Idea #1
Name of Your Project Computational complexity of the ‘Park Puzzle’
What research topic/question your project is going to address? The Park Puzzle is a game that involves splitting a n*n grid into different colored continuous ‘parks’. A solution to a park puzzle involves marking the position of a tree in each park such that no two trees share a row or a column. The research topic involves exploring various algorithms for solving the park puzzle, and determining some bounds on the computational complexity of the puzzle. The most involved version of this project could be to prove those bounds. (The initial, intuitive hypothesis is that the park puzzle cannot be solved in polynomial time)
Update: I am looking into the proofs of NP-completeness of other pencil puzzle problems to gain an understanding of the techniques involved in proving such complexity bounds.

Idea #3
Name of Your Project Fast Integer multiplication
What research topic/question your project is going to address? There have been significant improvements in algorithms for fast multiplication of integers within the last decade (see Fast integer Multiplication by Anandya et al), approaching O(nlogn), which use modular arithmetic and computation on p-adics  The research project would be to explore the theory behind these algorithms and verify their results.
Update: I am looking into the minimal mathematical background required to make sense of how the algorithms work, without necessarily getting into all of the details.

Idea#3 Iterative solvers for systems of Linear Equations
Description Large and sparse systems of linear equations appear in the solution to some partial differential equations by finite difference methods, and in general many day-to-day problem areas such as in engineering and scientific computing. As such the theory of developing methods for solving such large systems, and proving their correctness is an area of active research. in this project I would like to test the effectiveness of various iterative methods at finding solutions to systems linear equations with various characteristics by taking an experimental approach.
Update: no update for this idea