[ugrads] [Mathclub-members] Summer Research Opportunity

Michael Pelsmajer pelsmajer at iit.edu
Fri Apr 10 11:37:04 CDT 2009

This one is only for undergraduate minority* students that have taken
Matrices or another upper-level course.  Details below.

* The word "minority" refers to members of those groups
underrepresented in the mathematical sciences, including
African-Americans, Hispanics, American Indians, Alaska Natives, and
Pacific Islanders.

Documentation: You should include a short resume of no more than 2
pages and a letter of support from your department chair or an
administrator in a position to commit institutional resources. The
names, addresses, ethnicities, and home institutions of the minority
undergraduates must be specified.

The program is designed to introduce you to a mathematical research
problem and hone your presentation (written and verbal) skills. The
areas of research involved are graph theory/knot theory problems. We
are also going to include information about employment in the
mathematical sciences and getting into graduate school. I have
enclosed an approximate schedule below.

The dates would be mid-June to late July or late June to early August.

This would be a paid program - we requested a pre-tax stipend of
approximately $3500 for the six weeks and have included funding for
the dorms at McKendree (so you would not have to pay for room and
board).  We have also budgeted for supplies (textbooks) and some
weekend activities.  (I certainly remember how tight money was as an
undergraduate for me!)

We are still waiting on the details about the financial information
from the MAA.

Please feel free to contact me if you have any other questions.

Thank you,

Heather A. Dye
Knots at McKendree University
J. Alan Alewine (Associate Professor of Mathematics) and Heather A.
Dye (Assistant Professor of Mathematics) at Mckendree University are
applying for a grant to sponsor a research experience for
undergraduates (REU) in pure mathematics at McKendree University
through the NREUP program sponsored by the Mathematical Association of
America. McKendree University offers an ideal location for a summer
research program for undergraduates: McKendree is located in the small
town of Lebanon, IL but St. Louis is only 25 minutes away. We would be
able to provide both a stimulating academic program as well as a
variety of sponsored weekend activities.

Problem Description
In this six week program students will explore one of two possible
problems. The first problem is motivated by two papers by Jozef
Przytycki and Qi Chen: The Gram matrix of a Temperly-Lieb algebra is
similar to the matrix of chromatic joins (arxiv:0806.0878) and The
Gram determinant of the type B Temperly-Lieb Algebra. The second
problem is based on a paper by Louis Kauffman and Sam Lomonaco:
Quantum knots and Mosaics (arxiv:0805.0339v1).  In this paper, the
authors define knots based on a grid structure. To study these
problems, students should complete the following courses or their
equivalent:  Introduction to proof and Linear algebra. These courses
should introduce the students to the concepts of equivalence classes,
bilinear pairings, and determinant. The motivating problems involve
number theory, algebra, and knot theory so that the research focus can
be directed based on student interest. We describe the general
research plan for each question:
1. Mosaic Knots
a. What size grid do you need to construct all 4 or 5 crossing mosaic knots?
b. What is the maximum number of crossings that an  mosaic knot can contain?
2. Temperly-Lieb Algebras in Surfaces
a. Consider a disk with m punctures and 2n points around the outermost
boundary. We denote this surface as  . Find the set of all diagrams in
  (up to isotopy) with n non-crossing chords.
b. Form a surface with genus m by gluing two copies of   together.
Tabulate the number of curves that bound a disk and the number of
curves that do not bound a disk for each pair of diagrams from
question 1.
c. Compute the determinant of the bilinear pairing found in part 2.
How does the determinant change as m or n changes?
The general problem can be modified for individual students by
changing the values of m or n, allowing the surfaces to be glued in
different ways, or studying Mobius bands.  Over the course of the
summer, students will be introduced to 2 dimensional surface theory,
algebras, and the Gram determinant. We plan to meet with students
individually for 2 hours per week and have planned a variety of
lectures and interactive activities (a summary of these activities and
a schedule are shown below). During weeks 1-2, students will work
preliminary problems based on the lectures. Students will present, in
both written and oral formats, on their solutions (and their attempts
to solve) these preliminary problems.  Students will be expected to
keep a research journal during the program and prepare a presentation
on their research. We have allotted funds for the students to travel
to a mathematics conference and present their work. We plan for the
students to present their work at either MAA's Mathfest, the Illinois
section meeting of the MAA, or the Rose-Hulman Institute of Technology
mathematics conference.


Week 1 - 2 Morning 9:30-11 AM Afternoon 2:30-4 PM
Monday Lecture Lecture
Tuesday Professional Development
Wednesday Lecture Lecture
Thursday Writing and Speaking Mathematics
Friday Lecture Lecture
Week 3-4 Morning 9:30-11 AM Afternoon 2:30-4 PM
Monday Student Mini-Presentation Student Discussion
Tuesday Professional Development
Thursday Writing and Speaking Mathematics
Friday  Student Mini-Presentations
Week 5-6
Monday Student Mini-Presentation Student Discussion
Tuesday Professional Development
Thursday Writing and Speaking Mathematics
Friday  Mini Conference

1. 6 Lectures on Introduction to Surfaces and Topology
2. 3 Lectures on Graphical Representations
3. 3 Lectures on Algebras

Writing and Speaking Mathematics
* Week 1: Problem solving strategies
* Week 2: Describing, formulating and reformulating a problem - abstracts
* Week 3: Writing definitions and proofs
* Week 4: Presentations
* Week 5: Putting it all together
* Week 6: Revisions
Professional Development
* Week 1: What can I do with a math degree?
* Week 2: How do I get into grad school?
* Week 3: The GRE
* Week 4: What should I do in grad school?
* Week 5: Networking!
* Week 6: Employment in the mathematical sciences

Weekend Activities:
1. St. Louis Art Museum
2. Shakespeare Festival
3. St. Louis Zoo
4. City Museum/ St. Louis Arch
5. St Louis Science Center
6. Local festival in Lebanon, Trenton, or O'Fallon
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