I will review the current status of 2D Coulomb gas with $beta = 1$ (hence "normal matrices" in the title). 

When the potential is nice enough (algebraic Hele-Shaw) we can apply the method that has been developed to study complex orthogonal polynomials.

I will explain such method (called Riemann-Hilbert method) and its usefulness in studying the partition function of the Coulomb system. 

Since the early 2000s, it has turned out that well developed mathematical tools can play crucial roles in quantum information theory. One monumental work was made by Hastings in 2009. He disproved a long standing conjecture in quantum information theory, which is called the additivity conjecture of Holevo capacities. A natural way to prove this result will be covered in this talk based on the theory of i.i.d. random unitary matrices. Another outstanding application of random matrix theory in quantun information theory is to provide a systematic way to produce PPT entangled states in high dimensional tensor product spaces. This construction comes from i.i.d. random Gaussian matrices and I will try to explain why this application is important in view of quantun information theory. 

Directed polymer models are well known Gibbs measures on random walk paths. Canonically they are defined so as to tilt the path distribution towards regions of space-time where an independent random field happens to be large, and as a result the paths tend to exhibit superdiffusive Kardar-Parisi-Zhang type fluctuation exponents, somehow betraying their random walk upbringing. Constructing these models on in the discrete space-time setting with a finite time horizon is straightforward, but extending them to infinite time horizons is difficult even in the fully discrete setting. I will review some relatively recent progress in the discrete and semi-discrete setting by myself and several other authors, some previous work of myself, Khanin, and Quastel on constructing continuous space-time models in the finite time horizon setting, and some attempts in progress to connect the two.