# Projects

Here are some of the topics and books studied by past DRP participants. If you are looking for project ideas, you might also find it helpful to look at this list of books and their prerequisites.

## Spring 2019

Project | Resources |
---|---|

An Introduction to the p-adic Numbers | P-adic numbers: An Introduction by Gouvea |

Line Bundles on Elliptic Curves (presentation) | Vector Bundles Over an Elliptic Curve by Atiyah |

Stochastic Processes: Martingales | — |

Algorithmic/High-Frequency Trading | — |

Automated Market Making | — |

Eigenvalues for Odd-dimensional Real Vector Spaces | Elementary Number Theory, Group Theory, and Ramanujan Graphs by Davinoff, Sarnak, and Valette |

Representations of p-adic Groups | Berstein's lecture notes |

Convex Optimization | — |

Optimal Control for PDE Systems/Linear Systems | — |

Winning Strategies for Nim (presentation) | On Numbers and Games by Conway |

Four Coin Welter's Game | On Numbers and Games by Conway |

Jordan Canonical Form | — |

Commutative Algebra and Point Set Topology Towards Category Theory | A Term of Commutative Algebra by Altman and Kleinman, Topology by Munkres, Foundations in Algebraic Geometry by Vakil |

Stochastic Gradient Descent | — |

Sheaves and Algebraic Curves | Geometry of Algebraic Curves by Griffiths, et. al. |

Fiber Sequences in Topology | — |

Introduction to Quantum Mechanics and the Kochen-Specker Theorem | Incompleteness, Non-localism, and Reality by Michael Redhead |

Number Theory/Prime Number Theorem | — |

Groups and Rubik's Cubes | — |

Using Knot Theory to Untangle a Lightbulb Cord | The Knot Book by Colin Adams |

Random Matrix Theory and Wigner's Semicircle Law | — |

The Projective Plane | Rational Points on Elliptic Curves by Silverman |

Elliptic Curves and Fruit Algebra | Rational Points on Elliptic Curves by Silverman |

Generalized Cantor Functions | — |

## Spring 2018

Project | Resources |
---|---|

Basic Topology | An Introduction to Topology and Homotopy by Allan Sieradski |

Introduction to Logic | — |

Modular Arithmetic and Quadratic Reciprocity | Elementary Number Theory by Jones and Jones |

An Introduction to Computational Algebra and Geometry | Ideals, Varieties, Algorithms by Cox, Little and O'Shea |

Applications of Monte Carlo Methods | — |

Monte Carlo Methods for Finance | Monte Carlo Methods in Financial Engineering by Paul Glasserman |

p-adic numbers | p-adic Numbers: an Introduction by Gouvea |

An Introduction to the Theory of Numbers | An Introduction to the Theory of Numbers by Hardy and Wright |

Combinatorics and Problem Solving | Combinatorics Through Guided Discovery by Kenneth Bogart |

Number Theory and Ramanujan Graphs | Elementary Number Theory, Group Theory and Ramanujan Graphs by Davidoff, Sarnak, Valette |

Finite Volume Methods for Linear and Non-linear Conservation Laws | Finite Volume Methods for Hyperbolic Problems by Leveque |

Tangent Spaces of Algebraic Curves | — |

Group theory and Burnside's Lemma | — |

## Fall 2018

Project | Resources |
---|---|

Coherent Cohomology of Sheaves | — |

Computational Algebra | Ideals, Varieties, Algorithms by Cox, Little and O'Shea |

Random Sampling | — |

An Introduction to Number theory and Algebraic Geometry | — |

Linear Algebra | Linear Algebra Done Right by Sheldon Axler |

Fundamental Groups and Covering Spaces | Algebraic Topology by Hatcher |

Singular Functions | Principles of Mathematical Analysis by Rudin |

Differential Equations | Finite Difference Methods for Ordinary and Partial Differential Equations by Leveque |

How to read a terse book on Commutative Algebra | A Term of Commutative Algebra by Altman and Kleiman |

Introduction to Topology | Shape of Space by Jeff Weeks |

Algorithmic Information Theory | An introduction to Kolmogorov complexity and its applications by Li and Vitanyi |

Analytic Number Theory | Introduction to Analytic Number Theory by Apostol |

Representation Theory | Representations and Characters of Groups by James and Liebeck |

Combinatorics | A Walk Through Combinatorics by Bona |

TheRiemann Hurwitz Theorem | Riemann Surfaces and Algebraic Curves by Cavalieri and Miles |

Introduction toLogic | Foundations of Mathematics by Kunen |

Random Walk | Random Walk and the Heat Equation by Lawler |

Category Theory | — |

Reading Tate's Thesis | — |

Mathematics for Finance | Essentials of Stochastic Processes by Rick Durrett |

## Spring 2017

Project | Resources |
---|---|

History of Mathematical Ideas | Mathematics and it's History by Stillwell |

Stochastic Math Biology | — |

The Hairy Ball Theorem | Topology from a Differentiable Viewpoint by John Milnor |

Quantitative Investment Strategies | — |

Dirichlet's Theorem on Arithmetic Progressions | Introduction to Analytic Number Theory by Apostol |

Chaotic Dynamics | — |

Category Theory | Abstract and Concrete Categories - The Joy of Cats by J.Adámek, H.Herrlich, G. Strecker |

Young Tableaux | Young Tableaux by Fulton |

Combinatorics and Graph Theory | — |

Square the Circle | — |

Probabilistic Methods in Combinatorics | — |

Matrix Factorization | — |

Nonstandard Analysis | — |

Braid Groups | — |

## Fall 2017

Project | Resources |
---|---|

The fundamental group and covering spaces | — |

Introduction to probabilistic methods and heuristics | — |

Introduction to stochastic calculus and stochastic differential equations | — |

Quantum computation | — |

Orthogonal polynomials | — |

Set theory | Elements of Set Theory by Herbert Enderton |

Bezout's Theorem | Undergraduate Algebraic Geometry by Miles Reid |

Introduction to group theory | Groups and Symmetry by M. A. Armstrong |

p-adic numbers | p-adic Numbers, p-adic Analysis, and Zeta-Functions by Koblitz |