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.
In this article, we study the family of elliptic curves $E_{-2pq}: y^2=x^3-2pqx$, where $p$ and $q$ are distinct odd primes. Using a $2$-isogeny methods and some elementary techniques, we obtain explicit possibilities for the Mordell--Weil ranks, conditional on the Parity Conjecture. Moreover, in the rank-one case, we are also able to derive explicit conditions that are independent of the parity conjecture. Moreover, the main results depend only on the residue classes of $(p,q)$ modulo $8$ and the Legendre symbols $\legendre{p}{q}$.
A well-known conjecture of Chern, do Carmo, and Kobayashi asserts that, for $n,m \geq 1$, the scalar curvature of a closed, minimally immersed $n$-submanifold of $\mathbb{S}^{n+m}$ with second fundamental form of constant length takes values in a discrete set. This property holds in every codimension when $n \in \{1,2\}$. We disprove this conjecture for all $n \geq 3$ with $m \geq 4$, and for even $n \geq 4$ with $m \geq 3$.
We prove that every full-dimensional compact convex body in $\mathbb R^n$ whose barycenter is its unique interior lattice point and whose volume is $(n+1)^n/n!$ is a unimodular image of the simplex $(n+1)\Delta_n-(1,\dots,1)$. This resolves the equality case of Ehrhart's volume conjecture, as a counterpart of the inequality part recently proved by OpenAI. The main result of this paper is obtained by generative AI, particularly GPT-5.6-sol, Fable 5, and the Danus system.
A well-known conjecture of Artin states that if $a$ is an integer not equal to $0, \pm 1$ or a perfect square, then there exist infinitely many primes $p$ such that $a$ is a primitive root $(\text{ mod } p)$. In this article, we study a generalization of the classical (abelian) large sieve inequality in non-abelian settings. Assuming the non-abelian large sieve inequality, we provide a proof of Artin's primitive root conjecture. Further, using duality techniques, we derive unconditional results towards the conjectured non-abelian large sieve inequality.
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 $2$.
In $2011$, Osin and Thom conjectured that the first $\ell^2$-Betti number of a torsion-free discrete group is bounded above by the normal rank of the group minus one. The conjecture has surprising consequences for some fundamental problems in group theory and topology. These include the Wiegold problem on perfect groups, the Levin conjecture, the torsion-free case of the Kervaire conjecture, and an important special case of the Whitehead asphericity conjecture. In this article, we construct for each $n\in \mathbb{N}$ a countable torsion-free group $\Gamma_n$ such that $\beta^{(2)}_1(\Gamma_n)=n$ and so that the normal rank, $n(\Gamma_n)$, equals one. This disproves the conjecture. Our counterexamples are locally free and hence locally indicable. However, they are not finitely generated.