Study on the topology of ordered disc configurations, revealing nontrivial homotopy classes.
problem Topology of ordered disc configurations and their homotopy types.
method Analysis of ordered configuration spaces of hard discs, focusing on homotopy types and nontrivial classes.
result Exhibit nontrivial classes in π_{n-3} for all n, and their persistence in deformed ambient discs.
Study shows challenges in reinforcement learning math problems, proposing enhancements and a hardness measure.
problem Challenges in reinforcement learning finding rare high-reward instances.
method Combining combinatorial group theory, algorithmic enhancements, and topological hardness measure.
result Resolved mathematical questions and proposed enhancements for reinforcement learning.
Topological quantum computers use hyperbolic knots for computations.
problem The difficulty of calculating quantum invariants of knots.
method Using hyperbolic knots to compute topological quantum computer invariants.
result The hyperbolic geometry of knots is unlikely to be useful for topological quantum computation.
New proof shows a link problem is hard without complex links.
problem Deciding if a link contains a trivial sublink is hard.
method Reduces from Independent Set Problem, avoiding Brunnian links.
result The Trivial Sublink Problem is NP-hard due to mod 2 linking.
Researchers prove quantum invariants remain hard even when restricted.
problem Computing quantum invariants on 3-manifolds with specific restrictions.
method Using Heegaard splittings and Hempel distance, they construct a hyperbolic 3-manifold with same invariant.
result Proving hardness of computing quantum invariants is preserved under specific restrictions.
The Hard Lefschetz Theorem extends to certain Kähler Lie Algebroids with ellipticity.
problem Extending the Hard Lefschetz Theorem to Kähler Lie Algebroids.
method Analyzing a specific class of Kähler Lie Algebroids with ellipticity requirements.
result A class of Kähler Lie Algebroids satisfy the Hard Lefschetz Theorem with ellipticity.
We prove that the problem of deciding whether a 2- or 3-dimensional simplicial complex embeds into R3 is NP-hard. Our construction also shows that deciding whether a 3-manifold with boundary tori admits an S3 filling is NP-hard. The former stands in contrast with the lower dimensional cases wh…
Algorithm calculates quantum invariants of 3-manifolds with polynomial time complexity.
problem Computing quantum invariants from Tambara-Yamagami categories is #P-hard.
method Fixed-parameter tractable algorithm with first Betti number as parameter.
result Existence of FPT algorithm for Tambara-Yamagami invariants.
We investigate the complexity of finding an embedded non-orientable surface of Euler genus g in a triangulated 3-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance of embeddability of complexes into 3-manifolds. We prove that the problem is NP…
Quantum algorithm approximates Khovanov homology ranks.
problem Efficient computation of Khovanov homology ranks.
method Novel quantum algorithm with pre-thermalization procedure.
result Additive approximations to Khovanov homology ranks are hard problems.
We prove that deciding if a diagram of the unknot can be untangled using at most k Riedemeister moves (where k is part of the input) is NP-hard. We also prove that several natural questions regarding links in the 3-sphere are NP-hard, including detecting whether a link contains a trivial sublink with n componen…
Polynomial invariants classify molecular chains based on their contact arrangements.
problem No established invariants for molecular chains with both hard and soft contacts.
method Developed polynomial invariants for circuit topology of molecular chains.
result Polynomial invariants efficiently classify chains with various contact types.
Study monopole h-invariants from a topological viewpoint.
problem Understanding the h-invariants of 3-manifolds.
method Using Lidman and Manolescu's description of monopole Floer homology.
result Prove several properties of the h-invariants.
The paper explores conditions for topological rigidity in quotients of the Davis complex.
problem Understanding when quotients of the Davis complex are topologically rigid.
method Analyzing quotients of the Davis complex of right-angled Coxeter groups and conditions on defining graphs.
result Introduction of infinitely many infinite topologically rigid subclasses.
Constructing compact non-Kähler manifolds with and without the Hard Lefschetz Condition
problem Symplectic non-Kähler manifolds
method One-parameter family of symplectic forms on orbifold
result Symplectic manifolds with HLC and non-HLC structures
The study of higher-order homology embeddings for manifold topology.
problem Understanding the structure of higher-order homology embeddings to disclose geometric or topological information.
method Analysis of the null space of the k-th order Laplacian and proposing an algorithm to factorize the homology embedding. result The proposed spectral loop detection algorithm is more efficient and effective on various data types.
