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 *L*^{2} (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**/*n***Z**, *k*th 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