UNDERGRADUATE PROGRAM

Junior Colloquium

In the Fall semester of 2002 the department began a new weekly afternoon seminar called the “Junior Colloquium” on Tuesdays. It is intended for talks of broad mathematical and scientific interest, given by both faculty and graduate students, at a level that bright undergraduate math majors can follow. Undergraduates, graduate students, and faculty are all encouraged to attend. Refreshments are served before the colloquium.

The offerings in this inaugural semester were deliciously diverse: celestial mechanics, the Banach-Tarski paradox, Latin squares, nonlinear waves, Fibonacci numbers, pattern formation in oscillator arrays, computer object recognition, Morse theory, Fourier transforms, homotopy theory, coin-tossing and Brownian motion, the Langlands program, and Penrose tilings.

So far, the core audience has consisted of lower-level graduate students, with occasional strong showings by undergraduates. We would like to see the undergraduates take over! On many Tuesdays the Junior Colloquium is followed by meetings of the new Undergraduate Math Club, meeting in the new Math Majors Lounge in Vincent Hall 116.

The Junior Colloquium is co-organized by Vic Reiner (faculty), and graduate students Jon Rogness and James Swenson. The current schedule can be found by a link from the main math page or Reiner’s web page. Contact any of the organizers if you are interested in speaking.

Vic Reiner, Professor of Mathematics

New Honors Sequence

This fall, the mathematics department initiated a new honors math sequence in multivariable calculus. This three course sequence is designed to help develop the mathematical potential and ability of promising undergraduates (and area high-school students). The sequence has a much higher requirement of mathematical rigor explicitly including proofs in the treatment of topics than in the analogous IT sequence. The students are excellent. They are both talented and hardworking. Our biggest challenge is to increase our pipeline of promising math majors.

Steve Sperber, Professor of Mathematics

The New Undergraduate Math Club

The Undergraduate Math Club meets on Tuesdays, after the Junior Colloquium, in the new Undergraduate Lounge located in 116 Vincent Hall. The goal of the club is to provide students with a forum to discuss and share information on topics such as graduate school, curricular issues, careers in mathematics, summer internships and participation in math competitions. To have fun is also one important goal of the Club; pizza is served at most meetings of the Club. The Club meetings are organized by faculty members Carme Calderer and Jackie Shen. Every meeting features a guest visitor to hold informal discussions with students. Several faculty members from the School of Mathematics visited the Club in the Fall of 2002. Students took the opportunity to ask questions about the research field of each visitor. We are always on the lookout for speakers.

Carme Calderer, Professor of Mathematics

Research Experiences for Undergraduates

The summer 2002 REU program involved 18 students from the U.S. and Canada. We had weekly presentations of students’ work, after Friday pizza lunch, in addition to presentations within groups during the week. Faculty mentors were Professors Carme Calderer, Paul Garrett, Rachel Kuske, Vic Reiner, and Sasha Voronov.

Carme Calderer supervised two participating undergraduate students: Phil Mendelsohn (Univ. of Minnesota) and Pearl Sandwick (New York Univ.). Phil Mendelsohn did numerical simulations of time dependent problems of phase transitions. For special geometries, the problems lead to nonlinear ordinary differential equations. Phil studied long-time behavior and bifurcation of solutions. Pearl Sandwick studied nonlinear elasticity and calculus of variations. Such topics provided her with mathematical and physical background to analyze models of elastomers. These are anisotropic materials that experience large deformations in the presence of electric fields. Such materials are investigated in connection with modeling of artificial tissues. Pearl studied extensional elastic deformations resulting from an applied electric field.

The topic of Vic Reiner’s group, consisting of Scott Hirschman (Univ. of Northern Iowa), Brian Jacobson (Univ. of Minnesota), Minseung Kim (Univ. of Pennsylvania), and Andy Niedermaier (Harvey Mudd College), was “Trees, determinants and Pfaffians”. They explored some results related to the enumeration of spanning trees and Kirchoff’s celebrated Matrix-Tree Theorem which counts spanning trees in a graph via a determinant. S. Hirschman and Professor Reiner found a new, direct proof of the recent Pfaffian Matrix-Tree Theorem of Masbaum and Vaintrob, which counts spanning trees in 3-graphs via a Pfaffian. They submitted their work as a note to “Graphs and Combinatorics”. A. Niedermaier and B. Jacobson worked on computing the structure of a certain finite abelian group (“the critical group”) associated to a graph, whose order equals the number of spanning trees. For some families of graphs, when they were able to calculate this number of trees, they could determine the structure of this group very explicitly. These results are likewise being submitted for publication.

Paul Garrett’s group, Jacob Bernstein (Univ. of Michigan), Anna-Marie Bohmann (M.I.T.), Lee Dicker (McGill Univ.), Amanda Febey (St. Olaf’s College), Joshua Green (Univ. of Arizona), Roxanne Johnson (Northland College), Michael Lieberman (Reed College), Stephen Lu (Princeton Univ.), Ben Rosenfield (Middlebury College), and Kristen Shaw (Univ. of British Columbia), investigated a variety of issues related to number theory, modular forms, and computations, both practical as well as theoretical. Projects included Amanda’s C++ code for some cryptographic applications, Jacob’s survey of Siegel modular forms, Kristen’s discussion of Goodstein sequences and inaccessible cardinals, Ben’s overview of Tate’s thesis, Anna-Marie’s treatment of Sylow theorems and related ideas about group actions, Roxanne’s cryptology course development outline, Michael’s introduction to Hecke L-functions, Stephen’s examples of Rankin-Selberg integrals, and Josh’s investigation of how one would or wouldn’t prove the Riemann Hypothesis. Sasha Voronov’s group shared Scott Hirschman (Univ. of Northern Iowa) and Brian Jacobson (Univ. of Minnesota) with Vic Reiner’s group. First, the students learned some basics of topology, algebra, and combinatorics, necessary to study graph homology. Then they studied ribbon graph homology, which produces a combinatorial description of the homology of the moduli spaces of algebraic curves. Scott Hirschman looked at different ways of defining orientation on a graph and showed they all were equivalent. Brian Jacobson wrote a computer program in GAP (Groups, Algorithms and Programming) which lists all ribbon graphs with a given number of edges, computes the automorphism group of a ribbon graph and the ribbon graph homology. If not for the natural time limitations of the algorithm, this would solve the tantalizing problem of computing the homology of moduli spaces. In a 30 minute computer time, the program was able to correctly compute the homology of some low-dimensional moduli spaces, in several known cases. This program is a first step towards computer computation of graph homology. Further improvement of the algorithm should not only produce new computations, but also gather evidence for a number of conjectures on moduli spaces, which have been motivating research in this new and exciting area of mathematics.

Paul Garrett, Professor of Mathematics and REU Coordinator