MathFest 2008 Invited Paper Sessions
CLASSICAL EUCLIDEAN GEOMETRY
Paul Yiu, Florida Atlantic University
Thursday, July 31, 1:00 p.m. – 4:00 p.m.
The speakers and the titles of their talks will be: Eisso J. Atzema, University of Maine, who will speak on “The (history of the) use of oriented angles and their extension to n-sections”; Clark Kimberling, University of Evansville, “High points in the Encyclopedia of Triangle Centers”; James L. Parish, Southern Illinois University at Edwardsville, “Special Quadrilaterals and Special Conics”; Steve Sigur, The Paideia School, Atlanta, Georgia, Algebraic ramifications in triangle geometry; Bogdan Suceava, California State University at Fullerton, “The equation of Euler's line yields a Tzitzeica surface” and Paul Yiu, “The circles of Lester, Evans, Parry and their Generalizations.”
ISOPERIMETRIC PROBLEMS AND MANIFOLDS WITH DENSITY
Frank Morgan, Williams College
Thursday, July 31, 2:00 p.m. – 5:00 p.m.
Members and alums of the Williams College NSF SMALL Undergraduate Research Geometry Group and perhaps others will report on work on isoperimetric problems and open questions in various settings, including Riemannian manifolds and manifolds with density (which appeared in Perelman's papers on the Poincaré Conjecture).
Fifteen-minute talks with five-minute breaks.
Program and abstracts:
2:00 Frank Morgan, Williams
"Isoperimetric problems, manifolds with density, and the SMALL undergraduate research project"
Abstract: Giving a manifold a density as in freshman physics, as occurs naturally throughout mathematics and applications, opens up new fundamental questions in geometry. The classical isoperimetric problem seeks the least-perimeter way to enclose a given amount of volume; in Euclidean space, the solution is a round ball. In a manifold with density, the isoperimetric problem seeks the least-weighted-perimeter way to enclose a given amount of weighted volume; the solutions are often surprising. The Williams College NSF SMALL undergraduate research project is now celebrating its 20th anniversary.
2:20 Alex Dubbs, Harvard '09
"The Isoperimetric Problem on Surfaces with Density"
Abstract: The isoperimetric problem on a surface seeks the least-perimeter way to enclose a given amount of area. What if you give the surface a density weighting both area and perimeter? Work of the 2008 NSF SMALL undergraduate research Geometry Group.
2:40 Hung Tran, Berea College '09
"The Isoperimetric Problem in the Plane with Density r^p"
Abstract: The least-perimeter way to enclose a given amount of area in the plane with density r^p is not a circle about the origin as you might expect. We report on our efforts to find the answer. Work of the 2008 NSF SMALL undergraduate research Geometry Group.
3:00 Jon Dahlberg, Williams '09
"The Geometry of Surfaces with Density"
Abstract: Providing a surface with a density function leads to new concepts of geometry and curvature. (Such densities appear in Perelman's original paper on the Poincaré Conjecture.)
3:20 Edward Newkirk, Williams '09
"Optimal Partitions of the Sphere"
Abstract: What is the least-perimeter way to partition the sphere into a small number n of equal areas? For n=2, the answer is an equator.
The solution is also known for n=3 and n=12. We discuss progress on the case n=4. Work of Conor Quinn and the NSF SMALL undergraduate research Geometry Group.
3:40 Joseph Corneli, PlanetMath.org
"Hypertext Platforms"
Abstract: We will describe a new tool for building hackable multi-user interactive semantic hypertext platforms.
GRAPH THEORY WITH CONNECTIONS TO GEOMETRY AND TOPOLOGY
Joshua Laison, Willamette University
Thursday, July 31, 3:15 p.m. – 5:45 p.m.
This session will feature the following presentations: Ellen Gethner, University of Colorado Denver, "An Adventurer's Guide to the Colorful Tale of Thickness-Two Graphs"; Stephen Hartke, University of Nebraska Lincoln, "Visibility Graphs: Different Models of Sight"; Alice Dean, Skidmore College, "Recent Rectangle Visibility Results";
Michael Pelsmajer, Illinois Institute of Technology, "Crossing Numbers"; and Art Busch, University of Dayton,"Recognition Algorithms for some Generalizations of Interval Graphs".
MATHEMATICAL BIOLOGY
Julie Mitchell, University of Wisconsin
Gheorghe Craciun, University of Wisconsin
Friday, August 1, 1:00 p.m. – 3:00 p.m.
