Research Paper Guidelines
  • Your paper must be original work of your own writing.
  • Your paper must be double spaced! This is helpful for grading purposes. The exceptions to double spacing are math environments and code listings, these should remain single spaced.
  • It should be approximately 14 - 20 pages long. Some variation is acceptable. Your paper will be judged on its quality not quantity.
  • It should discuss some piece of mathematics using mathematical notation where appropriate.
  • You must typeset your paper with LaTeX and submit a paper copy.
  • You may use the report or article document class. It is your choice.
  • You must have a minimum of three sources, at least two of which are books or journal articles.
  • Your paper must have at least one figure in it.
  • You should properly cite your sources, and include a bibliography or references section.
  • Web pages are acceptable sources. Please cite them properly. For example, Wikipedia has a special link on every page for citation purposes.
  • You should write your paper at your level of understanding. That is, it should be written such that you and your classmates can understand it, perhaps with some effort.
  • You do not need to show every step of every derivation or calculation, but you should show enough that your classmates could fill in the gaps.
  • A starter LaTeX template which demonstrates double spacing as well as how to generate filler text is available at: rp.tex,  chart.png
  • Use footnotes sparingly.
  • Figures, tables and code listings should have captions.

Research Paper Topics

Useful Journals (All at the undergraduate level)

Applied Math

  • Matrix Factorization Techniques for Recommendation Systems (Netflix Prize)
  • The Mathematics of Google's Page Rank Algorithm
  • Fourier Series (decomposition of periodic signals into sums of sines and cosines)
  • Numerical Solutions of Differential Equations (Euler, Leapfrog, Runge-Kutta)
  • Game Theory (Analyze a specific game of your choice)
  • Linear Programming (The name is a little misleading because is not directly related to computer programming, it is a method of optimization. However, the algorithms are very amenable to implementation in computer code.)
  • Public Key Cryptography (This forms the basis of nearly all financial transactions that occur over the Internet.)
  • Solving Equations Numerically (Newton-Raphson Method) (one can also explore fractals via Newton's method when it is applied to complex numbers.)

Analysis

  • The Contraction Mapping Principle (Banach Fixed-point Theorem)
  • Constructing R from Q: (two ways, pick one)
    1. Construction of the Real Numbers Via Cauchy Sequences (Completion, Contraction Mapping Principle)
    2. Construction of the Real Numbers Via Dedekind Cuts
  • The Lebesgue Integral and Measure Theory (Chapter 14 in "The Way of Analysis" by Robert S. Strichartz)
  • Cauchy's Theorem (See Chapter 2 of "Basic Complex Analysis" by Jerrold E. Marsden and Michael J. Hoffman)

Calculus

  • Elliptic Functions (used in Cryptography!)
  • Celestial Mechanics: Newton's Derivation of Kepler's Laws. (The famous physicist Richard Feynman wrote an excellent article on this!)
  • Stokes' Theorem ("div grad curl and all that: an informal text on vector calculus" 3rd ed. by H. M. Schey also see: "Advanced Calculus: A Differential Forms Approach" by Harold M. Edwards)
  • The Calculus of Variations (Brachistochrone Problem, Lagrangian Dynamics)

Geometry and Topology

  • Pappus' Theorem
  • Projective Geometry
  • Gauss' Formulation of Curvature
  • Julia Fractals and the Mandlebrot Set
  • Euler Characteristic of a Surface
  • Book: "From Geometry to Topology" by H. Graham Flegg, Dover, 2001.
  • The Fundamental Group of a Surface
  • Knot Theory
  • Algebraic Graph Theory

Algebra

  • The Solution of Cubic Polynomial Equations (Cardano/Bombelli)
  • The Binomial Theorem
  • Braid Groups
  • Cayley Graphs (John Meier, "Groups, Graphs and Trees: An Introduction to the Geometry of Infinite Groups", Cambridge, 2008)
  • The Chinese Remainder Theorem
  • Factorization of Polynomials and Eisenstein's Irreducibility Criterion (Eisenstein's Irreducibility Criterion can be implemented in computer code)
  • Tesselations of the Plane (The 17 symmetries of the plane)
  • Permutations and Games (I. N. Herstein and I. Kaplansky, "Matters Mathematical", New York: Chelsea, 1978. Chapter 3 of this book discusses several interesting applications of permutations to games.)
  • Quaternions (Discovered by William Rowan Hamilton, they are an extension of the complex numbers, and there are two more extensions beyond them)
  • J. Alperin, "Groups and Symmetry," Mathematics Today, New York: Springer-Verlag. 1978.
  • S. Winters, "Error-Detecting Schemes Using Dihedral Groups," UMAP Jounal 11, no. 4 (1990): 299-308
  • Matrix Groups
  • Symmetric Matrices, Quadratic Forms, and Conic Sections
  • The Fundamental Theorem of Algebra (Gauss, D'Alembert)

Number Theory

  • The Riemann Hypothesis (J. Derbyshire, "Prime Obsession")
  • Quadratic Reciprocity by Eisenstein's proof (counting lattice points)
  • Fermat's Little Theorem

Logic and Foundations

  • Godel's Incompleteness Theorem
  • Cantor's Diagonalization Argument and The Countability of The Rationals
  • Computability (Turing, Post)