arXiv research

TopInG improves graph interpretability using persistent homology.

There is an interpretation of open string field theory in algebraic topology. An interpretation of closed string field theory can be deduced from this open string theory to obtain as well the interpretation of open and closed string field theory combined.

Study topological quantum mechanics on orbifolds with geometric interpretation.

GeoTop resolves topological ambiguity in diagnostic imaging using geometric-topological analysis.

STS clarifies chaos and stochastic dynamics, linking algebraic topology and physics.

Method quantifies disentanglement of generative models using manifold topology.

We introduce a notion of topological quandle. Given a topological quandle $Q$ we associate to every classical link $L$ in $\R ^3$ an invariant $J_Q(L)$ which is a topological space (defined up to a homeomorphism). The space $J_Q(L)$ can be interpreted as a space of colourings of a diagram of the link $L$ with colours f…

Paper lifts Artin's representation to loop braid groups topologically.

Python library for integrating TDA with machine learning.

Topological method detects Hopf bifurcations from time series.

This paper introduces TDA and TSI for better business analytics.

This paper studies the canonical Chow quotient of a smooth projective variety by a reductive algebraic group. The main purpose is to give some topological interpretations and characterization of Chow quotient which have the advantage to be more intuitive and geometric. This is to be done over the field of complex numbe…

While a wide range of interpretable generative procedures for graphs exist, matching observed graph topologies with such procedures and choices for its parameters remains an open problem. Devising generative models that closely reproduce real-world graphs requires domain knowledge and time-consuming simulation. While e…

This paper calculates Dijkgraaf-Witten invariants from Chern classes.

Method interprets deep learning models using topological data analysis.

We make a precision test of a recently proposed conjecture relating Chern-Simons gauge theory to topological string theory on the resolution of the conifold. First, we develop a systematic procedure to extract string amplitudes from vacuum expectation values (vevs) of Wilson loops in Chern-Simons gauge theory, and then…

PHLP uses persistent homology to interpret graph link prediction.

We study the algebraic and geometric properties of stated skein algebras of surfaces with punctured boundary. We prove that the skein algebra of the bigon is isomorphic to the quantum group ${\mathcal O}_{q^2}(\mathrm{SL}(2))$ providing a topological interpretation for its structure morphisms. We also show that its sta…

New compactification for character varieties with good topological properties.

SOM-VQ tokenizes discrete models with semantic structure and navigable topology.

We indicate how to combine some classical topology (Thom's work on the Steenrod problem) with some modern topology (simplicial volume) to show that every map between certain manifolds must have degree zero. We furthermore discuss a homotopy theoretic interpretation of parts of our proof, using Thom spaces and Steenrod …

STRAND: A single representation for hypothesis testing and vectorisation of persistence diagrams

Paper introduces a framework for diagnosing Alzheimer's disease using higher-order topological features from fMRI.

We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot homology groups and suggests that they can be interpreted directly in topological st…

On the Geroch-Kronheimer-Penrose future completion $IP(X)$ of a spacetime $X$, there are two frequently used topologies. We systematically examine $τ_+$, the stronger (metrizable) of them, which is the coarsest causally continuous topology, obtaining a variety of novel results, among them a complete characterization of…

GISST interprets GNNs by combining attention and sparsity for graph structure and node feature importance.

A novel method extracts topological features from word embeddings for text classification.

Paper introduces topological eigenvalue theorems for tensor analysis in multi-modal data.

This work analyzes PPR-based node embeddings and their topological information.

With the rapid adoption of machine learning techniques for large-scale applications in science and engineering comes the convergence of two grand challenges in visualization. First, the utilization of black box models (e.g., deep neural networks) calls for advanced techniques in exploring and interpreting model behavio…

Bayesian optimisation with graph kernels improves neural architecture search and provides interpretability.

DiSeNE generates interpretable node embeddings without supervision.

TopoNTK kernel captures higher-order interactions in simplicial complexes.

This paper is an exposition of the new subject of String Topology. We present an introduction to this exciting new area, as well as a survey of some of the latest developments, and our views about future directions of research. We begin with reviewing the seminal paper of Chas and Sullivan, which started String Topolog…

We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a di…

We calculate the asymptotic behavior of the Kashaev invariant of a twice-itarated torus knot and obtain topological interpretation of the formula in terms of the Chern--Simons invariant and the twisted Reidemeister torsion.

We give a topological interpretation of the space of $L^2$ harmonis forms of some QALE manifolds introduced by D. Joyce. We introduce a analytical criterium which make possible the used of Mayer-Vietoris sequence.

This paper provides a topological interpretation for number theoretic properties of quantum invariants of 3-manifolds. In particular, it is shown that the p-adic valuation of the quantum SO(3)-invariant of a 3-manifold M, for odd primes p, is bounded below by a linear function of the mod p first betti number of M. Shar…

Study topological G₂ and Spin(7) strings at 1-loop using double complexes.

Study of spectral invariants on CR contact manifolds with circle action.

CAMEL enhances manifold embedding and learning with curvature metrics.

The paper generalizes Yang-Mills theory to study four-manifold topology.

Study approximates top Lyapunov exponents for surface mapping classes.

We introduce the concept of community trees that summarizes topological structures within a network. A community tree is a tree structure representing clique communities from the clique percolation method (CPM). The community tree also generates a persistent diagram. Community trees and persistent diagrams reveal topol…

Topological Flow Matching: A Generative Modeling Framework for Structured Spaces

We define and study spectral data associated to U(m,m)-Higgs bundles through the Hitchin fibration. We give a new interpretation of the topological invariants involved, as well as a geometric description of the moduli space.

We give a topological interpretation of the space of L2-harmonic forms on finite-volume manifolds with sufficiently pinched negative curvature. We give examples showing that this interpretation fails if the curvature is not sufficiently pinched and that our result is sharp with respect to the pinching constants. The me…

Survey on Allen-Cahn equations and systems, focusing on multiplicity results and geometric interpretation.