Several alternative notions of convexity arise in different areas of mathematics. One particularly interesting and intricate example is rank-one convexity, or more generally directional convexity. ...

Let $G_d$ be a uniformly random $d$-regular graph on $n$, an even number of nodes. Throughout this post we will think of $d$ as a fixed constant and are interested in understanding the setting in w...

Given a computation problem on a random input, one natural question is whether there is some polynomial time algorithm solving it with high probability. To argue that such a problem is hard, we usu...

The isoperimetric problem dates back to the ancient Greeks who conjectured that in $\mathbb{R}^2$, the shape with fixed boundary length that maximizes is given by a circle. While this has been pro...

Given a prime number $p$ such that $p \equiv 1 \pmod{4}$ the Paley graph on $p$ nodes $G_p$ is the graph where nodes $i$ and $j$ are connected if $i-j$ is a quadratic residue modulo $p$. This graph...

Mutually Unbiased Bases Finite frame theory is the source of many beautiful questions. A remarkable example concerns the existence of Mutually Unbiased Bases, and object of interest in Quantum Phy...

Throughout this post $\mathbb{K}$ will stand for either $\mathbb{R}$ or $\mathbb{C}$. Given $A\in\mathbb{K}^{N\times M}$, the Phase Retrieval Problem aims to recover a vector $x\in\mathbb{K}^M$ fr...

For a graph $G = (V, E)$, the clique number $\omega(G)$ and the chromatic number $\chi(G)$ are fundamental properties studied in combinatorics and graph theory. It is known that computing these qua...

Consider symmetric deterministic tensors $T_1, \ldots, T_n \in (\mathbb{R}^d)^{\otimes r}$ and let $g_1, \ldots, g_n$ be i.i.d. standard gaussian random variables. We are interested in determining ...

In recent years, significant progress has been made in understanding the mixing behavior of Glauber dynamics for Ising models, particularly in the Sherrington-Kirkpatrick (SK) model. Despite the cl...