Random Matrix Theory

The field of random matrices is a key area in probability theory, combining probability, linear algebra, free probability, complex analysis, and combinatorics. I will start by presenting famous results for large symmetric matrices. Despite its technical nature, I will focus on conveying the main ideas without delving into unnecessary complexity.

We will then move to non-Hermitian matrices, the subject of my thesis, where the spectrum becomes complex. I will present a recent simple result obtained with my supervisors on the simplest possible matrix with independent, identically distributed entries. This result illustrates that asymptotic behavior in high dimensions can often be described by surprisingly elegant formulas, making the development of conjectures in this field both an intellectually stimulating and unexpectedly enjoyable task.