Past DRP Projects

Mathematics Undergraduate Program

Below you will find somewhat detailed information about some of the DRP projects that we've had in the past.

Spring 2011

Riemannian Geometry
Mentee: Michael Boemo
Mentor: Brent Young
Texts: 1. Manfredo do Carmo, Riemannian Geometry. 2. John M. Lee, Introduction to Smooth Manifolds. 3. R. Creighton Buck, Advanced Calculus.
Topics: differentiable manifolds, Riemannian metrics, affine connections, Riemannian connections, the Levi-Civita connection, geodesics
Presentation topic: definition of regular surface (in R^3); definition of differentiable manifold; construction of the tangent bundle TM of a differentiable manifold M and verification that TM is a differentiable manifold

Fall 2010

Abstract Algebra and the Philosophy of Mathematics
Mentee: Daniel Cunha
Mentor: Humberto Montalvan-Gamez
Texts: 1. Birkhoff, A Survey of Abstract Algebra; 2. Bertrand Russell, Principles of Mathematics; 3. Philip J. Davis & Reuben Hersh, The Mathematical Experience
Topics: rings, integral domains, the integers, composition of functions, group of symmetries of a polygon, abstract groups, Russell's logicism
Presentation topic: a group-theoretic proof of Euler's theorem (from elementary number theory)

Hypergeometric Function Summation
Mentee: Koushik Dasika
Mentor: Emilie Hogan
Text: Herbert Wilf et al., A = B
Topics: hypergeometric functions, recurrences, summation, hypergeometric summation techniques, WZ theory
Presentation topic: Sister Celine's algorithm; the sum of (n choose k) over k as an example of how the algorithm works

Axiomatic Set Theory and the Construction of Number Systems
Mentee: David Feinblum
Mentor: Michael Marcondes de Freitas
Text: Claude Burrill, Real Numbers
Topics: axiomatic development of set theory, construction of natural numbers, integers and rationals, construction of the real numbers straight from the integers, real numbers via Cantor's construction, real numbers via Dedekind cuts
Presentation topic: the Cauchy sequences approach versus the Dedekind cuts approach to the construction of the real numbers

Summer 2010

Mathematics & Music + Elementary Number Theory
Mentee: Daniel Cunha
Mentor: Humberto Montalvan-Gamez
Texts: 1. J. Douthett et al., Music Theory and Mathematics: Chords, Collections and Transformations; 2. G. Andrews, Number Theory
Topics: signature transformations, well-formed scales, divisibility, congruences
Presentation topic: a musical piece composed using mathematics

Set Theory, Equivalence Classes, and the Hopf Fibration
Mentee: Pratik Desai
Mentor: David Duncan
Topics: basic set theory, construction of the natural numbers and integers, the algebra of complex numbers and quaternions, equivalence classes, construction of S^2 from the action of S^1 over S^3 (the Hopf fibration) from the viewpoint of equivalence classes
Presentation topic: equivalence classes and projective geometry

Spring 2010

Introduction to Mathematical Finance
Mentee: Barry Ickow
Mentor: Camelia Pop
Texts: 1. Steven Shreve, Stochastic Calculus for Finance II - Continuous Time Models; 2. Oksendal, Stochastic Differential Equations
Topics: general probability theory, information and conditioning, Brownian motion

Summer 2009

Basic Analysis
Mentee: Vyacheslav Kiria
Mentor: Humberto Montalvan-Gamez
Text: R. Creighton Buck, Advanced Calculus 3rd Edition
Topics: theory of integration, vector-valued functions, differential forms, Fourier analysis.

Spring 2009

Primes and Arithmetic Functions
Mentee: Ari Blinder
Mentor: Sarah Blight
Texts: 1. Tom M. Apostol, Introduction to Analytic Number Theory; 2. Benjamin Fine & Gerhard Rosenberger, Number Theory: an Introduction via the Distribution of Primes
Topics: properties of the distribution of primes, bounds on partial sums of arithmetic functions

Group Theory
Mentee: Mark Kim
Mentor: Robert McRae
Text: David S. Dummit & Richard M. Foote, Abstract Algebra
Topics: groups, subgroups, quotient groups, group actions, direct and semi-direct products, abelian groups, p-groups, nilpotent groups, solvable groups, applications of group theory to other disciplines.

Fall 2008

Fractal Geometry
Mentee: Daniel Greene
Mentor: Andrew Baxter
Texts: 1. Gerald Edgar, Measure, Topology, and Fractal Geometry; 2. 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: Brian Chellas, Modal Logic: An Introduction
Topics: Propositional modal logic, normal systems, standard models, soundness and completeness of logic systems, decidability.

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
Texts: 1. Fisher, Complex Variables; 2. 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, primenumber theorem.

Fall 2005

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

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

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