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

• The paper articulates 16 significant open problems in areas such as tensor concentration, graph theory, phase retrieval, and convex geometry.
• It leverages advanced tools including PAC-Bayesian methods, Fourier analysis, and sum-of-squares relaxations to tackle complex conjectures.
• The work highlights far-reaching implications for coding theory, quantum information, and high-dimensional data analysis while outlining promising future directions.

Authoritative Summary of "Randomstrasse101: Open Problems of 2025" (2603.29571)

Overview

"Randomstrasse101: Open Problems of 2025" records a curated selection of sixteen significant open problems in probability, computation, combinatorics, statistics, and related areas. This series continues previous efforts by the authors to publicize and stabilize referenceable versions of outstanding questions in mathematical research, with an emphasis on foundational and technically intricate problems. Each entry presents context, relevant background, precise conjectures or questions, and connections to fields such as high-dimensional probability, convex geometry, spectral graph theory, matrix and tensor analysis, and computational complexity.

Tensor Concentration Inequalities

The "Tensor Concentration Inequalities" problem centers on extending non-commutative Khintchine-type inequalities to injective norms of random tensor sums. The motivating conjecture proposes that for symmetric deterministic tensors $T_1,\ldots,T_n \in (\mathbb{R}^d)^{\otimes r}$ and i.i.d. Gaussian weights, the expected injective $\ell_p$ norm exhibits a scaling of $$\tilde{\mathcal{O}}_{r,p}\left(d^{1/2-1/p} \sqrt{\sum_i \|T_i\|_{\mathcal{I}_p}^2}\right)$$ for $p \geq 2$ and $r \geq 2$. The analogy with the matrix ($r=2$) case breaks due to the lack of tractable moment methods for general tensors, and the conjecture remains open except in special "volumetric" regimes such as $p \geq 2r$ (Bandeira et al., 2024).

Recent progress includes (i) PAC-Bayesian concentration removing logarithmic factors for certain structured randomness (Aden-Ali, 13 Mar 2025); (ii) sharper inequalities in the injective norm for tensors with independent entries (Boedihardjo, 2024); and (iii) nearly tight norm results in the regime $p=2$ and independently sampled matrix/tensor entries (Latała et al., 4 Feb 2025). However, in the general setting, achieving optimal polylog-free bounds and understanding the metric entropy in relevant Gaussian process suprema is still out of reach. The wider implications span random coding, Banach space geometry, optimal spectral bounds for non-homogeneous tensors, and applications in high-dimensional data analysis.

The Lovász Number in Random Circulant Graphs

The Lovász $\vartheta$ function, a key semidefinite programming (SDP) relaxation for clique and Shannon capacity estimation in graphs, is considered in the context of random circulant graphs—Cayley graphs over $\mathbb{Z}_n$ with random adjacency structure induced by randomizing the generator set. The principal conjecture states that, analogous to the Erdős-Rényi $\ell_p$0 model, the expected value of the $\ell_p$1 function for a random circulant graph is $\ell_p$2.

While the Erdős-Rényi case remains unresolved up to constant factors [coja2005lovasz], the authors establish upper bounds $\ell_p$3 and precise lower bounds for random circulant graphs (Bandeira et al., 22 Feb 2025), utilizing Fourier analytic tools and the Restricted Isometry Property (RIP) for subsampled DFT matrices. These results reinforce the pseudorandom model hypothesis for Cayley-type graph ensembles and suggest that algebraic structure might not facilitate large deviations in $\ell_p$4, despite the inherent regularity. The resolution of these questions would have significant implications for coding theory, quantum information, and extremal combinatorics.

Phase Retrieval: Injectivity and Stability

The phase retrieval problem investigates when a real or complex signal $\ell_p$5 can be reconstructed, up to a global phase, from intensity measurements $\ell_p$6. While over the real field generic injectivity holds at the information-theoretic limit $\ell_p$7, the complex setting is subtler. The established threshold for generic injectivity is $\ell_p$8 [conca2015generico4Mm4], but explicit constructions reveal that fewer measurements can sometimes suffice [vinzant2015small].

The leading conjecture posits that with $\ell_p$9, the probability of injectivity is always less than 1 and tends to 0 as $$\tilde{\mathcal{O}}_{r,p}\left(d^{1/2-1/p} \sqrt{\sum_i \|T_i\|_{\mathcal{I}_p}^2}\right)$$0. For stability, the sharp lower bounds for the worst-case conditioning of all possible support sets—especially for random $$\tilde{\mathcal{O}}_{r,p}\left(d^{1/2-1/p} \sqrt{\sum_i \|T_i\|_{\mathcal{I}_p}^2}\right)$$1—remain largely unknown. The problem intersects with frame theory, algebraic geometry, and numerical analysis, with practical ramifications for imaging and signal processing [Bandeira_etal_Savingphase, balan2013invertibility].

Mutually Unbiased Bases (MUBs), Equiangular Tight Frames (ETFs), and Zauner’s Conjecture

The search for maximal sets of Mutually Unbiased Bases (MUBs) in complex vector spaces is a persistent question with direct connections to quantum state tomography and design theory. The fundamental open case is to determine whether seven mutually unbiased bases exist in $$\tilde{\mathcal{O}}_{r,p}\left(d^{1/2-1/p} \sqrt{\sum_i \|T_i\|_{\mathcal{I}_p}^2}\right)$$2, challenging the belief that $$\tilde{\mathcal{O}}_{r,p}\left(d^{1/2-1/p} \sqrt{\sum_i \|T_i\|_{\mathcal{I}_p}^2}\right)$$3 [BDK-2022-DualBoundsMUB]. Expressing this as a nonexistence certificate amenable to the Sum-of-Squares (SoS) proof system, preferably at degree 4, opens a new path at the interface of algebraic optimization and combinatorics.

