Paper proves Hamilton-Tian conjecture using partial C0-estimate.
problem Existence of Kähler-Einstein metrics on Fano manifolds.
method Using Liu-Székelyhidi's partial C0-estimate for polarized Kähler metrics with Ricci bounded below.
result Proves Hamilton-Tian conjecture for Kähler-Ricci flow.
Smooth 2-tori in R^4 can be approximated by polyhedral Lagrangian or isotropic tori.
problem Approximating smooth 2-tori in high-dimensional spaces.
method Polyhedral approximation using Lagrangian and isotropic tori.
result Smooth 2-tori can be approximated by polyhedral Lagrangian or isotropic tori in C0 or C1 sense.
Study shows Hamiltonian diffeomorphisms form a connected component in C0-topology for most symplectic rational surfaces.
problem Understanding the C0-topology of symplectic diffeomorphisms on rational surfaces. method Combining techniques from symplectic mapping class groups and C0-symplectic topology, establishing C0-distance estimates. result Hamiltonian diffeomorphisms form a connected component in C0-topology for all but a few exceptions on rational surfaces. The paper examines robustness of topological entropy in geodesic flows.
problem Entropy robustness in geodesic flows under C0 perturbations. method Study of topological entropy on Riemannian metrics with C0 topology. result Metrics with contractible closed geodesics have robust entropy.
Proves some flat spacetimes can't be extended smoothly.
problem Proving smooth extension of certain FLRW spacetimes.
method Utilizing Sbierski's C0-inextendibility techniques. result Proves C0-inextendibility of spatially flat FLRW spacetimes. New rigidity result for metrics with positive scalar curvature and specific decay.
problem Understanding metrics with positive scalar curvature and C0 decay. method Analyzing metrics with non-negative scalar curvature and C0 decay properties. result Metrics with specific decay properties must be flat.
We consider the problem of detecting a tight community in a sparse random network. This is formalized as testing for the existence of a dense random subgraph in a random graph. Under the null hypothesis, the graph is a realization of an Erdös-Rényi graph on N vertices and with connection probability p0; under the …
A spacetime can be embedded in an enveloping space with all its extensions.
problem Existence and uniqueness of C0-maximal extensions in globally hyperbolic conformally flat spacetimes.
method Proving conformal embedding into an enveloping space containing all extensions.
result Existence and uniqueness of C0-maximal extensions proven.
In this paper we study the stochastic evolution equation (1.1) in martingale-type 2 Banach spaces (with the linear part of the drift being only a generator of a C0-semigroup). We prove the existence and the uniqueness of solutions to this equation. We apply the abstract results to the Heath-Jarrow-Morton-Musiela (HJMM)…
Proves existence and uniqueness of solutions to a quaternionic Monge-Ampère equation.
problem Solving the quaternionic Monge-Ampère equation for (n−1)-quaternionic plurisubharmonic functions on a hyperKähler manifold. method Proves existence and uniqueness of solutions using a Cherrier-type inequality and C1 and C2 estimates. result Obtains smooth solutions to the quaternionic Monge-Ampère equation.
Study shows a mass quantity for C0 metrics that agrees with ADM mass.
problem Understanding ADM mass for C0 metrics and its behavior under Ricci-DeTurck flow. method Developed a C0 mass quantity and analyzed its behavior under Ricci-DeTurck flow. result The C0 mass at infinity is independent of coordinate charts and has controlled distortion under Ricci-DeTurck flow. Simplified proof and new C0 estimate for Kähler-Einstein metrics.
problem Existence of Kähler-Einstein metrics on Calabi-Yau manifolds.
method Alternative C0 a priori estimate for the Monge-Ampère equation. result Established a new uniform bound for the solution of the Monge-Ampère equation.
