arXiv subjects

Victor Jaeck

Publications and source records attributed to Victor Jaeck.

5 recordsLinked to original sources

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↗

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

Real Spectrum Compactifications of Universal Geometric Spaces over Character Varieties

We construct universal geometric spaces over the real spectrum compactification $\Xi^{\mathrm{RSp}}$ of the character variety $\Xi$ of a finitely generated group $\Gamma$ in $\mathrm{SL}_n$, providing geometric interpretations of boundary points. For an algebraic set $Y(\mathbb{R})$ on which $\mathrm{SL}_n(\mathbb{R})$ acts by algebraic automorphisms (such as $\mathbb{P}^{n-1}(\mathbb{R})$ or an algebraic cover of the symmetric space of $\mathrm{SL}_n(\mathbb{R})$), the projection map $\Xi \times Y \rightarrow \Xi$ extends to a $\Gamma$-equivariant continuous surjection $(\Xi \times Y)^{\mathrm{RSp}} \rightarrow \Xi^{\mathrm{RSp}}$. The fibers of this extended map are homeomorphic to the Archimedean spectrum of $Y(\mathbb{F})$ for some real closed field $\mathbb{F}$, which is a locally compact subset of $Y^{\mathrm{RSp}}$. The Archimedean spectrum is naturally homeomorphic to the real analytification, and we use this identification to compute the image of the fibers in their Berkovich analytification. For $Y=\mathbb{P}^1$, the image is a real subtree.

math.GR↗

arXiv2024

Real spectrum and oriented Gromov equivariant compactifications of character varieties

The character variety $\Xi$ of a finitely generated group $\Gamma$ in $\mathrm{PSL}_2(\mathbb{R})$ has many compactifications. We construct a continuous surjection from the real spectrum compactification $\Xi^{\mathrm{RSp}}$ to the oriented Gromov equivariant compactification. Our construction is based on a geometric interpretation of the elements of $\partial \Xi^{\mathrm{RSp}}$ as $\Gamma$-actions by isometries on $\mathbb{R}$-trees. We endow these $\mathbb{R}$-trees with an orientation induced by the standard orientation on the circle, which we characterize by a semialgebraic equation. Moreover, we describe the $\Gamma$-actions by orientation preserving isometries on oriented $\mathbb{R}$-trees, which arise in both compactifications, as limits of $\Gamma$-actions on the oriented hyperbolic plane, via asymptotic cones endowed with an ultralimit orientation.

math.GT↗

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↗