transversality arguments

This work addresses the challenge of tracking topological features in time-varying persistence diagrams (vineyards) when parameters traverse closed loops, leading to feature permutations due to monodromy. Focusing on vineyards induced by radial persistence transforms of planar one-dimensional manifolds, the paper establishes a novel connection between symmetry sets and monodromy. By integrating singularity theory, geometric properties of distance functions, and persistent homology, it demonstrates that monodromy arises from specific singularities of the distance function. The study provides a complete classification of scenarios in which monodromy occurs and establishes necessary and sufficient conditions for its emergence along sufficiently small loops. These results lay a theoretical and algorithmic foundation for detecting, exploiting, or avoiding monodromy in topological data analysis.

This work investigates monodromy in persistent homology vineyards and its deep connections to knot theory. First, it establishes an equivalence between periodicity and monodromy for vineyards in arbitrary dimensions and periods—yielding the first characterization of this relationship. Second, it explicitly constructs *l*-persistent homology vineyards exhibiting nontrivial monodromy. The central result rigorously proves that, for any knot or link in ℝᵈ (*d* ≥ 3), there exists a point-cloud shape whose vineyard realizes that knot/link as its monodromy invariant. This demonstrates that persistent homology vineyards encode maximal topological complexity. Crucially, the work bridges persistent homology vineyard analysis with classical knot classification—establishing the first rigorous connection between these domains. It thereby provides a novel algebraic-topological framework and constructive paradigm for high-dimensional topological data analysis.

This work addresses the reliable computation of Galois and monodromy groups for parametrized polynomial systems. To this end, it introduces a novel framework that integrates certified homotopy path tracking with homotopy graphs, enabling—for the first time—the rigorous numerical verification of monodromy group actions. By combining certified numerical algorithms with techniques from numerical algebraic geometry, the proposed method guarantees the mathematical correctness of its computational results. The approach has been successfully validated on a range of examples drawn from both pure and applied mathematics, demonstrating its effectiveness, reliability, and practical utility in analyzing the group-theoretic structures of complex polynomial systems.

This work addresses the challenge of formally verifying widely accepted yet non-rigorous results in theoretical physics in the absence of strict mathematical proofs. It presents the first axiomatization of physical reasoning embedded within the Lean 4 interactive theorem prover, using Seiberg–Witten’s solution to N=2 SU(2) supersymmetric Yang–Mills theory as a case study. The approach explicitly articulates physical axioms, derives machine-checkable conclusions, and precisely tracks the assumptions underlying each result. The project successfully formalizes the genus-one case and establishes an axiomatic framework for higher-genus SU(N) theories, thereby providing a verifiable, reproducible paradigm for evaluating AI-generated or AI-inspired physical arguments.

This work proposes a pedagogical framework for introducing topological data analysis to students of mathematics and computer science, balancing mathematical rigor with accessibility. Departing from conventional metric-space-based approaches, the framework models data as information-carrying functions and foregrounds the role of the observer along with symmetry constraints. It naturally bridges persistent homology and symmetry-aware modeling in machine learning through group equivariant non-expansive operators (GENEOs). By integrating persistent homology, algebraic topology, and monodromy theory from two-parameter persistence, the approach forms a self-contained instructional system that significantly enhances conceptual clarity and cross-disciplinary applicability, making it well-suited for advanced undergraduate and graduate instruction.

This study investigates Helly-type theorems and their variants from a topological perspective, with a focus on combinatorial and geometric intersection problems. By introducing tools from algebraic topology, the work systematically develops two core proof strategies: one based on the nerve lemma and the other on non-embeddability arguments. Beyond providing a comprehensive survey of recent advances in the field, the paper establishes a unified theoretical framework that substantially extends the applicability of the classical Helly theorem to topological settings. This contribution fosters a deeper integration of combinatorial geometry with topological methods, broadening the scope of intersection theorems in non-classical contexts.

This work addresses the absence of rigorous formalizations of abstract simplicial complexes and their stellar subdivisions in existing proof systems. It presents the first purely combinatorial formal framework for abstract simplicial complexes grounded in combinatorial topology, implemented in the Lean theorem prover. The framework encompasses fundamental operations such as morphisms, links, and joins, and systematically investigates their interaction with stellar subdivision. Key contributions include the first formalization of stellar subdivision in any proof assistant, the verification of several crucial identities—some previously undocumented in the literature—for the study of triangulated manifolds, and the proof of significant theorems such as the invariance of links under subdivision. This development establishes a reliable formal foundation for computational topology.

This work resolves the long-standing conjecture on the sharp growth rate for gradient Hölder continuity in non-uniformly elliptic variational problems, particularly addressing the regularity boundary issues in multiphase physical systems that have remained open due to the absence of a differentiable Euler–Lagrange structure. We introduce an innovative synthesis of the “ghost equation” analytical framework with a neuro-symbolic Large Reasoning Model (LRM) grounded in slice topos theory, modeling the reasoning process as a categorical colimit. This leads to the first formally verifiable Safe and Typed Chain-of-Thought framework (PC-CoT), which enables a machine-checkable proof of the precise threshold \( q/p < 1 + \alpha/n \). Our approach endows AI systems with the capability to autonomously explore the “dark side” of the calculus of variations.

Existing symmetry-enhanced resolution proof systems suffer from prohibitive computational complexity, limiting their practicality, and the theoretical limits of dynamic symmetries have long remained unclear. This work introduces the notion of “small symmetries,” which restricts the number of variables involved in each symmetry operation, yielding a new proof system that balances theoretical tractability with practical potential. The paper establishes, for the first time, a strict hierarchy between local and global small symmetries based on symmetry size, proving exponential separations in proof length across different levels. Notably, even the weakest level of this hierarchy substantially surpasses both standard resolution and constant-depth Frege systems in proof strength. Additionally, this work resolves the long-standing open problem of achieving an exponential separation between the SRCI and SRII proof systems.

This study addresses the local-to-global consistency problem for finite closure systems over overlapping domains: given a collection of local closure operators, can they be coherently extended to a global conservative closure system? To this end, the paper introduces the “atlas-induced closure” construction and establishes a semantic correspondence between indexed truth spaces and closure systems, revealing an intrinsic connection between regional inclusion relations and closure derivations. The central contribution is a finite, directly computable criterion—termed “local visibility obstructions”—that precisely characterizes the necessary and sufficient conditions for the existence of a global conservative realization. In the absence of such obstructions, the atlas-induced closure yields the unique global conservative extension, and an efficient algorithm is provided for detecting these obstructions.