Sidhanth Mohanty

Spencer's theorem via Brownian motion

Mon, 27 Apr 2026 00:00:00 +0000

A fact that was initially surprising to me is that stochastic differential equations can actually be useful in solving discrete problems. The Lovett–Meka proof of Joel Spencer’s classic discrepancy theorem is an elegant showcase of this idea. Today we will discuss Spencer’s theorem and a proof based on Brownian motion!

Spencer's theorem.
Let $A$ be an $n \times n$ matrix with entries in $[-1,1]$.
Then there is a vector $x \in \{\pm 1\}^n$ such that

John's ellipsoid theorem

Sat, 25 Apr 2026 00:00:00 +0000

John’s ellipsoid theorem is a clean high-dimensional convex geometry fact that shows up in a lot of different places. Informally, it says:

Every $n$-dimensional symmetric convex body is an ellipsoid, up to a $\sqrt n$ scaling.

This is quite nice and convenient because ellipsoids are basically “$\ell_2$” objects, which makes them a lot easier to do math with than arbitrary convex bodies. There is a long list of applications of John’s theorem that I won’t cover here, but it recently came up for me in three different contexts.

Why Northwestern is a fun place for TCS ∩ Probability Theory

Mon, 01 Jan 0001 00:00:00 +0000

If you are a prospective student applying to PhD programs and are interested in the intersection of theoretical computer science with probability and statistics, do consider applying to Northwestern!

We have a vibrant community of researchers in these areas. Within the Computer Science department, in addition to me, Aravindan Vijayaraghavan and Konstantin Makarychev do exciting work in algorithmic statistics and theoretical machine learning with a “beyond worst-case” bent, and Miklos Z. Racz works on very interesting inference problems on random graphs and networks. More broadly, this page contains all my colleagues who work on a wide range of cool topics in theoretical computer science.