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Raphael Appenzeller

Publications and source records attributed to Raphael Appenzeller.

11 recordsLinked to original sources
arXiv2026

Genus-zero links with prescribed knots as components

We prove that any finite collection of at least three isotopy classes of knots in a 3-manifold $M$ is realizable as the components of a genus-zero link in $M$, provided that an obvious requirement on their conjugacy classes in $\pi_1(M)$ is met. This condition is vacuously satisfied for $M = \mathbb S^3$, and in this case we also control the pairwise linking numbers of the components. Replacing the 3-genus with the 4-genus, we obtain an analogous result where only two knot isotopy classes are prescribed.

math.GT↗
arXiv2026

All knots are trivial: a "proof" by sleight of hand

We take a close look at a classical magic trick performed with a string, where a trivial knot is seemingly isotoped into a trefoil, and generalize it to a family of magic tricks for transforming the unknot into other knots. We encode such a trick by depicting the target knot as a special type of knot diagram, which we call a "knotholder diagram". By proving that all knots admit knotholder diagrams, we obtain variants of the trick for producing every knot.

math.GT↗
arXiv2026

Semisimple algebraic groups over real closed fields

We give a self-contained introduction to linear algebraic and semialgebraic groups over real closed fields, and we generalize several key results about semisimple Lie groups to algebraic and semialgebraic groups over real closed fields. We prove that a torus in a semisimple algebraic group is maximal $\mathbb{R}$-split if and only if it is maximal $\mathbb{F}$-split for real closed fields $\mathbb{F}$. For the $\mathbb{F}$-points we formulate and prove the Iwasawa-decomposition $KAU$, the Cartan-decomposition $KAK$ and the Bruhat-decomposition $BWB$. For unipotent subgroups we prove the Baker-Campbell-Hausdorff formula, facilitating the analysis of root groups. We give a proof of the Jacobson-Morozov Lemma about subgroups whose Lie algebra is isomorphic to $\mathfrak{sl}_2$ for algebraic groups and a version for the $\mathbb{F}$-points, when the root system is reduced. We describe the rank 1 subgroups which are the semisimple parts of Levi-subgroups. We prove a semialgebraic version of Kostant's convexity theorem. The main tool used is a model theoretic transfer principle that follows from the Tarski-Seidenberg theorem.

math.GR↗
arXiv2026

Morphisms of generalized affine buildings

We define a notion of morphism for generalized affine buildings, also known as affine $\Lambda$-buildings, extending existing definitions and giving rise to a category of generalized affine buildings. For affine $\Lambda$-buildings equipped with a transitive group action, we provide sufficient conditions for the existence of morphisms between them. As an application, we investigate under which conditions morphisms or isomorphisms between various generalized affine buildings from the literature (defined via lattices, norms, non-standard symmetric spaces, or \`a la Bruhat-Tits) can be defined. For generalized affine buildings coming from non-standard symmetric spaces we further show functoriality for subgroups and under change of valued field.

math.GR↗
arXiv2026

Generalized affine buildings for semisimple algebraic groups over real closed fields

We use real algebraic geometry to construct an affine $\Lambda$-building $B$ associated to the $\mathbb{F}$-points of a semisimple algebraic group, where $\mathbb{F}$ is a valued real closed field. We characterize the spherical building at infinity and the local building at a base point. We compute stabilizers of various subsets of $B$ and obtain group decompositions.

math.GR↗
arXiv2025

IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation

As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 77 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can already solve a significant percentage of research-level questions. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.

cs.CL↗
arXiv2025

Cops and robbers for hyperbolic and virtually free groups

Lee, Mart\'inez-Pedroza and Rodr\'iguez-Quinche define two new group invariants, the strong cop number $\operatorname{sCop}$ and the weak cop number $\operatorname{wCop}$, by examining winning strategies for certain combinatorial games played on Cayley graphs of finitely generated groups. We show that a finitely generated group $G$ is Gromov-hyperbolic if and only if $\operatorname{sCop(G)} = 1$. We show that $G$ is virtually free if and only if $\operatorname{wCop(G)}=1$, answering a question by Cornect and Mart\'inez-Pedroza. We show that $\operatorname{sCop}(\mathbb{Z}^2) = \infty$, answering a question from the original paper. It is still unknown whether there exist finite cop numbers not equal to 1, but we show that this is not possible for CAT(0)-groups. We provide machinery to explicitly compute strong cop numbers and give examples by applying it to certain lamplighter groups, the solvable Baumslag-Solitar groups, and Thompson's group F.

math.GR↗
arXiv2024

Semialgebraic groups and generalized affine buildings

We develop the theory of algebraic groups over real closed fields and apply the results to construct a geometric object $\mathcal{B}$ and to prove that $\mathcal{B}$ is an affine $Λ$-building. We use a model theoretic transfer principle to prove generalizations of statements about semisimple Lie groups. In this direction we give proofs for the Iwasawa-decomposition $KAU$, the Cartan-decomposition $KAK$ and the Bruhat-decomposition $BWB$. For unipotent subgroups we prove the Baker-Campbell-Hausdorff formula and use it to analyse root groups. We give a proof of the Jacobson-Morozov Lemma about subgroups whose Lie algebra is isomorphic to $\mathfrak{sl}_2$ and we describe other rank 1 subgroups which are the semisimple parts of Levi-subgroups. We prove a semialgebraic version of Kostant's convexity. Over the reals, semisimple Lie groups are closely related to the symmetry groups of symmetric spaces of non-compact type. These symmetric spaces can be described semialgebraically, which allows us to consider their semialgebraic extension over any real closed field. Starting from these non-standard symmetric spaces we use a valuation (with image some non-discrete ordered abelian group $Λ$) on the fields to define a $Λ$-pseudometric. Identifying points of distance zero results in a $Λ$-metric space $\mathcal{B}$. Assuming that the root system of the associated Lie group is reduced, we prove that $\mathcal{B}$ is an affine $Λ$-building. The proof relies on a thorough analysis of stabilizers.

math.GR↗
arXiv2023

(In)dependence of the axioms of $Λ$-trees

A $Λ$-tree is a $Λ$-metric space satisfying three axioms (1), (2) and (3). We give a characterization of those ordered abelian groups $Λ$ for which axioms (1) and (2) imply axiom (3). As a special case, it follows that for the important class of ordered abelian groups $Λ$ that satisfy $Λ=2Λ$, (3) follows from (1) and (2). For some ordered abelian groups $Λ$, we show that axiom (2) is independent of the axioms (1) and (3) and ask whether this holds for all ordered abelian groups. Part of this work has been formalized in the proof assistant \textsf{Lean}.

math.GR↗
arXiv2022

An incomplete real tree with complete segments

Let $\mathbb{F}$ be the field of real Puiseux series and $\mathcal{T}_\mathbb{F}$ the $\mathbb{Q}$-tree defined by Brumfiel. We show that completing all the segments of $\mathcal{T}_\mathbb{F}$ does not result in a complete metric space.

math.GT↗