Defines twisted Blanchfield pairings for chain complexes and proves their properties.
problem Defining and analyzing twisted Blanchfield pairings for chain complexes.
method Defines twisted Blanchfield pairings for symmetric triad of chain complexes over a group ring, proving sesquilinearity, hermitian, and nonsingularity under certain conditions.
result Proves the twisted Blanchfield pairing is sesquilinear, hermitian, and nonsingular under certain conditions.
Reduces identity testing of reversible Markov chains to simpler symmetric chain tests.
problem Testing identity of reversible Markov chains from a single trajectory.
method Using lumping-congruent Markov embeddings, the problem is simplified to testing symmetric chains over a larger state space.
result Achieves state-of-the-art sample complexity for identity testing.
New mapping class group actions on Hochschild complexes for modular categories.
problem Understanding actions of mapping class groups on Hochschild complexes of modular categories.
method Construction of a symmetric monoidal functor with excision property.
result Homotopy coherent projective action of mapping class groups on Hochschild complexes.
We define an algebraic group comprising symmetric chain complexes which captures the first two stages of the Cochran-Orr-Teichner solvable filtration of the knot concordance group in a single invariant. To achieve this we impose additional structure on each chain complex which puts extra control on the fundamental grou…
We define an algebraic group comprising symmetric chain complexes which captures the first two stages of the Cochran-Orr-Teichner solvable filtration of the knot concordance group in a single obstruction. To achieve this we impose additional structure on each chain complex which puts extra control on the fundamental gr…
Clarifies mathematical aspects of Picture Changing Operators.
problem Understanding the geometric properties of PCOs.
method Showed PCOs are chain maps between differential and integral forms on supermanifolds.
result PCOs are chain maps, providing a new perspective on their structure.
We refine the Whitehead torsion of a chain equivalence of finite chain complexes in an additive category $\bA$ from an element of $\widetilde{K}^{iso}_1(\bA)$ to an element of the absolute group $K_1^{iso}(\bA)$. We apply this invariant to symmetric Poincaré complexes and identify it in terms of more traditional invari…
The study connects monopole chains to Higgs bundles and classifies symmetric chains.
problem Classifying symmetric monopole chains invariant under cyclic actions.
method Formulation of a correspondence between monopole chains and spectral data, using the Nahm transform.
result Classification of symmetric monopole chains of charge k.
Constructs a support-preserving homotopy for differential forms with boundary decay estimates.
problem Non-uniqueness of chain homotopies in de Rham complexes with boundary decay properties.
method Constructs a specific chain homotopy with desirable support propagation and boundary decay estimates.
result Obtains a support-preserving right inverse of the divergence operator with optimal decay estimates.
The paper bounds generalization errors for deep neural networks with Markov datasets.
problem Bounding generalization errors for deep learning with Markov datasets.
method Developed new symmetrization inequalities for Markov chains, using spectral gap of the infinitesimal generator.
result Derived upper bounds on generalization errors for deep neural networks with Markov datasets.
Paper introduces a diagnostic for approximate inference methods.
problem Estimating errors in probabilistic inference algorithms, especially for approximate methods.
method Repeatedly simulate datasets from the prior and perform inference on each, estimating a symmetric KL-divergence.
result A diagnostic for approximate inference methods can be estimated using symmetric KL-divergence.
In this paper we present a new theory of calculus over k-dimensional domains in a smooth n-manifold, unifying the discrete, exterior, and continuum theories. The calculus begins at a single point and is extended to chains of finitely many points by linearity, or superposition. It converges to the smooth continuum w…
New proof of chain duality for simplicial complexes.
problem Proving the existence of chain duality for chain complexes over simplicial complexes.
method Geometric and conceptual treatment of chain duality.
result Fundamental for Ranicki's surgery exact sequence.
New method adds user constraints to Markov chains for better data reduction.
problem No systematic framework to impose user-defined constraints on Markov chains.
method Path entropy maximization to derive transition probabilities with user constraints.
result Improved nonlinear dimensionality reduction with user-prescribed constraints.
Unified Morse-Bott-Smale chain complex, resolves well-definedness issue.
problem Well-definedness of Morse-Bott-Smale chain complex.
method Unified five degeneracy relations into a single condition.
result Quasi-isomorphic to Morse-Smale-Witten chain complex, alternative proof of Morse Homology Theorem.
