Fixed-point theorems for set-valued maps using homological methods.
problem Finding fixed points for specific types of set-valued maps.
method Homological selection theorems applied to finite-dimensional spaces.
result Established fixed-point theorems for usco homologically UV^n set-valued maps.
We analyze decision boundaries using topological data analysis.
problem Quantifying deep neural network complexity for model selection.
method We use labeled Čech complex, plain labeled Vietoris-Rips complex, and locally scaled labeled Vietoris-Rips complex to infer persistent homology of decision boundaries.
result We provide theoretical conditions and analysis for recovering the homology of a decision boundary from samples.
These notes from the 2014 summer school Quantum Topology at the CIRM in Luminy attempt to provide a rough guide to a selection of developments in Khovanov homology over the last fifteen years.
Given a multifunction from X to the k−fold symmetric product Symk(X), we use the Dold-Thom Theorem to establish a homological selection Theorem. This is used to establish existence of Nash equilibria. Cost functions in problems concerning the existence of Nash Equilibria are traditionally multilinear in the mixe…
Knot Floer homology is a knot invariant derived from Heegaard Floer homology.
problem Understanding and computing knot invariants.
method Derived from Heegaard Floer homology, with contributions from many researchers.
result Knot Floer homology is a useful tool for studying knots.
This paper studies representation stability in the sense of Church and Farb for representations of the symmetric group Sn on the cohomology of the configuration space of n ordered points in Rd. This cohomology is known to vanish outside of dimensions divisible by d−1; it is shown here that the Sn-…
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…
Ensemble method ranks homologous proteins robustly across various similarity metrics.
problem Ranking homologous proteins in a candidate set with high accuracy and robustness.
method Ensemble of models and assessment metrics, phalanxes, and aggregation of diverse metrics.
result Ensemble of phalanxes identifies strong and diverse subsets of feature variables for robust ranking.
Often, high dimensional data lie close to a low-dimensional submanifold and it is of interest to understand the geometry of these submanifolds. The homology groups of a manifold are important topological invariants that provide an algebraic summary of the manifold. These groups contain rich topological information, for…
This paper interprets critical scales in persistent homology for compact metric spaces.
problem Understanding critical scales in persistent homology for general compact metric spaces.
method Analyzing local minima of the distance function and their impact on persistence.
result Each decrease in zero-dimensional persistence and increase in one-dimensional persistence is induced by local minima of the distance function.
Persistent homology improves fingerprint classification accuracy.
problem Classifying fingerprints into arch, loop, and whorl groups.
method Applying topological data analysis (TDA) to 3D point clouds of oriented minutiae points and ink-roll images, using supervised learning.
result Achieves near state-of-the-art classification accuracy rates.
A new method for optimal filtration learning in time-series data analysis.
problem Finding an optimal filtration for analyzing topological properties of discrete data.
method Formulated an optimization problem and proposed an algorithm for solving it.
result Derivation of the exact formula of the gradient of the loss function with respect to filtration parameters.
This review explores TDA and TDL beyond persistent homology.
problem Limitations of persistent homology in capturing topological invariants and homotopic evolution.
method Spectral representations, sheaf theory, Mayer topology, interaction topology, differential topology, geometric topology.
result Review of topological tools for various data types.
The paper optimizes autoencoder latent spaces for one-class learning with controlled connectivity.
problem Learning representations with controllable connectivity for better upstream tasks.
method A novel loss function based on persistent homology controls the connectivity of autoencoder latent spaces.
result The controlled connectivity in latent space improves one-class learning performance, especially in low sample size scenarios.
The paper develops a new mathematical framework for group-equivariant operators in machine learning.
problem Developing a robust mathematical framework for group-equivariant operators in machine learning.
method The paper introduces group-equivariant non-expansive operators (GENEOs) and studies their topological and metric properties.
result The space of GENEOs is compact and convex, providing fundamental guarantees for machine learning.
TDA classifies MNIST digits with reduced feature set.
problem Classifying MNIST digits using machine learning.
method Persistent homology for feature generation and classification.
result 5x reduction in feature set size with similar accuracy.
