## Everyone Should Learn Optimal Transport, Part 1

In my opinion, optimal transport (OT) is a seriously underrated topic. I think part of the reason is the way OT is often introduced: as an optimization problem or a metric on probability distributions. While these are interesting to study for their own sake, OT presents a tool much more powerful and arguably even fundamental to the study of probability. In a series of two posts, I’m going to present a message that is well understood by experts but often missed by the uninitiated:

## An Unusally Clean Proof: Dyson Brownian Motion via Conditioning on Non-intersection

Dyson Brownian motion [Dy62] is best known to characterize the eigenvalues of special random matrices [Ta12]. Most interestingly, it is also equal in distribution to $$n$$ independent Brownian motions conditioned to not intersect [Gr99]. In a topics course by Bálint Virág, I came across a proof of this result that is just too clean for this type of calculations. After picking up my jaw from the ground months later, I finally decided to write up this surprisingly elegant proof.

## On Escape Time, Lyapunov Function, Poincaré Inequality, and the KLS Conjecture Beyond Convexity

Nobody has time to read an 80 page paper [LE20]. Therefore I doubt most readers realized the manifold Langevin algorithm paper actually contains a novel technique for establishing functional inequalities. And I really doubt anyone had time to interpret the intuitive consequences of such results on perturbed gradient descent, and definitely not extending the Kannan-Lovász-Simonovits (KLS) conjecture [LV18] - which brings me to write this blog post.

## The Auffinger-Chen Representation

Equivalent representation results contribute not only a connection between different concepts, but also a new set of proof techniques. Indeed, stochastic analysis has offered a number of alternative proofs to many problems. Occasionally the proof can simplify drastically. In this post, we will discuss a particularly elegant application by Auffinger and Chen (2015), for an otherwise very difficult problem in spin glass.