Motivated by an extra credit problem from our Linear Algebra class, we study the invertibility probability of binary matrices (the number of invertible binary matrices divided by the total number of binary matrices). Binary matrices are of interest in combinatorics, information theory, cryptology, and graph theory. It is known that the invertibility probability of n × n binary matrices goes to 1 as n→∞. We conjecture that this probability monotonically increases as the size of the binary matrix increases, and we investigate this by exploring how n×n binary matrices of rank n and rank (n−1) can be enlarged to (n + 1) × (n + 1) invertible binary matrices. Calculating this explicitly for the identity matrix, we obtain a probable bound that would show that, in a sense, our conjecture is asymptotically true. With the use of a computer, we also computed how many (n + 1) × (n + 1) invertible binary matrices can be enlarged from n×n matrices of rank n and rank (n−1) for small n. In addition, we study the invertibility probability of matrices with entries in Z_q.

Characterization of homogeneous polynomials with isolated critical point at the origin follows from a study of complex geometry. Yau previously proposed a Numerical Characterization Conjecture. A step forward in solving this Conjecture, the Granville-Lin-Yau Conjecture was formulated, with a sharp estimate that counts the number of positive integral points in ndimensional (n≥3) real right-angled simplices with vertices whose distance to the origin are at least n-1. The estimate was proven for n≤6 but has a counterexample for n = 7. In this project we come up with an idea of forming a new sharp estimate conjecture where we need the distances of the vertices to be n. We have proved this new sharp estimate conjecture for n≤7 and are in the process of proving the general n case.

This paper estimates the upper bound of the minimum prime quadratic residue module a prime and gives the asymptotic formula of &∑_{p≤x;f(p)=r} 1& (r is a prime).