Simultaneously, Zauner’s conjecture on the existence of $$\tilde{\mathcal{O}}_{r,p}\left(d^{1/2-1/p} \sqrt{\sum_i \|T_i\|_{\mathcal{I}_p}^2}\right)$$4-vector Equiangular Tight Frames (SIC-POVMs) in all dimensions remains open, with recent progress hinged upon deep connections to number theory and explicit candidate constructions contingent on the Stark conjectures (Appleby et al., 7 Jan 2025). These problems delineate the mathematical limits of quantum measurements and optimal packing in projective spaces.

The Clique Number of the Paley Graph and Convex Relaxations

The Paley graph $$\tilde{\mathcal{O}}_{r,p}\left(d^{1/2-1/p} \sqrt{\sum_i \|T_i\|_{\mathcal{I}_p}^2}\right)$$5 exemplifies a highly structured deterministic pseudorandom graph, and its clique number conjecture—that $$\tilde{\mathcal{O}}_{r,p}\left(d^{1/2-1/p} \sqrt{\sum_i \|T_i\|_{\mathcal{I}_p}^2}\right)$$6—remains unresolved. The best unconditional upper bound is $$\tilde{\mathcal{O}}_{r,p}\left(d^{1/2-1/p} \sqrt{\sum_i \|T_i\|_{\mathcal{I}_p}^2}\right)$$7 [HP-2021-SumsetsPaleyClique]. The authors focus on exploiting convex relaxations, notably the Lovász number and higher-degree SoS relaxations, especially on localizations of the Paley graph, to potentially break the square-root barrier. Numerical evidence indicates that degree-4 SoS relaxations could yield subpolynomial upper bounds for cliques [KY-2022-PaleyDegree4SOS], while theoretical results limit the possible improvements.

A further conjecture links the Paley ETFs—deterministically constructed equiangular tight frames using DFT submatrices—to the Restricted Isometry Property (RIP) for sparse recovery, seeking to definitively exceed the square-root sparsity barrier [Bandeira_ConditionalPaley].

KLS Conjecture: Isoperimetry in High Dimensions and Recent Developments

The Kannan-Lovász-Simonovits (KLS) conjecture is central in high-dimensional convex geometry, positing that all isotropic log-concave probability measures in $$\tilde{\mathcal{O}}_{r,p}\left(d^{1/2-1/p} \sqrt{\sum_i \|T_i\|_{\mathcal{I}_p}^2}\right)$$8 admit a uniform dimension-free lower bound on the Cheeger (isoperimetric) constant, up to covariance scaling. This conjecture, apart from its geometric essence, underpins fast mixing in Markov chains for sampling, concentration inequalities, and probabilistic inequalities for convex bodies.

Significant recent advances have reshaped the landscape: (i) stochastic localization techniques drove the lower bound for the isoperimetric constant from $$\tilde{\mathcal{O}}_{r,p}\left(d^{1/2-1/p} \sqrt{\sum_i \|T_i\|_{\mathcal{I}_p}^2}\right)$$9 to $p \geq 2$0 [chen2021almost], (ii) further improved to $p \geq 2$1 [klartag2023logarithmic], and (iii) the complete resolution of the Thin Shell and Bourgain's Slicing Conjectures [klartag2025thin, klartag2025affirmative]. The KLS conjecture remains unresolved in full generality, but the established equivalences demonstrate that resolving it would have immediate structural consequences for the geometry of log-concave measures.

Sharp Bounds for Graph Matrices in the Sum-of-Squares Framework

The analysis of random graph matrices, arising naturally in pseudo-expectation function constructions for SoS lower bounds on combinatorial optimization problems, focuses on determining the tight norm asymptotics of so-called "graph matrices" $p \geq 2$2 associated with fixed graph shapes $p \geq 2$3. These matrices take the form of matrix chaoses—random matrices with entries being degree-$p \geq 2$4 polynomials of i.i.d. random variables.

While linear ($p \geq 2$5) cases are completely understood via noncommutative Khintchine and tail bounds [BBvH-Free, universalityBRvH24], the extension to higher-degree polynomial cases remains challenging, with results only recently matching leading order up to polylogarithmic factors [bandeira2025matrix]. Whether iterated free probability tools can yield tight (polylog-free) matching bounds for all relevant shapes is unresolved; sharper analysis may directly impact lower bounds for SoS algorithms in problems such as planted clique and random constraint satisfaction [ahn2016graph, banderia2025matrix].

Implications and Future Directions

The open problems collected in this manuscript highlight frontier challenges that shape the development of modern probability, analysis, combinatorics, and theoretical computer science. Many of these problems, particularly those connecting high-dimensional geometry and convexity with spectral analysis and polynomial concentration, have practical relevance to algorithm design, quantum information, compressed sensing, random matrix theory, and the analysis of large complex systems.

A common theme is the quest for (i) universal inequalities, (ii) optimality of convex relaxations, and (iii) identification of fine transitions between tractable and intractable computational landscapes. Further progress may leverage a combination of geometric functional analysis, probabilistic methods, harmonic and Fourier analysis, algebraic geometry, and advances in proof complexity. Future research will require the resolution of these central conjectures, improvements in quantitative bounds, and the discovery of new combinatorial or analytic methodologies with robust algorithmic applications.

Conclusion

"Randomstrasse101: Open Problems of 2025" (2603.29571) serves as both a reference and stimulus for ongoing research at the critical interface of probability, geometry, information theory, and computation. The open problems are carefully selected to reflect deep unresolved technical questions, concrete conjectures with broad consequences, and challenging bridges between continuous and discrete mathematics. The continued resolution of these problems promises to yield impactful theoretical and algorithmic innovations.