Extends Khovanov homology spectral sequence using Heegaard Floer homology.
problem Relating Khovanov homology and Heegaard Floer homology of branched double covers.
method Involutive Heegaard Floer homology, bordered Floer homology, surgery exact triangle.
result Establishes spectral sequence connecting Khovanov homology and Heegaard Floer homology.
New homological action on sutured instanton homology defined.
problem Detecting link splitting in knot homology.
method Defining a homological action on sutured instanton Floer homology.
result Instanton knot homology detects link splitting for two-component links.
The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.
problem Constructing rational homology 3-spheres that bound rational homology 4-balls.
method Exploring plumbed 3-manifolds and using rational homology circles.
result Infinite families of rational homology 3-spheres that bound rational homology 4-balls.
Study on knot concordance and homology cobordism using Heegaard Floer homology.
problem Knot concordance and homology cobordism.
method Heegaard Floer homology.
result Recent results in knot concordance and homology cobordism.
RCLA reduces noise in topological data analysis, preserving essential structure.
problem Noise in large datasets obscures topological features in persistent homology.
method Grid-based RCLA integrates data reduction and denoising with a threshold parameter.
result RCLA provides a theoretical guarantee and automatic parameter selection.
Unified framework for proving functoriality of various link homology theories.
problem Functoriality and Reidemeister invariance of link homology theories.
method Unified framework leveraging Khovanov homology and various link homology theories.
result Stronger functoriality results by avoiding spectral sequences.
The paper uses Heegaard Floer homology to study homology cobordism groups.
problem Understanding the structure of homology cobordism groups.
method Defined an invariant using Heegaard Floer homology, analogous to Stoffregen's connected Seiberg-Witten Floer homology.
result Computed involutive correction terms for certain three-manifolds.
Khovanov and chromatic homologies show similar patterns, improving chromatic bounds and computing Jones polynomial.
problem Understanding similarities between Khovanov and chromatic homologies.
method Analyzing isomorphism and improving bounds using explicit formulas.
result Improved bounds for chromatic homology and explicit formula for chromatic homology rank.
Maps quandle homology to relative group homology.
problem Understanding the relationship between quandle and group homology.
method Introducing a chain map and constructing quandle cocycles.
result Relates quandle homology to relative group homology through triangulations.
Homological stability aids in computing group homology.
problem Computing homology of families of groups.
method Proving homological stability theorems and computing stable homology.
result Computation of Higman-Thompson groups' homology.
Study ribbon homology concordances using link Floer homology.
problem Understanding ribbon homology concordances and their effects on link Floer homology.
method Combining results from Daemi, Lidman, Vela-Vick, Wong, and Zemke, using link Floer homology and torsion submodules.
result Ribbon homology concordances induce split injections on HFL−. New symplectic annular Khovanov homology connects knot theory to Floer homology.
problem Understanding the relationship between knot theory and Floer homology.
method Introducing a new version of symplectic annular Khovanov homology and establishing spectral sequences.
result Established spectral sequences linking different knot homologies.
Grid homology confirms the Upsilon invariant in knot theory.
problem Verifying the equivalence of Upsilon invariants in knot theory.
method Reconstructed Upsilon invariant using grid homology and proved equivalence.
result Upsilon invariants in knot Floer and grid homology are equivalent.
The paper introduces representation homology of topological spaces and its connections to other homology theories.
problem Understanding the algebraic structure of topological spaces through representation homology.
method Developed a geometric relation between representation homology and higher Hochschild homology, constructed maps and spectral sequences, and computed explicit examples.
result Representation homology of the suspension of a space is isomorphic to its higher Hochschild homology.
The paper examines lattice homology invariants of Seifert homology spheres.
problem Understanding homology cobordism invariants for Seifert fibered integral homology 3-spheres.
method Utilizes lattice homology and Heegaard Floer homology to study invariants.
result Reproves and extends the invariance of Seifert homology spheres' d-invariants and maximal monotone subroots. Detects figure-eight knot using Khovanov homology.
problem Detecting the figure-eight knot.
method Using Dowlin's spectral sequence from Khovanov homology to knot Floer homology.
result Reduced Khovanov homology (over Q) detects the figure-eight knot.
