The article introduces a new non‑archimedean metric, the cophenetic metric, defined on persistent homology classes of all degrees. It demonstrates that zeroth persistent homology combined with this metric and various hierarchical clustering algorithms yields statistically verifiable, commensurate topological information on multiple datasets. The resulting clusters, evaluated by silhouette score and Rand index, perform well, and the metric enables visualization of inter‑relations among persistent homology classes across all degrees via rooted trees.
By \.Ismail G\"uzel, Atabey Kaygun
arXiv:2609. 17899v1 Announce Type: new Abstract: We extend the discrete complex complement quotient (DCCQ) framework from binary Bernoulli counts to multinomial count compositions.
By Y. Kenan Y{\i}lmaz
arXiv:2607. 05835v1 Announce Type: cross Abstract: For every loopless matroid $M$ and every Feichtner--Yuzvinsky building set $\mathcal{G}$ containing the top flat, we construct an integral tangent class $T_{M,\mathcal{G}}^{\mathbb{Z}}\in K_{\mathbb{Z}}(M,\mathcal{G})$; in the realizable case it specializes to the class of the tangent bundle of the corresponding wonderful compactification, it recovers the Hilbert series of the Chow ring through Hirzebruch--Riemann--Roch, and it satisfies the expected Chern-alpha lower bounds.
By Ronnie Cheng, Shurui Liu, Guoxiong Gao
arXiv:2607. 23732v1 Announce Type: cross Abstract: In Question~3.
By Xinan Dai, Wenhao Deng, Yingdong Shi, Tailin Wu, Yuchen Yang
arXiv:2512.23348v3 Announce Type: replace-cross
Abstract: We introduce a data-analysis framework based on filtrations of finite topological spaces. Starting from a finite metric data set, we construc...
By Sel\c{c}uk Kayacan
arXiv:2608. 18570v1 Announce Type: cross Abstract: Machine learning and data analysis techniques have recently emerged as powerful tools for identifying patterns and formulating conjectures in mathematical research, most notably in the field of low-dimensional topology.
By Brandon Robinson, Shimal Harichurn, Fabian Ruehle, Sergei Gukov, Rak-Kyeong Seong, Miranda C. N. Cheng
arXiv:2506. 15020v2 Announce Type: replace-cross Abstract: We propose persistent discrete homology as a tool for topological data analysis and discuss its advantages over the existing methods.
By Chris Kapulkin, Nathan Kershaw
arXiv:2607. 24847v1 Announce Type: cross Abstract: We introduce an extremal invariant associated with Chowla-type order conditions in finite groups.
By Mohsen Aliabadi, Keith Driscoll, Elliot Krop, Petar Sirkovic, Everett Sullivan, Elahe Vedadi
arXiv:2409. 15600v3 Announce Type: replace Abstract: A representation of a molecule or material should be invariant to the symmetries of physics, unique, continuous, efficient and general.
By Rahul Khorana, Marcus Noack, Jin Qian
The paper introduces Solver Agent, an AI framework that uses large language models to perform calculations and proofs in mathematics and theoretical physics, tracking the solution process via a persistent ledger. It applies this framework to study global F‑theory uplifts of Type IIB orientifolds and S‑folds, establishing conditions for Weierstrass models over projective threefolds with terminal ζ_k quotient singularities to yield Ε-factorial elliptically fibered Calabi‑Yau fourfolds. The authors derive fixed‑point contributions to Hodge data and Euler characteristics, demonstrate how these corrections determine localized D3‑brane charges for tadpole cancellation, and illustrate the results with toric hypersurface constructions and methods for four‑form flux analysis in Δ=1 compactifications.
By Eliott Morgensztern, Cesar Fierro Cota, Alessandro Mininno
arXiv:2503. 03156v4 Announce Type: replace-cross Abstract: We propose DiRe, a force-directed dimensionality reduction framework designed to preserve global structure and homological features while remaining practical on modern hardware.
By Alexander Kolpakov, Igor Rivin
The paper introduces persistent magnitude homology as a functorial invariant for quantitative equational theories, providing a barcode that captures the metric structure of the free algebra generated by a metric space of generators. It shows how this invariant combines graded magnitude homology with persistence, yielding stability estimates and a method to compare barcodes when theories are extended. Four concrete examples illustrate the theory in each homological degree.
By Luciano Melodia