Undergraduate Program
UNDERGRADUATE PROJECTS
Senior projects involve substantial one-on-one student-faculty interaction and enhance greatly the students’ learning experience. Below is a list of the faculty members who recently supervised such projects, together with the titles of the projects.
Stephen Agard: “Decision Theory based on Bayes’s Theorem”, “Statistical Inference and Estimation”, “On the reasons for Interest as an Institution”, and “Proof of the Black-Scholes Formula for European Put-Options”;
John Baxter: “Counting partition matrices” and “Rigid motion”;
Paul Garrett: “Independence Results in Set Theory”, and “The role of the history of mathematics in the curriculum”;
Tian-Jun Li: “Symplectic vector spaces and symplectic matrices”;
Willard Miller: “Constructing Grids using a Density Function to Determine Concentration”;
Victor Reiner: “Critical groups of graphs”, and “Critical groups and line graphs”;
Joel Roberts: “Pedagogy of Gifted Students and UMTYMP”, “Postulates and Theorems in High School Geometry and in Math 5335”, “Archimedes’ Method Proposition 14: Archimedes’ place in the history of calculus”, “Quadric Surfaces: theory and computer drawings”, and “Comparing and contrasting high school and college calculus”;
Dennis Stanton: “Extra accuracy in some alternating series”.
The theses supervised by Professor Reiner were follow-ups to the students’ REU (Research Experiences for Undergraduates) work done under Reiner’s guidance during the summer of 2003. Both theses dealt with the notion of the critical group of a graph. This is a certain finite abelian group naturally associated to the graph, whose order equals the number of spanning trees of the graph. One of the students solved some open problems about this critical group, including one posed by Greg Kuperberg of UC Davis. The other student, who in the meantime began graduate studies, obtained results that were suggested by his REU data on the critical groups of regular line graphs. This has led to a further conjecture which is a subject of ongoing investigations by the student, Professor Reiner, and another grad student of Reiner.
CRYPTOLOGY AND CODING THEORY COURSES
These two popular courses were developed by Professor Paul Garrett and taught by him since 1995. Since there were no completely satisfactory texts covering the subject matter in a manner appropriate for our undergraduates, Paul even wrote his own books—”Making, Breaking Codes, and Intro to Cryptology”, and “The Mathematics of Coding Theory: Information, Compression, and Error-correction”, both published by Prentice-Hall. The cryptology book was recently translated into Chinese for distribution in Asia. Because of the high student interest in these courses (well over one hundred students take each of the two courses each year), three other faculty members (Mark Feshbach, David Frank, and Vic Reiner) taught crypto in Fall 2004, with Paul serving as the Course Supervisor. Don Kahn and Gennady Lyubeznik are teaching the Coding course in the spring of 2005. Steve Sperber taught Coding two years ago. Crypto and coding theory are vital areas of application of pure mathematics to internet commerce and to data storage. Owing to Paul’s dedication, our department is one of the few mathematics departments in the country with such a substantial undergrad course offering in these subjects.
News about the Graduate Program
From Scot Adams, Director of Graduate Studies
This year there are 22 incoming students, the same number as last year. Ten are international; eight are women.This year’s orientation continued the long tradition of placement exams, begun by Paul Garrett. Results of these exams are a tool to help incoming students (with the aid of their advisors) decide which of the standard first year courses they are prepared to take, and which will likely require additional preparation.
We congratulate our graduating PhD students (Oct. 2003 to Dec. 2004):
Radek Erban ("From Individual to Collective Behavior in Biological Systems.", Hans Othmer, advisor; Oxford University)
Michael L Galbraith (“Geometric Optics, Convex Functions, Carleman Estimates and Interfaces in the Boundary Control of the Wave Equation.”, Robert Gulliver, advisor; National Security Agency)
Oleg Alexandrov (“Wave Propogation in Optical Fibers: Analysis and Optimization.”, Fadil Santosa, advisor; UCLA)
Youngae Han (“An Efficient Solver for Problems of Scattering.”, Fernando Reitich, advisor; CalTech)
Nicolae Tarfulea (“Constraint Preserving Boundary Conditions for Hyperbolic Formulations of Einstein’s Equations.”, Doug Arnold, advisor; Purdue University Calumet)
Kyeong-hun Kim (“On Stochastic Partial Differential Equations with Variable Coefficients in C1 domains.”, Nicolai Krylov, advisor; University of Utah, Salt Lake City)
Jae Hyouk Lee (“Geometrics Motivated from Normed Algebras.”,Naichung Conan Leung, advisor; Washington University, St.Louis)
Kijung Lee (“Lp Theory of Stochastic Partial Systems.”, Nicolai Krylov, advisor; University of Rochester)
Jens Rademacher (“A Mechanism for Periodic Secondary Wave Bifurcation of Pulses in Reaction-Diffusion Systems.”, Arnd Scheel, advisor; University of British Columbia)
We congratulate our graduating Master’s Degree students (Oct. 2003 to Sept. 2004):
Hyeung-Joon Kim (Naresh Jain, advisor)
Lin Fan (Stephen Agard, advisor)
Yang-Jin Kim (Hans Othmer, advisor)
Guang-Tsai Lei (Walter Littman and Peter Rejto, advisors)
Fan Yang (Stephen Agard, advisor)
Mingyan Lin (Mitchell Luskin, advisor)
Zahra Al Shamrani (Paul Garrett, advisor)
Lisa Rassel (Fadil Santosa, advisor)
Shagufta Nazir Ahmad (Bernardo Cockburn, advisor)
Leah P Prom (Harvey Keynes, advisor)
Adam Galambos (Victor Reiner, advisor)
Yang-Jin Kim, Shagufta Nazir Ahmad and Zahra Al Shamrani continue as graduate students in our department. According to an analysis done by the Graduate School, our graduate program’s completion statistics have a favorable comparison with the general results in Engineering, Physics, etc. For example, the six year completion rates for our PhD graduate students entering in 1994,1995, 1996 and 1997 are, respectively 58%, 67%, 85% and 73%. The general figures for the same years are 44%, 48%, 49% and 37%.