Study improves Heegaard Floer homology relations for knots in homology spheres.
problem Improving relations in Heegaard Floer homology for knots in homology spheres.
method Proved inequality for d-invariants, used reduced Floer homology rank relations.
result Degree one maps between aspherical Seifert homology spheres are homotopic to homeomorphisms if Heegaard Floer homologies are isomorphic.
New link detection results using knot and link Floer homology.
problem Detecting specific links and knots using Floer homology.
method Inspired by Khovanov homology, uses knot and link Floer homology.
result Detects specific links and knots with high precision.
Proves a rank inequality between Khovanov and knot Floer homologies.
problem Rank inequality between Khovanov and knot Floer homologies for knots.
method Oriented cube of resolutions construction for knot Floer homology.
result Proves Rasmussen's conjecture about Khovanov and knot Floer ranks.
Corrects a lemma in a 2009 paper about contact homology and Floer homology.
problem An error in a lemma about contact homology and Floer homology.
method None, as it is a correction of an existing lemma.
result Corrects an error in a previously published lemma.
Survey on proof of homology isomorphism between two complex theories.
problem Proof of isomorphism between Heegaard Floer homology and embedded contact homology.
method Survey of proof methods from multiple papers.
result Established isomorphism between Heegaard Floer homology and embedded contact homology.
Khovanov-Floer theories are shown to be invariant under mutation.
problem Invariance of Khovanov-Floer theories under Conway mutation.
method Spectral sequences from Khovanov homology and proofs of conjectures.
result Strong Khovanov-Floer theories are mutation-invariant.
Study shows infinite-rank summand in homology concordance group of knots.
problem Homology concordance of knots in integer homology three-spheres.
method Knot Floer homology to construct homology concordance homomorphisms.
result Homology concordance group modulo knots from S^3 contains an infinite-rank summand.
Lower bounds on ribbon distance using Bar-Natan and α-Homology.
problem Calculating the minimum number of ribbon operations to unknot a knot.
method Bar-Natan Homology and α-Homology approaches.
result Lower bounds on ribbon distance via both Bar-Natan and α-Homology.
Paper studies spectral invariants and monopole Floer homology for rational homology three-spheres.
problem Tackles the existence of positive scalar curvature metrics on ribbon homology cobordisms.
method Defines an R-filtration on the equivariant complex of monopole Floer homology via Chern-Simons-Dirac functional, leading to a spectral invariant.
result Shows that the spectral invariant provides an obstruction to the existence of positive scalar curvature metrics on ribbon homology cobordisms.
Study links with annuli using sutured Floer homology.
problem Characterize links with specific cable structures.
method Apply sutured Floer homology techniques.
result Characterizations of links with (n,nm)-cables and (2,2m)-cables. Spectral sequence connects knot homologies to quotient knots.
problem Distinguishing knots using homology.
method Construct spectral sequence relating Khovanov homology to quotient knots.
result Khovanov homology distinguishes certain slice disks.
We compare the homology groups HnIC(X) of the chain complex of integral currents with compact support of a metric space X with the singular Lipschitz homology HnL(X) and with ordinary singular homology. If X satisfies certain cone inequalities all these homology theories coincide. On the other hand, for…
In~\cite{rotvandervorst} a homology theory --Morse-Conley-Floer homology-- for isolated invariant sets of arbitrary flows on finite dimensional manifolds is developed. In this paper we investigate functoriality and duality of this homology theory. As a preliminary we investigate functoriality in Morse homology. Functor…
New link detection results using closures of 3-braids.
problem Link detection using homology theories.
method Closure operations on 3-braids and homology theories.
result Detection of specific links using link Floer homology, Khovanov homology, and annular Khovanov homology.
Coarse assembly maps generalize known results for K-homology.
problem Generalizing known results for K-homology to other coarse homology theories.
method Constructing coarse assembly maps as natural transformations between coarse homology theories.
result Explicit calculation of the domain of the coarse assembly map in terms of locally finite homology theory.
Khovanov homology detects trefoil knots.
problem Identifying knots using homology.
method Spectral sequence to knot Floer homology.
result Reduced 2-coloured Khovanov homology detects trefoil.