The advent of large-scale systems has led to many new and exciting computational and statistical challenges. A prominent example is the so-called “curse of dimensionality”, which informally suggests that complexity should scale exponentially with dimension. However, as is becoming more evident, this exponential dependency is often mitigated by emergent universality phenomena, which seem inherent in high dimensions, and allow for tractable analysis.

In this talk, I will survey some recent developments in the study of high-dimensional universality phenomena. The primary vehicle underlying these results is the introduction and combination of new techniques originating from stochastic analysis, infinite dimensional calculus, and the theory of stochastic mass transport. 

I will also discuss some examples of applications, such as:

• Inference in random geometric graphs.
• Approximation of neural networks by Gaussian processes. 
• Noise stability of Boolean functions.