Study vineyards linking TDA and knot theory, showing rich topological features.
problem Linking computational topology and knot theory through vineyards.
method Construct periodic functions to study evolution of persistence diagrams.
result Vineyards are as rich as possible in topological terms.
(1) We show that if a presentation of the trivial group is "hard to trivialize", in the sense that lots of Tietze moves are necessary to transform it into the trivial presentation, then the associated presentation complex (which is a contractible 2-dimensional cell complex) is "hard to embed in R3", in the …
Paper calculates topological complexity of robot movement in narrow aisles.
problem Determining minimum number of scenarios for robot movement in a narrow strip.
method Examined cohomology ring of ordered configuration space to find lower bound.
result Lower bound for minimum number of cases in robot movement program.
Computer experiments reveal complex knots that don't simplify.
problem Understanding the dynamics of complex knots under self-repulsion.
method Computer simulations of knot theory, focusing on rational knots and tangles.
result Discovered hard unknots and complexified knots that do not reduce to simpler forms under self-repulsion.
Complex manifolds with compatible metric have a naturally defined subspace of harmonic differential forms that satisfy Serre, Hodge, and conjugation duality, as well as hard Lefschetz duality. This last property follows from a representation of sl(2,C), generalizing the well known structure on the harmonic f…
FibeRed reduces complex data dimensions while preserving topology.
problem Hard embedding of topologically complex datasets in low-dimensional Euclidean space.
method Modeling datasets with vector bundles, reducing fibers while preserving topology.
result FibeRed learns topologically faithful embeddings in lower dimensions than existing methods.
The well-known Kähler identities naturally extend to the non-integrable setting. This paper deduces several geometric and topological consequences of these extended identities for compact almost Kähler manifolds. Among these are identities of various Laplacians, generalized Hodge and Serre dualities, a generalized hard…
This paper proposes a new method for learning covers of geometric datasets to improve topological inference and visualization.
problem Improving topological inference and visualization of large-scale geometric datasets.
method Proposes a method for learning topologically-faithful covers of geometric datasets using optimization.
result Simplicial complexes obtained from learned covers outperform standard methods in terms of size and representation of large-scale topology.
In this paper, we use an aerial base station (aerial-BS) to enhance fairness in a dynamic environment with user mobility. The problem of optimally placing the aerial-BS is a non-deterministic polynomial-time hard (NP-hard) problem. Moreover, the network topology is subject to continuous changes due to the user mobility…
We investigate the computational complexity of some problems in three-dimensional topology and geometry. We show that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. Using similar ideas, we show that deciding whether a curve in a metrized PL 3-manifold bounds a surface of area…
The increasing penetration of distributed energy resources poses numerous reliability issues to the urban distribution grid. The topology estimation is a critical step to ensure the robustness of distribution grid operation. However, the bus connectivity and grid topology estimation are usually hard in distribution gri…
Paper proposes a new topology for AML analysis using Poincaré embeddings.
problem Complex money laundering schemes and regulatory constraints hinder AML analysis and information sharing.
method Proposes a new topology for AML analysis using Poincaré embeddings.
result Demonstrates improved AML analysis and information sharing through Poincaré embeddings.
In this empirical paper, we investigate how learning agents can be arranged in more efficient communication topologies for improved learning. This is an important problem because a common technique to improve speed and robustness of learning in deep reinforcement learning and many other machine learning algorithms is t…
Paper tackles optimal network compression for financial systems.
problem Optimal network compression for financial systems under shocks.
method Formulated as an NP-hard problem, studied systemic risk measures, and analyzed specific networks.
result Systemic fragility results no longer hold generally under shocks and heterogeneous networks.
