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...

In September 2024, at a conference on Mathematical Aspects of Learning Theory in Barcelona, Dan Spielman gave a beautiful talk on discrepancy theory and some of its applications in clinical trials....

In 2014, Andrea Montanari and Emile Richard (Montanari & Richard, 2014) proposed a statistical model for understanding a variant of Principal Component Analysis in Tensor data. This is currentl...

In the study of average-case complexity, one is often interested in the critical threshold of the signal-to-noise ratio at which a problem becomes tractable. In this post, we will consider a multi...