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