Archive

On some ideas in Mathematics and Physics -
Max Lipyanskiy, Senior Thesis. Supervisor: Professor Patrick Gallagher).

Contents: An Elementary Proof of the Uncountability of Real Numbers;
The Cayley-Hamilton Theorem; Geometrization of Classical Mechanics for
Motion in One Dimension; A Geometric Perspective on Space-Time.

Computer-Assisted Application of Poincare's
Fundamental Polyhedron Theorem - Max Lipyanskiy, 2001-2002.

The main objective of this paper is the exposition of a series of computer
programs for determining when a subgroup of PSL(2,C) specified by a set of
generators is discrete. As an application, the program is used to construct
manifolds associated to each of the seven exceptional regions discussed in
the work of Champanerkar, Lewis, Lipyanskiy and Meltzer. The program and a
User's Guide can be downloaded
here.

Alternating sign matrices and tilings of Aztec
rectangles - David Anderson, Undegraduate Thesis 2001.
Supervisor: Professor Douglas Zare.

Here Dave derives a recurrence relation for the number of domino tilings of
Aztec rectangles with squares removed along one or both of the long edges.
A theorem of Lindstrom-Gessel-Viennot is used to express this number in terms
of determinants.

Continuous Choice of Summands in Convolution Sums
on a Lie Group - Alex Sotirov, Undegraduate Thesis 2001.
Supervisor: Professor Herve Jacquet.

This paper refines a prior result which
establishes a representation for any smooth function on a Lie Group (with
compact support) as a finite sum of convolutions of smooth functions with
compact support.

An Elementary Proof of the Uncountability of
Real Numbers - Max Lipyanskiy, Fall 2001.

A proof based on the least upper bound property of real numbers without use
of compactness, measure or decimal expansions.

The Casimir Effect on a 1-Dimensional Manifold of p
Periodicity - Dave Anderson, John A. Conley, Scott A. Meltzer and
Mark G. Jackson, Winter 2001.

Using Abel-Plana formula, they calculate the Casimir energy of a massless
scalar field on a one-dimensional twisted manifold of periodicity p.

A Review of the
Abel-Plana Formula
- Dave Anderson, Winter 2001.

An expository paper containing the derivation of the formula as well as
a short discussion of applications to the Casimir Effect.

Path Integrals in Quantum Mechanics
- Dave Anderson, Summer 2001.

This is a rough draft of a review paper summarizing some of Dave's work
on the mathematics of quantum mechanics and, in particular, the Lagrangian/path
integral formalism.

Algebraic and Transcendental Numbers VIGRE 2001
- Eric Patterson, Vladislav Shchogolev, Summer 2001.

An expository paper presenting material on number theory with detours to
other areas of mathematics. This paper summarizes the Summer 2001 Seminar
supervised by John Zuehlke.

Computational Algebraic Geometry VIGRE 2001 -
David Drescher, Bhavana Nancherla, Summer 2001. Supervisor: Prof. Henry Pinkham.

The group investigated some of the algorithmic issues connected with Groebner bases.

A cusp singularity with no Galois cover by a
complete intersection - Dave Anderson, 2001. Supervisor: Prof. Walter Neumann.

A paper on Algebraic Geometry based on Dave's research.

Exceptional Hyperbolic 3-Manifolds - Abhijit
Champanerkar, Jacob Lewis, Max Lipyanskiy, Scott Meltzer, Summer 2001.

This project was supervised by W. Neumann. The group investigated questions
of existence of exceptional
manifolds
which arise in a recent important
paper
establishing that "homotopy hyperbolic 3-manifolds are hyperbolic".

3-Manifolds and Topology
- John Bueti, Philip Ording, Ari Stern, Summer 2000. Supervisor: Walter Neumann.

This project computation of the A-polynomial of hyperbolic 3-manifolds. This very
important recent invariant for 3-manifolds appears to have deep connections with other
invariants. The emphasis of the project was computational geometry.

Application of Algebro-Geometric Method
to Generation of Pretty Pictures - David Kagan, Darya Krym, Philip Ording, 1999.
Supervisors: Profs. Igor Krichever and Anoton Dzhamay.

A paper decribing a method for generating surfaces and orthagonal co-ordinate
systems on them.