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Randomstrasse101: Open Problems of 2025

Afonso S. Bandeira, Daniil Dmitriev, Kevin Lucca, Petar Nizić-Nikolac, Almut Rödder

Abstract

Randomstrasse101 is a blog dedicated to Open Problems in Mathematics, with a focus on Probability Theory, Computation, Combinatorics, Statistics, and related topics. This manuscript serves as a stable record of the Open Problems posted in 2025, with the goal of easing academic referencing. The blog can currently be accessed at randomstrasse101.math.ethz.ch

Randomstrasse101: Open Problems of 2025

Abstract

Randomstrasse101 is a blog dedicated to Open Problems in Mathematics, with a focus on Probability Theory, Computation, Combinatorics, Statistics, and related topics. This manuscript serves as a stable record of the Open Problems posted in 2025, with the goal of easing academic referencing. The blog can currently be accessed at randomstrasse101.math.ethz.ch

Paper Structure

This paper contains 7 sections, 2 theorems, 32 equations, 1 figure, 1 table.

Table of Contents

  1. Tensor Concentration Inequalities (KL)
  2. The Lovász number of random circulant graphs (DD)
  3. Injectivity and Stability of Phase Retrieval (ASB)
  4. Mutually Unbiased Bases, ETFs, and Zauner's Conjecture (ASB)
  5. On the clique number of the Paley Graph (ASB)
  6. The KLS Conjecture and Implications (AR)
  7. Sharp Bounds for Graph Matrices (PNN)

Key Result

Theorem 13.2

For any isotropic log-concave measure $\mu$ on $\mathbb{R}^n$, the mass concentrates around a thin spherical shell of radius $\sqrt{n}$ and constant width, i.e. there exists a universal constant $C>0$ such that

Figures (1)

  • Figure 1: Top row: Circulant graphs on 9 vertices; Bottom row: Their corresponding adjacency matrices (with 0's represented by dots).

Theorems & Definitions (23)

  • Conjecture 16: Type-$2$ constant of Tensors
  • Conjecture 17
  • Conjecture 18
  • Conjecture 19
  • Definition 10.1
  • Conjecture 20
  • Definition 11.1: Mutually Unbiased Bases
  • Conjecture 22
  • Definition 11.2: Equiangular Tight Frame (ETF) in $\mathbb{C}^d$
  • Conjecture 24: Zauner's Conjecture
  • ...and 13 more