The paper proves the existence of infinitely many nodal solutions to a Paneitz-type equation.
problem Proving the existence of nodal solutions to a specific type of partial differential equation.
method First, a C0−estimate for positive f-invariant solutions is proven. Then, the existence of mountain pass solutions with arbitrarily large energy is established. result The existence of infinitely many nodal solutions to the equation Δ2u−αΔu+βu=uq is proven. We propose a simple change to existing neural network structures for better defending against gradient-based adversarial attacks. Instead of using popular activation functions (such as ReLU), we advocate the use of k-Winners-Take-All (k-WTA) activation, a C0 discontinuous function that purposely invalidates the neural …
Box Thirding identifies the best arm efficiently under limited samples.
problem Efficiently identifying the best arm with limited sampling.
method Iterative ternary comparison of arms, discarding the weakest and exploring the best.
result Achieves comparable performance to Successive Halving with less predefined parameters.
Study shows nonextendibility of warped spacelike singularities in specific spacetimes.
problem Nonextendibility of warped spacelike singularities in specific spacetimes.
method Establishes a local obstruction through integrability conditions and radial compression.
result Imply C0-inextendibility for the one-horizon Birmingham-Kottler family. The paper develops algorithms and topological invariants for distinguishing dynamic systems.
problem Distinguishing the topological type of surfaces and functions in dynamic systems.
method Construction of algorithms and topological invariants using discrete topological structures.
result The development of discrete topological structures for topological equivalence of dynamic systems.
Study of digital topology concepts like hyperspaces and function graphs.
problem Adapting classical topology concepts to digital topology.
method Define digital hyperspaces and function graphs, study their properties.
result Some relationships and graphical properties of digital hyperspaces and function graphs.
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.
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.
Novel theory combines combinatorial and topological elements.
problem Understanding combinatorial phenomena at the intersection of topology.
method Synthesizes combinatorial and topological approaches with a new framing concept.
result Framed combinatorial spaces exhibit better behavior than classical spaces.
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.
New proof for some knots being topologically slice.
problem Understanding which knots are topologically slice.
method Equivariant topological slice disks for strongly negative amphichiral knots.
result Strongly negative amphichiral knots with trivial Alexander polynomial are equivariantly topologically slice.
Constructs algorithms to recognize and classify 2D surfaces.
problem Recognizing and classifying 2D surfaces in dynamic systems.
method Discrete topological structures and algorithms for simplicial and CW-complexes.
result Determines the topological type of 2-manifolds.
Paper constructs continuous families of topological Morse functions.
problem Existence and deformability of topological Morse functions.
method Simple construction of continuous families of topological Morse functions.
result Gives a construction of continuous families of topological Morse functions.
Novel M-theory approach classifies topological phases of matter.
problem Classifying and understanding topological phases of matter.
method Establishing a correspondence between (2+1)d topological field theories and non-hyperbolic 3-manifolds, identifying topological phases from internal wrapped 3-manifolds.
result Paves a new route toward the classification of topological phases of matter, including fermionic and non-unitary phases.
New smoothing techniques for topological surfaces in 4-manifolds.
problem Topological isotopy of surfaces in smooth 4-manifolds.
method Combining Quinn's smoothing theory with Gabai's light bulb theorem and other developments.
result Proves topological = smooth results for certain disks and spheres.
Diffeological spaces are generalizations of smooth manifolds which include singular spaces and function spaces. For each diffeological space, Iglesias-Zemmour introduced a natural topology called the D-topology. However, the D-topology has not yet been studied seriously in the existing literature. In this paper, we…
Critiques incorrect fixed point assertions in digital topology.
problem Incorrect or incorrectly proven fixed point assertions in digital topology.
method Critical review of existing assertions.
result Identifies and critiques incorrect fixed point assertions.
We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack modules and principal rack bundles. Central extensions of topological racks are then…
Minimal topology on surface homeomorphisms proven.
problem Proving the compact-open topology is minimal for surface homeomorphisms.
method Combining Hausdorff group topology properties and automatic continuity results.
result Compact-open topology is unique Hausdorff separable group topology on surface homeomorphisms.
