A symmetric matrix has simple spectrum if all eigenvalues are different.
In 1980s, Babai conjectured that random graphs have simple spectrum with probability tending to $1$.
Confirming this conjecture, we prove the simple spectrum property for a large class of random matrices.
If time allows, we will discuss the harder problem of bounding the spacings between consecutive eigenvalues, with motivation from mathematical physics and numerical linear algebra.
Several open questions will also be presented.
Joint work with H. Nguyen (OSU) and T. Tao (UCLA).