Past Projects

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

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