Work

We construct, for every even $n \ge 2$, a positive cone on $F_n \times \mathbb{Z}$ that is finitely generated as a semigroup, extending the previously known construction for $n = 2$. Malicet, Mann, Rivas and Triestino proved that $F_n \times \mathbb{Z}$ admits an isolated left-order if and only if $n$ is even. Since every finitely generated positive cone determines an isolated left-order, for $n \ge 2$, the group $F_n \times \mathbb{Z}$ admits a finitely generated positive cone if and only if $n$ is even.

A regular left-order on finitely generated group $G$ is a total, left-multiplication invariant order on $G$ whose corresponding positive cone is the image of a regular language over the generating set of the group under the evaluation map. We show that admitting regular left-orders is stable under extensions and wreath products and give a classification of the groups whose left-orders are all regular left-orders. In addition, we prove that solvable Baumslag-Solitar groups $B(1,n)$ admits a regular left-order if and only if $n\geq -1$. Finally, Hermiller and Sunic showed that no free product admits a regular left-order, however we show that if $A$ and $B$ are groups with regular left-orders, then $(A*B) \times \mathbb{Z}$ admits a regular left-order.

We propose a criterion for the regularity of a formal language representation when passing to subgroups. We use this criterion to show that the regularity of a positive cone language in a left-orderable group passes to its finite index subgroups, and to show that there exists no left order on a finitely generated acylindrically hyperbolic group such that the corresponding positive cone is represented by a quasi-geodesic regular language. We also answer one of Navas’ question by giving an example of an infinite family of groups which admit a positive cone that is generated by exactly $k$ generators, for every $k \geq 3$. As a special case of our construction, we obtain a finitely generated positive cone for $F_2 \times \mathbb{Z}$.

Recent progress in autoformalisation of mathematics using AI has been impressive. Measuring this progress requires datasets of formal statements that are both hard and trustworthy. We present AnnalsChallenge: 50 formalised theorem statements written by hand in Lean 4 using mathlib. Each is the main result of a paper published in the Annals of Mathematics since 2020. Every formalisation was manually reviewed and then audited by AI. We release them without formalised proofs.

• Minimal chat server: a Python TCP chat server using threads and non-blocking sockets.
• Sum of odd numbers animation: a p5.js visual proof that $1+3+\cdots+(2n-1)=n^2$, with Sukanya Aneja.
• Magic tmux box: a Bash launcher that checks for an existing tmux session before starting one.
• OK Bot: a Python Twitter bot that posts reassurance with links to matching tweets.

The goal of this presentation is to set up the framework for $\ell^2$ - Betti Numbers from the point of view of von Neumann Algebras.

The goal of this document is to give the undergraduate reader an overview of tripartite quantum entanglement under SLOCC, with no background assumed.

Keywords: growth in groups, nilpotent groups, Malcev normal form, Carnot groups, Gromov, growth in the polynomial range, rational growth, Duchin, Shapiro, geodesics, Cayley graph, CC metric, complexity, polynomial-time algorithm.