Speakers and their topics will include: Daniel Beard, Medical College of Wisconsin, “Physical Chemical Principles for Mathematical Analysis and Simulation of Large-Scale Biochemical Systems”; Gheorge Craciun, University of Wisconsin - Madison, “Mathematical Methods for Understanding Biochemical Reaction Networks”; Isabel Darcy, University of Iowa, “Modeling Protein-DNA Complexes Using Tangles”; and Bo Su, Iowa State University, “Mathematical Modeling of Immune Response in Tissue.”
RAMANUJAN’S IMPACT ON NUMBER THEORY – THEN AND NOW
James Sellers
Pennsylvania State University
Friday, August 1, 1:00 p.m. – 3:00 p.m.
For the past 90 years, Srinivasa Ramanujan's profound impact on number theory has been easily seen. It is clearly demonstrated by the wide variety of related mathematical endeavors that have been undertaken by many since his brief years at the University of Cambridge with G.H. Hardy. The goal of this session will be to share historical and contemporary perspectives on Ramanujan's results as well as significant work that has been accomplished in recent years in areas for which Ramanujan was fond - from arithmetic properties of integer partition functions to series approximations of 1/π to continued fractions and so much more. The speakers will include Krishna Alladi, University of Florida; George Andrews, Penn State University, Bruce Berndt, University of Illinois at Urbana-Champaign; and Ken Ono; University of Wisconsin. All are well-known for their excellent research in the subject as well as their craftsmanship as speakers. The session promises to be truly enlightening and enjoyable.
IMPLICATIONS FOR TEACHING OF RESEARCH ON LEARNING
Patrick Thompson, Arizona State University
Friday, August 1, 3:15 p.m. – 5:15 p.m.
This session will feature the following presentations:
Marilyn Carlson, Arizona State University "The role of quantitative reasoning in learning precalculus mathematics"; Sean Larsen, Portland State University "From individual interviews to whole-class experiments: Reconstructing a course in abstract algebra"; Keith Weber, Rutgers University "Effective and ineffective learning strategies in real analysis"; and Pat Thompson, Arizona State University "Learning without understanding and its implications for the mathematics that teachers teach".
RESEARCH WITH UNDERGRADUATES
Mario Martelli, Claremont-McKenna College
Saturday, August 2, 1:00 p.m. – 3:00 p.m.
Speakers will present research in Mathematics completed in collaboration with undergraduates and, possibly, submitted for publication to a professional journal. In some cases the articles may have already been accepted. Each speaker will describe in detail how the research was done and will highlight the undergraduates' participation.
1. Saturday August 2, from 1 PM to 1.30 PM
Author
Prof. Annalisa Crannell
Department of Mathematics
Franklin & Marshall College
Lancaster, PA 17604-3003
Email: annalisa.crannell@fandm.edu
Title
From Chaos to Colleagues
Abstract:
In the spirit of Paul Halmos' advice on writing ("There is no recipe, and what it is"), I will share some of the idiosyncrasies of my own successful research collaborations with students. The advice I will share includes these dubious gems: Start collaborating with students before you start collaborating with them. Get students to work independently by gathering them into small groups. By doing lots of work, you can get students to do all your work. Carefully choose just the right problem, and then don't stick with it.
Author
Prof. Colin Adams
Thomas T. Read Professor of Mathematics
Bronfman Science Center
Williams College
Williamstown, MA 01267
Email: Colin.C.Adams@williams.edu
Title
Knot Theory Research with Undergraduates
Abstract
Knot theory is an area that lends itself to research by students. One can state open problems with almost no need for background and students can play with their shoelaces the first day. We will discuss the kinds of results students have already obtained and directions for further research by students.
3. Saturday August 2, from 2.00 PM to 2.30 PM
Author
Prof. Asuman Aksoy
Department of Mathematics
Claremont McKenna College
Claremont, CA 91711
Email: aaksoy@cmc.edu
Title
Compactness in metric trees
Abstract
A metric tree (M,d), also known as R-tree or T-Theory, is a metric space such that any two distinct points P and Q of M are connected by a unique arc and that arc is isometric to an interval in . I will present some fundamental properties of metric trees and characterize their compact subsets. These results will be used to show that a map
is k-set contractive if and only if T is k-ball contractive.
4. Saturday August 2, from 2.30 to 3.00
Author
Prof. Marc Chamberland
Department of Mathematics and Statistics
Grinnell College, Grinnell, IA 50112
Title
Ramanujan's Dream
Abstract
One can only imagine what Ramanujan would have discovered had he lived a century later with a computer to help him explore his ideas. Today's computing tools allow even undergraduates to make mathematical discovery easier and accessible. Some recent student projects will be detailed, including generalizing a beautiful Ramanujan identity
and factoring Hankel matrices.
