Let $k\ge1$ be an integer and let $\lambda$ be the Liouville function. In 1965, Chowla gave a conjecture that the values of $\lambda(n+h_1),\dots, \lambda(n+h_k)$ are asymptotically unrelated for any distinct natural numbers $h_1, \dots, h_k$. In this article, motivated by the recent work of Bergelson and Richter on the dynamical generalizations of the prime number theorem, we will show a dynamical generalization of Chowla's conjecture on average. In the proof, we follow an approach of Qi and Zheng who established a variant of Bergelson and Richter's theorem over irreducible binary cubic forms. Moreover, we will use this approach to show an analogue of the dynamical Chowla's conjecture along the primes on average as well.
Bergeron's conjecture states that, if $1\leq a<b<c<d$ are integers with $ad=bc$, then one has the coefficient-wise inequality ${\binom{b+c}b}_q \ge {\binom{a+d}a}_q$ among two Gaussian polynomials. It originated in algebraic combinatorics and is wide open. The corresponding inequality for binomial coefficients (i.e., the case $q=1$) must be known to experts, but we could not find it in the literature. We give two proofs, each generalizing the statement in a separate direction. Binomial coefficients (and Gaussian polynomials) are fundamental combinatorial objects, and so one naturally hopes to see a combinatorial proof of this inequality. However, this seems hard to come by. We nevertheless give a combinatorial proof of a special case.
In these notes, we introduce the 50-year-old $K(\pi, 1)$ conjecture alongside Coxeter and Artin groups. Roughly speaking, the conjecture states that the complement in $\mathbb{C}^n$ of a"symmetric"configuration of hyperplanes is a $K(\pi, 1)$ space. Our end goal is to present a proof of the conjecture in the so-called spherical case, where only a finite number of hyperplanes are removed, through methods from combinatorial topology. This proof draws inspiration from the original proof of the spherical case, which is a special case of a celebrated 1972 theorem by Pierre Deligne.
Based on a conjecture of Loughran and the second author, we give an explicit prediction for the leading constant in Malle's conjecture for Galois $\mathcal{H}$-extensions of $\mathbb{Q}$ ordered by discriminant, where $\mathcal{H}$ is the $3\times 3$ Heisenberg group over $\mathbb{F}_4$. The predicted leading constant is not a single Euler product, but rather a sum of two distinct Euler products. Our methods also give an efficient algorithm for computing the conjectural Loughran-Santens leading constant for many $2$-groups of nilpotency class $2$.
Let $G=(V,E)$ be a simple graph of order $n$ and let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ be the eigenvalues of its Laplacian matrix. Brouwer conjectured that for every $1\le k\le n$, $\sum_{i=1}^k\lambda_i(G)\le |E|+\binom{k+1}{2}$, which was recently confirmed by Kothari and Tudose. Before Brouwer's conjecture was proved, Lew (JCT-B, 2026) established a weaker form of Brouwer's Laplacian eigenvalue inequality and proposed two conjectures for upper bounds on the sum of the $k$ largest Laplacian eigenvalues, one in terms of the matching number and the other in terms of the vertex-cover number. Using Brouwer's Laplacian inequality, we prove both conjectures.
We present a short proof of Feige's conjecture: for $n$ independent nonnegative random variables with expectation one, the probability that their sum is less than $n+1$ is at least $\left(\frac{n}{n+1}\right)^n\ge \frac{1}{e}$. The proof was obtained with the assistance of GPT-5.6 Sol and builds on the recent breakthrough of Vlassis and Thomas establishing Gaffke's conjecture on the finite-sample validity of a distribution-free $p$-value. We also discuss the implications of the subsequent work of Ming, Ramdas, Shen, Wang, and Waudby-Smith.
For a convex body $K \subset \mathbb R^d$ let $\Delta(K)$ be the expected distance between two independent uniform points of $K$, and let $\theta(K)$ be the corresponding expectation for normalized surface measure on $\partial K$. The Zaporozhets-Tarasov conjecture asserts $\Delta(K) \le \theta(K)$. We prove this conjecture in case $d=2$. In addition, we give a six-vertex convex polytope in $\mathbb R^3$ for which the reverse strict inequality holds, and obtain counterexamples in every dimension $d \ge 3$ by taking products with segments.