## Past DRP Projects

This page contains a list of general topics for past DRP projects. If you’re looking for a project idea, it might be helpful to look through this list for inspiration. If you see any topics that sound cool but that you don’t know anything about, try Googling them to get more information. Wikipedia is also a valuable resource for general information about topics. Remember, you don’t need to have more than a general direction or topic for your project at the outset, and it’s totally fine if you don’t have any background in your topic.

### Fall 2021

- Partitions of Unity in Manifolds
- Model Theory and o-minimality
- Hyperbolic Geometry: Discovering a Strange New World
- Hash Functions, Bit Commitment, and Zero-Knowledge Proofs
- Numerical methods for solving PDEs
- Line bundles on elliptic curves
- Consistency of the continuum hypothesis with ZFC
- Proof of Kronecker-Weber Theorem for Quadratic Extensions
- Algebraic Combinatorics: Non-crossing partitions and Dyck paths

### Spring 2017

- Catalan Numbers
- Machine Learning
- Chip Firing
- Enriched Category Theory
- Quantum Computation
- Elementary Number Theory – Diophantine Equations
- Elliptic Curves
- Peano Space Filling Curve
- Filters and Ultrafilters
- The Word Problem for Coxeter Groups
- Vector Bundles and Projective Modules
- Information Theory
- Law of Quadratic Reciprocity
- Bootstrap Argument
- Markov Chains
- The Moduli Stack of Elliptic Curves
- Algebraic Topology
- The Classification of Semisimple Lie Algebras
- Factoring Sums of Squares in Z[i]

### Fall 2016

- Computation of the Class Number Formula
- Category Theory for the Sciences: Databases
- A Geometric Proof of the Quadratic Gauss Sum
- Gauss Winding Number
- Free Groups and Trees
- Brief Overview of Homology Theories
- Finite Difference Method for PDE and ODE
- Polynomial Hierarchy
- Homotopy Type Theory
- Representation Theory
- Integral extension and Convex Hull
- End Compactifications
- Universal Covering Space
- Weyl’s Equidistribution Theorem in Dynamical Systems
- The Uncertainty Principle
- On Hausdorff Gaps
- Algebraic Tools for Reverse Engineering Biological Systems
- Integral Closures and Nonsingular Plane Cuves

### Spring 2015

- Ax-Grothendieck Theorem: Basic Model Theory
- Algebraic Statistics in Biology
- Tensor & Fluid Mechanics
- Morley Rank
- Elliptic Curve Digital Signature Algorithm
- Quadrature Schemes
- Dynkin Diagram
- Catenary Degree
- Riemann Mapping Theorem
- Galois Analysis of Fifth Degree Polynomials
- Hilbert’s Nullstellensatz
- Gaussâ€“Bonnet theorem
- Harmonic Analysis
- Hall’s Theorem
- Differential Forms
- Topics in Cryptography
- Counting Flags
- Demystifying the Fundamental Group with Category Theory
- Infinite Games and Determinacy
- Convolution inequalities

### Fall 2015

- Determinacy: Let’s Play a Game
- Dynkin diagrams
- The Kneser Graph
- Rational Points on Elliptic Curves
- Congruent Number Problem and Elliptic Curves
- Primes of form x^2 + ny^2 for infinitely many n
- Topologies
- Weierstrass Uniformization of Elliptic Curves
- Peter-Weyl Theorem
- Tate’s Thesis
- Geometric Classification of Semisimple Lie Algebras
- The Uniform Convergence of Fourier Series
- Dirichlet’s theorem on arithmetic progressions
- Elliptic curve cryptography
- Degree Structure of Turing Reducibility
- Self-adjoint operators and Observables in Quantum Mechanics
- A Continuous but Nowhere Differentiable Function
- Maxwell’s Equations on Manifolds
- A Basic Result on Maps of Riemann Surfaces
- Differential Forms and Stokes Theorem
- A quick proof of Leray-Hirsch via hypercohomology