multiplication error with interesting consequences in Dr. Poss's Calculus II class, which Haoqi took in Spring 2009, his first semester at SNC. He continued to think about the problem for all of winter break and the collaborative research with Dr. Carroll began while Haoqi was a student in her Advanced Foundation of Math course that following spring. Haoqi has given two different talks on the results.
Summer 2011: Egyptians Fractions, the Koch Curve, and a Hiring Problem
Balanced Sequences and Egyptians Fractions
by Haoqi Chen under the direction of
Dr. Teena Carroll.
This summer Haoqi and Dr. Carroll received a student/faulty grant from the St. Norbert College Collaborative to write a
paper based on the results of Summer 2010 (above). Finding three integers which can be the lengths of sides of a right triangle is an idea that has interested mathematicians from ancient to modern times. We found a way to generate infinitely many disjoint sequences encoding Pythagorean triples.The recurrence relation which generates these
sequences is the same recurrence relation used to find the Egyptian
fraction representation of one of a given length using the largest
denominator. This connection to Egyptian Fractions is particularly
striking because it combines ideas developed separately in Ancient
Greece and Ancient Egpyt.
Haoqi Chen has investigated this particular subject area since his first semester at St. Norbert College. The project took roots in the Summer Undergrate Research Program in Mathematics at the College in 2010 (see year above).
Generalizations of the Koch Curve and Their Dimensions
by Caesar (HanQin) Cai (bottom left in picture above) under the direction of
Dr. John Frohliger (top left) and Dr. Kevin Murphy (top right).
Unlike traditional geometric shapes, fractals may have non-integer dimensions. The Koch curve is one of the basic fractals, which has dimension log4/log3. We show how to generate a fractal having any dimension between 1 and 2 by modifying the Koch curve with a focus on a Dimension Equation. This modification encouraged us to examine three questions: how can the Koch curve be generalized, when can we use the Dimension Equation, and what dimensions are possible with generalizations of the Koch curve. Special attention is paid to space-filling variation. The number of pre-images required to parameterize the generalized Koch curve with dimension 2 is explored.
Caesar (HanQin) Cai is from Shanghai, China. He is a double major in mathematics and computer science, and plans to graduate in 2012. He is an active math speaker who has given four math talks in the past three years. Notably, one of these presentations won second place in Wisconsin Mathematical Modeling Challenge in Fall 2010. Ceasar is also a Mathematics Teaching Assistant, a College tutor, and the President of the St. Norbert Sigma Nu Delta Math Club (elected twice from 2010 through 2012).
Parameterizing the Koch Curve
by Brian Pietsch (bottom right in picture above) under the direction of
Dr. John Frohliger and Dr. Kevin Murphy.
One of the most famous fractals is the self-similar Koch curve. It is known for having infinite length, and it is generated by infinite iterations of four affine transformations. We use these transformations along with a base four addressing scheme to create a well-defined, continuous parameterization of the Koch curve.
Brian Pietsch is from Green Bay, WI, and is a double major in mathematics and physics with plans to graduate in 2012. The research project began after Brian's junior year at SNC. He currently has plans to give several talks on this research, and he also plans to pursue a graduate degree in mathematics. At MathFest 2011 in Lexington, Kentucy, Brian's presentation was judged to be one of the best Pi Mu Epsilon National Honorary Society student talks. In recognition, he was prized a financial award from the American Mathematical Society and the American Statistical Association.
An Extension of the Hiring Problem
by Nicole Harp (center in picture above) under the direction of
Dr. John Frohliger and Dr. Kevin Murphy.
A classic probability problem, the Hiring Problem, also known as the Secretary Problem, looks at the optimal strategy for hiring the best-qualified candidate from a pool of applicants. If the best applicant is hired it is considered a success. it is dependent on the fact that an applicant cannot be revisited once he or she has been interviewed. We extended this problem from originally hiring the best applicant, to instead hiring one of the top (pre-determined) applicants. The probability of success cannot be maximized using basic techniques so we used numerical approximations to explore specific cases. Further, we found that by allowing one to hire from the top number of applicants, the maximum probability of a success will increase.
Nicole Harp is from Appleton, WI. This summer research project began just prior to her senior year. Nicole is pursuing a double major in mathematics and economics. She plans to complete her degree program in three and a half years. Nicole has been an active member of the Sigma Nu Delta Math Club since her first year at SNC. She also teaches tennis during the summers at her former high school.
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