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Logic Blog 2023-2024

Andre Nies

TL;DR

This Logic Blog volume surveys deep links between logic and diverse mathematical areas, from effective questions about the Tits alternative and torsion in linear groups to model-theoretic analyses of automorphism structures and meet groupoids for oligomorphic groups. It develops topology-aware descriptions of Aut(G) and explores how the meet groupoid W(G) encodes sufficient information to recover the underlying group and its Chabauty space, connecting model theory, topology, and dynamics. The volume also treats fractal and computability aspects in profinite and oligomorphic contexts, ergodic theory with computability, and foundational questions in category theory and set theory, highlighting independence phenomena and diagrammatic invariants. It concludes with AI-assisted attempts to rederive classical results (e.g., the Basel problem), illustrating the potential and limits of automated reasoning in pure mathematics. Overall, the collection advances cross-disciplinary methods and provides new tools for understanding symmetries, dynamics, and logical structure across algebra, topology, and foundations.

Abstract

The logic blogs 2023 and 2024 have been joined. The present file contains a lot on particular classes of groups and their relationship with logic, as well as entries on ergodic theory and on foundations. There is also a bit on AI proving at the end.

Logic Blog 2023-2024

TL;DR

This Logic Blog volume surveys deep links between logic and diverse mathematical areas, from effective questions about the Tits alternative and torsion in linear groups to model-theoretic analyses of automorphism structures and meet groupoids for oligomorphic groups. It develops topology-aware descriptions of Aut(G) and explores how the meet groupoid W(G) encodes sufficient information to recover the underlying group and its Chabauty space, connecting model theory, topology, and dynamics. The volume also treats fractal and computability aspects in profinite and oligomorphic contexts, ergodic theory with computability, and foundational questions in category theory and set theory, highlighting independence phenomena and diagrammatic invariants. It concludes with AI-assisted attempts to rederive classical results (e.g., the Basel problem), illustrating the potential and limits of automated reasoning in pure mathematics. Overall, the collection advances cross-disciplinary methods and provides new tools for understanding symmetries, dynamics, and logical structure across algebra, topology, and foundations.

Abstract

The logic blogs 2023 and 2024 have been joined. The present file contains a lot on particular classes of groups and their relationship with logic, as well as entries on ergodic theory and on foundations. There is also a bit on AI proving at the end.

Paper Structure

This paper contains 34 sections, 32 theorems, 63 equations, 1 figure.

Table of Contents

  1. Group theory and its connections to logic
  2. Algorithmic questions around the Tits alternative
  3. Torsion in $\mathop{\mathrm{GL}}\nolimits_d(K)$
  4. Khelif,Wilson: On ultraproducts of alternating groups
  5. Khelif, Nies: f.g. groups acting on Boolean algebras
  6. Nies: The meet groupoid of an oligomorphic group
  7. A model theoretic view of $\mathcal{W}(G)$
  8. Nies: The quotient of $N_G$ by $C_G$ for $1$-transitive oligomorphic $G$
  9. Nies: $\mathop{\mathrm{Aut}}\nolimits(G)$ and $\mathop{\mathrm{Out}}\nolimits(G)$ for closed subgroups $G$ of $\mathrm{Sym}({\mathbb{N}})$
  10. A Polish topology on $\mathop{\mathrm{Aut}}\nolimits(G)$ with continuous action on $G$
  11. Polish topology on $Aut(G)$ for locally Roelcke precompact $G$
  12. Recovering $G$ from its meet groupoid $\mathcal{W}(G)$.
  13. Some Borel classes of closed subgroups
  14. When is $\mathop{\mathrm{Inn}}\nolimits(G)$ closed in $\mathop{\mathrm{Aut}}\nolimits(G)$?
  15. The Chabauty space of a t.d.l.c. group via its meet groupoid
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $K$ be a field, and let $\Gamma \le \mathop{\mathrm{GL}}\nolimits_d(K)$ be finitely generated. Then either $\Gamma$ contains a non-abelian free subgroup, or $\Gamma$ has a solvable subgroup of finite index (i.e., $\Gamma$ is virtually solvable).

Figures (1)

  • Figure 1: Some Borel classes of infinite closed subgroups of $\mathrm{Sym}({\mathbb{N}})$

Theorems & Definitions (102)

  • Theorem 1.1: Tits, 1972
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof : Proof of the theorem
  • Lemma 3.1
  • proof
  • Claim 3.2
  • ...and 92 more