Finding optimal correction of errors in generic stabilizer codes is a computationally hard problem, even for simple noise models. While this task can be simplified for codes with some structure, such as topological stabilizer codes, developing good and efficient decoders still remains a challenge. In our work, we syste…
Researchers use discrete Morse theory to improve the topology of matching complexes of complete graphs.
problem Understanding the topology of matching complexes of complete graphs, especially for small n.
method Developed gradient vector fields to simplify the computation of homology groups.
result Computed the homology groups of M7 efficiently and conjectured an optimal gradient vector field. This paper poses some basic questions about instances (hard to find) of a special problem in 3-manifold topology. "Important though the general concepts and propositions may be with the modern industrious passion for axiomatizing and generalizing has presented us...nevertheless I am convinced that the special problems …
Study shortest non-separating curves on non-orientable surfaces, proving NP-hardness and tractability.
problem Computing shortest non-separating simple closed curves on non-orientable surfaces.
method Developed tools for computing shortest curves, proving NP-hardness and tractability.
result Proved NP-hardness and fixed-parameter tractability for computing shortest orienting curves, and polynomial-time algorithm for non-orienting curves.
Paper proves hardness of learning various complex models under local pseudorandom generators.
problem Hardness of learning various complex models.
method Existence of local pseudorandom generators.
result Proves hardness of learning shallow ReLU neural networks and other models.
This work connects hardness of approximation and learning.
problem Hardness of approximation and learnability in machine learning.
method Shows a single hardness property implying both approximation and learning hardness.
result Obtains new results on hardness of approximation and learnability of specific functions.
Study on hard Legendrian unknots using normal rulings.
problem Understanding the complexity of Legendrian unknots in knot theory.
method Using normal rulings to obstruct and construct hard unknot diagrams.
result Construction of infinitely many smoothly hard max-tb unknot diagrams with bounds on minimum possible writhe.
Constructs real algebraic functions with specified preimages.
problem Reconstructing smooth functions with prescribed preimages.
method Using real algebraic functions and techniques from singularity theory and differential topology.
result Constructs examples of real algebraic functions with specified preimages.
New method finds large counterexamples by selectively exploring triangulations.
problem Finding small counterexamples in 3-manifold triangulations. method Selective enumeration of triangulations using heuristics.
result Found counterexamples to three conjectures about vertex triangulations.
Moving between 3-manifold triangulations is NP-hard
problem Moving between two triangulations of a 3-manifold
method Showing that the number of bistellar moves and sparse degree-two edge collapses is NP-hard
result First NP-hardness result concerning moves between two triangulations of a 3-manifold
Hard instances, which require a long time for a specific algorithm to solve, help (1) analyze the algorithm for accelerating it and (2) build a good benchmark for evaluating the performance of algorithms. There exist several efforts for automatic generation of hard instances. For example, evolutionary algorithms have b…
Tackles the computational hardness of HPC detection, conjecturing equivalence to PC detection.
problem Computational hardness of hypergraphic planted clique detection.
method No specific method mentioned; focuses on conjecturing equivalence.
result Equivalence of computational hardness between HPC and PC detection.
The main goal of this work is to present a detailed study of the foundations of Complex Geometry, highlighting its geometrical, topological and analytical aspects. Beginning with a preliminary material, such as the basic results on holomorphic functions in one or more variables and the definition and first examples of …
Three hard diagrams of the unknot require extra crossings to simplify.
problem Finding diagrams of the unknot that require many crossings to simplify.
method Applying previously proposed methods to construct diagrams and using computational resources to prove their hardness.
result Three hard diagrams of the unknot require at least three extra crossings.
In this article, we introduce a fixed parameter tractable algorithm for computing the Turaev-Viro invariants TV(4,q), using the dimension of the first homology group of the manifold as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parame…
Hardness proven for learning neural networks with polynomial size and Gaussian inputs.
problem Learning one hidden layer ReLU neural networks with polynomial size and Gaussian inputs.
method Based on the hardness of the Continuous Learning with Errors (CLWE) problem.
result Hardness of learning neural networks is proven under standard cryptographic assumptions.
This paper gives infinitely many examples of unknot diagrams that are hard, in the sense that the diagrams need to be made more complicated by Reidemeister moves before they can be simplified. In order to construct these diagrams, we prove theorems characterizing when the numerator of the sum of two rational tangles is…
Study categorizes knots and links as rigid or shaky based on Reidemeister moves.
problem Classifying knots and links as rigid or shaky based on adaptability to Reidemeister moves.
method Categorization of hard diagrams as rigid or shaky, investigation of rigid and shaky hard diagrams for specific knots and links.
result Every link has a rigid hard diagram, and there is an upper limit for the number of crossings in such diagrams.