Study topological constraints for minimal hyperbolic surface laminations.
problem Understanding which topological surfaces can be leaves in minimal hyperbolic surface laminations.
method Analyzes topological obstructions and embeddings of Cantor tree-like surfaces.
result All possible topological types of leaves can be simultaneously embedded in a lamination.
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.
Proposes TSBP for matching topological signal distributions.
problem Matching signal distributions on topological domains.
method Topological Schrödinger Bridge (TSBP) with linear topology-aware stochastic dynamics.
result Derives closed-form topological SB (TSB) for Gaussian boundary distributions.
A groupoid is a small category in which each morphism has an inverse. A topological groupoid is a groupoid in which both sets of objects and morphisms have topologies such that all groupoid structure maps are continuous. The notion of monodromy groupoid of a topological groupoid generalises those of fundamental groupoi…
Unified framework for T-duality in both trivial and non-trivial topologies.
problem Unified description of T-duality for metrics and B-fields in non-trivial topology.
method Developed a new unifying framework for T-duality.
result Unified description of T-duality for metrics and B-fields in non-trivial topology.
Paper introduces a new topological loss for better convergence.
problem Optimizing topological losses for model's desired topological behavior.
method Introduces a new regularized topology-aware loss function.
result Guarantees efficient optimization of the new loss function.
Survey on hyperplane arrangements and their topology.
problem Topology of hyperplane arrangements.
method Focus on the relationship between topology and real structure.
result Relationship between topology and real structure of hyperplane arrangements.
I review few conceptual steps in analytic description of topological interactions, which constitute the basis of a new interdisciplinary branch in mathematical physics, "Statistical Topology", emerged at the edge of topology and statistical physics of fluctuating non-phantom rope-like objects. This new branch is called…
Lectures on topological field theories and differential cohomology.
problem Exploring topological field theories and their connections to differential cohomology.
method Introduction to topological field theory and generalized Abelian gauge theories.
result Explains the relationship between topological field theories and differential cohomology.
A novel method extracts topological features from word embeddings for text classification.
problem High dimensional and noisy text representations in natural language processing.
method Persistent homology for topological data analysis on word embeddings.
result Topological features outperform conventional text mining features on long textual documents.
Quantum physics model uses knot theory for fragile topology.
problem Modeling quantum physics' fragile topology.
method Knot theoretic algorithm.
result Quantum physics' fragile topology modeled.
Study topological hyperbolicity of moduli spaces of elliptic surfaces.
problem Characterize the largeness of the topological fundamental group of complex varieties.
method Introduce topological hyperbolicity and provide supporting evidence for moduli spaces of elliptic surfaces.
result Establish a weak form of topological hyperbolicity for moduli spaces of elliptic surfaces of Kodaira dimension one.
Topology aids in solving machine learning classification problems.
problem Machine learning classification problems.
method Classical topology applied to neural networks.
result Topology guides neural network architecture and training.
The paper explores the topology of polygonal meshes and their properties.
problem Understanding the topological properties of polygonal meshes.
method Overview of topological concepts, definitions of intrinsic and extrinsic topology, proofs of Euler and Euler-Poincaré formulas, and discussion on cutting meshes.
result Detailed understanding and definitions of polygonal mesh topology, including intrinsic and extrinsic properties.
New method constructs Stein surfaces using topological isotopy.
problem Creating Stein surfaces in complex surfaces.
method Combining Freedman's topology with Eliashberg's holomorphic theory.
result Every tame 2-complex can be isotoped to a Stein surface.
Topology-GS improves 3D GS for better structural and feature integrity.
problem Compromised pixel-level and feature-level integrity in 3D GS.
method Incorporates Local Persistent Voronoi Interpolation (LPVI) and PersLoss based on persistent homology.
result Topology-GS outperforms existing methods in PSNR, SSIM, and LPIPS metrics.
Novel tRSA combines geometry and topology for brain and model analysis.
problem Traditional RSA overlooks topological information in neural representations.
method Topological RSA (tRSA) using nonlinear monotonic transforms.
result Robust model comparisons and novel insights into neural computation.