Geometrically interprets a duality theorem linking cochain and chain complexes.
problem Understanding a complex duality theorem in geometric terms.
method Introduces a chain isomorphism involving simplicial and cellular complexes.
result Establishes a geometric interpretation of Ranicki duality.
Identity testing for reversible Markov chains without symmetry assumption.
problem Identity testing of reversible Markov chains.
method Using distance notion from Daskalakis et al. [2018a], testing without symmetry assumption.
result It is possible to perform identity testing under weaker assumption of reversibility.
In this paper, we introduce the notion of Reidemeister torsion for quasi-isomorphisms of based chain complexes over a field. We call a chain map a quasi-isomorphism if its induced homomorphism between homology is an isomorphism. Our notion of torsion generalizes the torsion of acyclic based chain complexes, and is a ch…
We introduce some chain maps between Khovanov complexes. Each of the chain maps commutes with a chain homotopy map and a retraction maps which obtain a Reidemeister invariance of Khovanov homology.
Procedure tests if unknown Markov chain matches a reference chain.
problem Testing if an unknown Markov chain matches a reference chain.
method An efficient procedure based on a single long state sequence.
result Nearly matching upper and lower sample complexity bounds for total variation distance.
Fix an integer N>1. To each diagram of a link colored by 1,...,N, we associate a chain complex of graded matrix factorizations. We prove that the homotopy type of this chain complex is invariant under Reidemeister moves. When every component of the link is colored by 1, this chain complex is isomorphic to the chain com…
We give a new proof of the Morse Homology Theorem by constructing a chain complex associated to a Morse-Bott-Smale function that reduces to the Morse-Smale-Witten chain complex when the function is Morse-Smale and to the chain complex of smooth singular N-cube chains when the function is constant. We show that the ho…
Estimates covariance matrices using Markov chain Monte Carlo with improved sample complexity.
problem Complexity of covariance matrix estimation for Gibbs distributions.
method Uses Markov chain Monte Carlo with conditions on the chain's spectral gap and Poincaré inequality.
result Achieves similar sample complexity as i.i.d. samples with better query complexity.
New quantum code lacks sparse lift.
problem Existence of sparse lifts for quantum codes.
method Constructed a sparse Z2 chain complex without a sparse lift. result Found a quantum code without a sparse lift.
Characterizes real holomorphic chains on complex manifolds.
problem Representing homology classes by algebraic cycles.
method Characterization of real holomorphic chains; application to homology classes.
result Real holomorphic chains are characterized by local properties.
Python package cegpy models processes with asymmetries.
problem Leveraging CEGs for processes with structural asymmetries.
method Developed cegpy, a Python package for CEGs with Bayesian model selection and probability propagation.
result First CEG package in any language that can model symmetric and asymmetric structures.
New algorithm reduces autocorrelation in HMC for lattice field theories.
problem Reduction of autocorrelation in HMC for lattice field theories.
method Hybrid Monte-Carlo algorithm with restricted Boltzmann machine.
result Reduction of autocorrelation in both symmetric and broken phases.
Study nonparametric estimator for Markov chain transition matrices in offline setting.
problem Estimating transition matrices of finite controlled Markov chains from logged data.
method Developed sample complexity bounds and conditions for minimaxity.
result Achieving certain statistical risk requires balancing mixing properties and sample size.
The paper constructs Morse complexes for orbifolds and shows their homologies are orbifold invariants.
problem Understanding the homology of orbifolds.
method Constructing invariant and coinvariant Morse chain complexes for orbifolds.
result The homology of coinvariant Morse complexes computes the singular homology of the underlying space.
Computes homology of an obstruction chain complex in grid homology.
problem Computing the homology of an obstruction chain complex in grid homology.
method Defined and computed the homology of the obstruction chain complex of the full grid.
result Results about the existence of sign assignments in grid homology.
Link Floer homology is split into snake complexes and local systems.
problem Classifying link Floer complexes over specific rings.
method Classifying isomorphism and chain homotopy equivalence classes of free chain complexes over a specific ring, then applying these results to link Floer complexes.
result Link Floer complexes split uniquely into snake complexes and local systems.
Smooth knots in complex hyperbolic plane limit sets to chains or R-circles.
problem Characterizing limit sets of knots in complex hyperbolic geometry.
method Analyzing embeddings of knots as limit sets of discrete subgroups of PU(2, 1).
result Knots are either chains or R-circles as limit sets.