5. Saturday August 2, 3.00 PM to 3.30 PM
Prof. Carlos Castillo-Chavez
Department of Mathematics and Statistics
Arizona State University,
Tempe, AZ 85287
Email: ccchavez@asu.edu
Title
Building Communities through REU Programs
Abstract
Despite tremendous national efforts to entice American students into graduate programs in the mathematical science the numbers do not seem to add up. As a nation, we continue to have difficulties recruiting and keeping American students in the mathematical sciences despite the efforts instigated by IGERT, VIGRE, AGEP, LSAMP and NIH and NSF Training Grants. Tradition is often the engine behind the growth and survival of scientific and Mathematical communities. The birth and growth of American Schools in pure and applied mathematics are closely tied in to the migration of large numbers of extraordinary mathematicians before, during and after WWII. How do we build such communities? How do we establish cultures that promote the growth and vitality of the mathematical sciences? REU programs play a significant role. In this talk, I will describe the model that the Mathematical and Theoretical Biology Institute (MTBI) has developed over the past 14 years and the way it has enhanced through its collaborations with the Institute for Strengthening Understanding of Mathematics and Science (SUMS) that was established in AZ by Joaquin Bustoz Jr, 23 years ago.
6. Saturday August 2, from 3.30 PM to 4.00 PM
Author
Prof. Jacqueline Jensen
Department of Mathematics and Statistics
Sam Houston State University
410 Lee Drain Bldg.
Huntsville, TX 77341-2206
Email: Jensen@shsu.edu
Title:
How to Juggle Seven Undergraduate Student Projects Without Dropping Any
Abstract
Sam Houston State University (SHSU) provides a Methods of Research course. Undergraduate students participating in this class individually choose research topics of interest to them. Under the guidance of the instructor, students investigate this area, with their results presented as talks at the meeting of the Texas Section of the MAA and as posters displayed at the SHSU Honors Consortium. Individual topics range from expository (The Existence of Infinity Through Transfinite Numbers) to pure research (Fourier Transforms and 4-D Tensor Based Wave Equations). We will discuss some of the projects researched during the Spring 2008 semester, as well as suggestions for other schools interested in implementing such a program. Particular attention will be paid to juggling techniques.
7. Saturday August 2, 4.00 PM to 4.30 PM
Author
Prof. Michael Orrison
Department of Mathematics
Harvey Mudd College
Claremont, CA 91711
Email: orrison@hmc.edu
Title
The Applied Representation Theory Group at Harvey Mudd College
Abstract
Each year, I lead a research group of three to five undergraduates. Our group is called the Applied Representation Theory (A.R.T.) group, and is made up of students doing senior thesis, summer research, and independent research projects. In this talk, I'll survey some of the projects we have tackled in recent years (e.g., "Algebraic Voting
Theory" and "FFTs for the Symmetric Group"), and describe some of the mechanics involved in sustaining what has become one of the most enjoyable ways in which I interact with students.
8. Saturday August 2, 4.30 PM to 5.00 PM
Author
Prof. Aihua Li
Department of Mathematics
Montclair State University
Montclair, New Jersey, 07043
Title
Tracing Certain n - Dimensional Space Points
Abstract
Given a timely ordered set of n-dimensional space points P1, P2, … , Pm+1, we look for a mathematical function f to trace the points so that f(Pi) = Pi + 1 for i = 1, 2, ... , m. We study methods to construct linear tracing functions (linear models) for certain special types of space point sets, called “Jordan-like” point sets (a set of points with `Jordan-like' column-echelon form). Linear algebra techniques are applied to investigate conditions on the existence of homogeneous linear models for such point sets and to develop methods of constructing homogeneous linear models when they exist. In addition, every linear model generates a sequence that has a period point. Periodic behavior of such extended sequences is discussed.
In this presentation, I will discuss the development and results of an undergraduate research project on discrete time series modeling. Michael Wilson, a math senior, has been working on this project since fall of 2007. A joint paper is to appear in the Pi Mu Epsilon Journal.
GEMS IN NUMBER THEORY
Sarah Mabrouk, Framingham State College
Thomas Koshy, Framingham State College
Saturday, August 2, 3:15 p.m. – 5:15 p.m.
Art Benjamin, Harvey Mudd College; Ezra Brown, Virginia Polytechnic Institute and State University; Richard K. Guy, University of Calgary; and Margaret Robinson, Mount Holyoke College.
HISTORY OF MATHEMATICS
Amy Shell-Gellasch, Pacific Lutheran College
Shawnee L. McMurran, California State University at San Bernardino
Saturday, August 2, 1:00 p.m. – 3:00 p.m.