Application for admission to our graduate program shows a favorable trend. According to the Graduate School, we had 176 completed applications in 2000-2001, then 207 in 2002-2003 and then 249 in 2003-2004. We are hoping for another great year!
We have begun a long process of trying to determine the current whereabouts of our 285 Mathematics PhD graduates since 1981. For 184 of these 285, we have a last-known email address, and will soon begin testing to see how many of these addresses are currently functional.
NEW GRADUATE COURSE OFFERINGS
Professors George Sell and Tian-Jun Li developed interesting new graduate level courses on Global Climate Modeling and on Symplectic Structures on Manifolds. We thank them both for sharing with our readers their descriptions of these innovative courses. Professor Sell is presenting his course for the first time during the Spring 2005 Semester. It is his intention to make this course a regular graduate level offering in the School of Mathematics. The description of Professor Sell’s course appears in the section “FEATURED COLLEAGUES”. Professor Li taught the course for the first time during the 2003-04 academic year. He has written a book on the subject, based on the lecture notes of his course. The title is “Moduli spaces of symplectic structures” and it will be published by the World Scientific Publishing Company.
SYMPLECTIC STRUCTURES ON MANIFOLDS
The course covers an important and rapidly developing new area of mathematics and mathematical physics. Symplectic structures originated in classical Hamiltonian mechanics, including the mechanics of the solar system as well as rigid body systems. Over the years, they have become important in many branches of mathematics. This development is an excellent example of how physics inspires mathematics. In modern terms, a symplectic structure is given by means of a non-degenerate closed differential form of degree 2. Every 2-manifold has such a structure, via a volume form. One rich source of symplectic manifolds comes from a beatiful branch of mathematics, namely algebraic geometry. In the last few years it has become apparent that symplectic geometry is also closely linked to low dimensional topology.
The course organization is as follows. In the fall semester (and part of the spring) the aim is to develop basic concepts and describe various important constructions. Prerequisites include some familiarity with differential forms and manifolds. The rest of the course focuses on recent directions of research such as applications of pseudo-holomorphic curves and Seiberg-Witten theory to classifying symplectic structures on four-dimensional manifolds.
Tian-Jun Li
GRADUATE STUDENT TOPOLOGY CONFERENCE
A topology conference intended specifically for graduate students was held at the School of Mathematics on April 24th, 2004. Twenty-one students from around the nation gathered to meet their colleagues, give research talks, and listen to the keynote address by Professor Fred Cohen of the University of Rochester.
The first Graduate Student Topology Conference was held at the University of Notre Dame in 2003; Northwestern University has offered to host it in 2005. Based on the success of the topology student conference, the School of Mathematics will host the first-ever Graduate Student Combinatorics Conference in April of 2005 (see below).
The organizers would like to thank Harry Singh and Kathy Swedell for their help, and to the department itself for providing funds for this unique conference.
James Swenson, Jonathan Rogness
GRADUATE STUDENT COMBINATORICS CONFERENCE
On the weekend of April 15-17, the department will host the first-ever Graduate Student Combinatorics Conference. This conference — modeled after the annual Graduate Student Topology Conference which was held here in 2004 — will feature over 30 participants and 25 graduate student speakers from across the U.S. and Canada. The conference will have two keynote speakers: Richard Ehrenborg and Margaret Readdy, both of the University of Kentucky. The conference is being organized by Dan Drake, John Hall, Ning Jia, Sangwook Kim, and Muge Taskin. Funding has been provided by the department and the Institute of Technology. The organizers hope that this conference will follow in the footsteps of the Student Topology Conference and become an annual event hosted at universities across the country.
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