Let M be a closed connected manifold, f be a Morse map from M to a circle, v be a gradient-like vector field satisfying the transversality condition. The Novikov construction associates to these data a chain complex C∗=C∗(f,v). There is a chain homotopy equivalence between C∗ and completed simplicial cha…
Estimates Markov chain parameters from a single long sequence, analyzing complexity based on mixing properties.
problem Estimating parameters of a discrete-state Markov chain kernel from a single long sequence of observations.
method Characterizes minimax sample complexity in finite and countably infinite cases, focusing on mixing properties.
result Sample complexity is governed by mixing properties, with finite-sample estimators available for finite-state cases.
Review of sigma models on flag manifolds, linking to spin chains and integrable theories.
problem Understanding phase transitions and anomalies in spin chains and sigma models.
method Analyzing topological angles, discrete 't Hooft anomalies, and integrable models.
result Gapless phases in certain spin chains can be explained by discrete anomalies in continuum theories.
Four constructions of constant mean curvature (CMC) hypersurfaces in the (n+1)-sphere are given, which should be considered analogues of `classical' constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multipl…
New algorithm for Riemannian barycentre from Markov chain samples.
problem Computing Riemannian barycentre from non-i.i.d. Markov chain samples.
method Proposes a new Markov chain Monte Carlo algorithm for Riemannian barycentre computation.
result Algorithm converges to Riemannian barycentre of stationary distribution.
New Markov chains defined on simplicial complexes for understanding their topology.
problem Understanding the topology of simplicial complexes and hypergraphs.
method Defining new Markov chains on simplicial complexes and studying their properties.
result The generator of the new Markov chain is the upper Laplacian, and the Markov chain is positive recurrent.
A new approach uses circuit topology to study complex polymer interactions.
problem Understanding structural phase transitions in entangled polymer systems.
method Braided circuit topology framework for multiple-chain systems.
result Circuit topological motif fractions are effective order parameters for structural transitions.
Study hypothesis testing for noisy Markov chain samples.
problem Hypothesis testing between two discrete distributions via noisy Markov chain samples.
method Derive instance-dependent minimax rates and analyze spectral properties of the Markov chain.
result Wide statistical window in sample complexity for different initial distributions.
Complete integrability proven for Poisson-Nijenhuis systems on compact hermitian symmetric spaces.
problem Proving complete integrability for a specific class of Poisson-Nijenhuis systems.
method Defined a class of Poisson-Nijenhuis systems on compact hermitian symmetric spaces, determined their spectrum, and introduced an abelian algebra of collective Hamiltonians.
result Proved complete integrability with respect to both Poisson structures.
A method for learning with autoregressive chain-of-thoughts.
problem Learning prompt-to-answer mappings from sequence-to-next-token generators.
method Iterating a fixed, time-invariant generator for multiple steps to generate a chain-of-thought, then taking the final token as the answer.
result Universal representability and computationally tractable chain-of-thought learning for a simple base class.
Legendrian invariant studied in knot lattice homology.
problem Legendrian invariant in knot lattice homology.
method Defined Alexander grading and used filtered chain homotopy.
result Alexander grading of Legendrian invariant is invariant under blow-ups.
Let f:M→R be a Morse-Bott function on a finite dimensional closed smooth manifold M. Choosing an appropriate Riemannian metric on M and Morse-Smale functions fj:Cj→R on the critical submanifolds Cj, one can construct a Morse chain complex whose boundary operator is…
We analyze a functor from cyclic operads to chain complexes first considered by Getzler and Kapranov and also Markl. This functor is a generalization of the graph homology considered by Kontsevich, which was defined for the three operads Comm, Assoc, and Lie. More specifically we show that these chain complexes have a …
Alternative proof of uniform boundary condition using geometric Følner arguments.
problem Uniform boundary condition in normed chain complexes.
method Geometric Følner arguments on the chain level.
result Integral refinements of the uniform boundary condition.
Khovanov homology for links in S^3 via 1-tangle diagrams in annulus.
problem Computing Khovanov homology for links in S^3.
method Constructing a chain complex from a 1-tangle diagram in the annulus, using a cube of resolutions.
result A spectral sequence converging to reduced Khovanov homology.
New MCMC method for complex models with large variables.
problem Inference on posterior model probabilities in large model spaces.
method Reversible genetically modified mode jumping Markov chain Monte Carlo (GMJMCMC).
result Introduced a proper MCMC with correct limiting distribution.