Past DRP Projects

Mathematics Undergraduate Program

Below is a listing of older DRP projects.

Fall 2008

Fractal Geometry
Mentee: Daniel Greene
Mentor: Andrew Baxter
Text: Gerald Edgar Measure, Topology, and Fractal Geometry; Yamaguti, Hata, Kigami Mathematics of Fractals
Topics: fractal geometry, Cantor set, Sierpinski gasket, topology of metric spaces, topological dimension, fractal dimension, self-similarity

Modal Logic
Mentee: William Gunther
Mentor: Jay Williams
Text: Chellas Brian Modal Logic: An Introduction
Topics: propositional modal logic, normal systems, standard models, soundness and completeness of logic systems, decidability

Modern Number Theory
Mentee: Angel Martinez
Mentor: Brandon Bate
Text: Ireland and Rosen Introduction to Modern Number Theory; J.P Serre A Course in Arithmetic

Group Theory
Mentee: Michael Ratner
Mentor: Wesley Pegden
Text: Herstein Topics in Algebra
Topics: group theory and applications, including topics in graph theory and the Banach-Tarski paradox

Riemann Zeta Function
Mentee: Vaibhav Sharma
Mentor: David Duncan
Text: Fisher Complex Variables; Patterson An Introduction to the Theory of Riemann Zeta-Function
Topics: Riemann zeta function, Riemann hypothesis, complex analytic functions, infinite sums and products, analytic continuation, prime number theorem

Fall 2005

Commutative Algebra
Mentee: Charles Siegel
Mentor: Sarah Genoway

Elementary Number Theory
Mentee: Mark Labrador
Mentor: Eric Rowland
Text: Dudley, Elementary Number Theory
Topics: congruence, unsolvability of some Diophantine equations, primitive roots, quadratic reciprocity, arithmetic functions, Dirichlet convolution, Mobius inversion

Elementary Number Theory
Mentee: Yifan Lin
Mentor: Dan Staley

The Gamma Function
Mentee: Daniel Ahn
Mentor: Sara Blight

Geometry of Surfaces
Mentee: Abraham Rashin
Mentor: Catherine Pfaff
Text: Stillwell, Geometry of Surfaces
Topics: Euclidean and spherical geometry

Hilbert Spaces and Fourier Analysis
Mentee: Eric Wayman
Mentor: Jared Speck
Text: Folland, Real Analysis
Topics: inner products; Schwarz inequality; parallelogram law; Pythagorean theorem; closed subspace decomposition theorem; Riesz representation theorem for Hilbert spaces; best approximation theorem; orthonormal Hilbert bases; completeness; Parseval's Identity; separability of Hilbert spaces with a countable orthonormal basis; Stone-Weierstrass theorem; Fourier analysis on L² (torus)

Metric Spaces
Mentee: Paul Geyer
Mentor: Paul Ellis
Text: Kaplanksy, Set Theory and Metric Spaces
Topics: basic properties of metric spaces, continuity, separability, compactness

Quadratic Reciprocity
Mentee: Christopher Sadowski
Mentor: John Bryk
Text: Ireland & Rosen, A Classical Introduction to Modern Number Theory
Topics: unique factorization in PIDs, Chinese remainder theorem, solving congruences, unit group structure of Z/nZ, kth power residues, quadratic reciprocity and applications

Primality Testing Algorithms
Mentee: Marla Slusky
Mentor: Wes Pegden

Summer 2005

Algebraic Number Theory
Mentee: Michael Hall
Mentor: Eric Rowland
Text: Esmond and Murty, Problems in Algebraic Number Theory
Topics: basic Galois theory, number fields, algebraic integers, norm and trace, ramification, integral bases, unique factorization of ideals

Classical Mechanics
Mentee: Eric Wayman
Mentor: Jared Speck
Text: Arnold, Mathematical Methods of Classical Mechanics
Topics: Newtonian mechanics, one- and two-body central force problems, Lagrangian formulation of mechanics, Euler-Lagrange equations

Elliptic Curve Cryptography
Mentee: Nathan Melehan
Mentor: Saša Radomirović
Text: Koblitz, A Course in Number Theory and Cryptography
Topics: addition of points on an elliptic curve, number of points on a curve over a finite field, Hasse's theorem, the discrete logarithm problem, attacks on elliptic curve cryptosystems

Geometry of Surfaces
Mentee: Aron Samkoff
Mentor: Catherine Pfaff
Text: Stillwell, Geometry of Surfaces
Topics: isometries and group actions on Euclidean space, quotient surfaces, Three Reflections Theorem, classification of Euclidean isometries, Killing-Hopf Theorem

Riemann Surfaces
Mentee: Charles Siegel
Mentor: Catherine Pfaff
Text: Miranda, Algebraic Curves and Riemann Surfaces
Topics: basics of the theory of Riemann Surfaces, maps between surfaces, theory of finite group actions on a Riemann surface, basics of monodromy theory

Set Theory
Mentee: Paul Geyer
Mentor: Paul Ellis
Text: Kaplansky, Set Theory and Metric Spaces
Topics: basic set theory, cardinal numbers, ordinal numbers, the axiom of choice, basic properties of metric spaces, continuity, separability, compactness

Topology
Mentee: Alex Conway
Mentor: Mike Richter
Text: Munkres, Topology
Topics: topologies and metric spaces, connectedness, compactness, homotopy equivalence, the fundamental group, covering space theory