Paper introduces moves to simplify framed flow categories.
problem Simplifying framed flow categories for easier study.
method Inspired by Morse-Smale moves, introduces moves to change framed flow categories without altering their stable homotopy type.
result Finite sequence of moves can connect two framed flow categories representing the same stable homotopy type.
We pursue the analogy of a framed flow category with the flow data of a Morse function. In classical Morse theory, Morse functions can sometimes be locally altered and simplified by the Morse moves. These moves include the Whitney trick which removes two oppositely framed flowlines between critical points of adjacent i…
A calculus modifies flow categories without changing their homotopy type.
problem Modifying flow categories without altering their homotopy type.
method A calculus of moves to modify framed flow categories.
result Two flow categories with stable homotopy type give move equivalent categories.
Researchers find a Steenrod square for link Floer homology.
problem Computing the second Steenrod square for link Floer homology.
method Explicitly framing moduli spaces and constructing a framed 1-flow category.
result An algorithm for computing the second Steenrod square for all grid homology versions.
Efficiently learns object representations for FPS games.
problem Learning to play FPS games with limited training data.
method Detects salient segments, clusters them, and uses their importance for classification.
result Improves performance of DRQN by focusing on relevant object categories.
Generators and relations found for foam categories.
problem Understanding categories of framed tangled webs and foams.
method Multijet transversality techniques.
result Presentation by generators and relations of categories.
We construct a pairing, which we call factorization homology, between framed manifolds and higher categories. The essential geometric notion is that of a vari-framing of a stratified manifold, which is a framing on each stratum together with a coherent system of compatibilities of framings along links between strata. O…
A new class of 3-manifold invariants is constructed from representations of the category of framed tangles.
We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with certain algebraic properties determine topological invariants. We prove that fusion categories of nonzero global dimension are 3-dualizable, and therefore provide 3-di…
Constructs a spectrum for knot Floer homology without holomorphic geometry.
problem Computing knot Floer homology without using holomorphic geometry.
method Combinatorial definition and inductive construction of models for moduli spaces.
result Conjectures that the filtered homotopy type of the spectrum is an invariant of the knot.
Extracts object-centric frames from unlabeled images.
problem Extracting abstract models of 3D objects from visual measurements.
method Viewpoint factorization and dense equivariant labelling neural network.
result Extracts dense object-centric coordinate frames invariant to deformations.
Floer homotopy theory connects symplectic geometry with homotopy theory.
problem Is there a spectrum whose homology matches Floer homology?
method A framed flow category is used to construct a spectrum.
result New applications and deeper understanding of Floer homotopy theory.
New spin frame transformations affect Dirac equations without preserving metric structures.
problem Extending spin structures to spin manifolds with fixed signature.
method Defining spin frames and studying their effects on connections and Dirac equations.
result New transformations affect Dirac equations more generally than usual spin transformations.
New progress on frame flow ergodicity for nearly pinched manifolds.
problem Ergodicity of frame flow on negatively-curved manifolds.
method New ideas leading to ergodicity for nearly 0.25-pinched manifolds.
result Achieved progress towards Brin's conjecture.
New flow generates surfaces with constant curvature.
problem Creating surfaces with specific curvature properties.
method Framed curvature flow, analyzing trajectory surfaces.
result Trajectory surfaces of constant mean or Gaussian curvature.
The study shows ergodicity of unitary frame flows on Kähler manifolds with specific curvature conditions.
problem Ergodicity of unitary frame flows on Kähler manifolds with negative holomorphic sectional curvature.
method Analysis of the unitary frame flow on the principal U(m)-bundle of unitary frames.
result For even-dimensional Kähler manifolds with negative λ(m)-pinched holomorphic sectional curvature, the unitary frame flow is ergodic and mixing.
Proves spectral gap for frame flows on hyperbolic manifolds.
problem Exponential mixing of frame flows on hyperbolic manifolds.
method Resolvent estimates and Borel-Weil calculus.
result Optimal essential spectral gap property for the generator.
Study connects spectral properties to frame flows on curved manifolds.
problem Spectral properties and frame flows on curved manifolds.
method Link between spectral properties, frame flows, and polynomial maps between spheres.
result Ergodicity of frame flows on low-rank bundles.
Study higher arity self-distributive operations and their cohomology.
problem Understanding cohomology groups of higher arity operations.
method Introduced mutually distributive n-ary operations and defined a cohomology theory.
result Geometric interpretation of cohomology in terms of framed links.
New flow category for contact manifolds from Reeb orbits.
problem No direct problem stated; focuses on new construction.
method Adapting Kuranishi charts to contact setting, associating flow category based on Reeb orbits and pseudo-holomorphic buildings.
result Lifts contact homology and associates flow bimodule to exact symplectic cobordisms.
We consider the question: "If the zero-framed surgeries on two oriented knots in the 3-sphere are integral homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?" We show that this question has a negative answer in the smooth category, even for topologically sl…
The paper studies the ergodicity of frame flow on even-dimensional manifolds.
problem Understanding the ergodicity of frame flow on even-dimensional manifolds.
method Analyzing pinching conditions to determine ergodicity.
result The frame flow is ergodic under specific pinching conditions for even-dimensional manifolds.
We construct modular categories from Hecke algebras at roots of unity. For a special choice of the framing parameter, we recover the Reshetikhin-Turaev invariants of closed 3-manifolds constructed from the quantum groups U_q sl(N) by Reshetikhin-Turaev and Turaev-Wenzl, and from skein theory by Yokota. We then discuss …
We provide combinatorial realizations, according to the usual objects/moves scheme, of the following three topological categories: (1) pairs (M,v) where M is a 3-manifold (up to diffeomorphism) and v is a (non-singular vector) field, up to homotopy; here possibly the boundary of M is non-empty and v may be tangent to t…
Floer theory uses categories to construct 3-manifold invariants.
problem Constructing 3-manifold invariants via decomposition and functors.
method Floer field theory via decomposition in bordism category and functor to symplectic category.
result Floer theory has potential 4-dimensional extensions.
The paper studies posets from decompositions in symmetric monoidal categories.
problem Understanding posets from decompositions in symmetric monoidal categories.
method Defining decompositions and partial decompositions, complexes of frames, partial bases, and ordered versions.
result Unified approach to combinatorics and homotopy type of posets and complexes.
Quantum invariant derived from ternary cohomology of self-distributive structures.
problem Defining and proving a quantum invariant from ternary cohomology.
method Constructing a ribbon category from a TSD set, showing it coincides with the cocycle invariant.
result The ribbon cocycle invariant is a quantum invariant.
A new category generates 1D tangle invariants.
problem Developing a new category for 1D tangles.
method Proving a new (∞,1)-category has universal mapping property. result The new category generates link invariants.
This paper proves exponential mixing for frame flows on hyperbolic manifolds with cusps.
problem Establishing exponential mixing for frame flows on geometrically finite hyperbolic manifolds with cusps.
method Symbolic coding of geodesic flow, Dolgopyat's method, large deviation property, combinatorics of cusp excursions, renewal theorem.
result Frame flows for geometrically finite hyperbolic manifolds of arbitrary dimensions are exponentially mixing.
Generalizes Morse theory to n-categories using critical points and moduli spaces.
problem Constructing n-categories from Morse theory.
method Extending Cohen & Jones & Segal's flow category to n-categories by Morse theory on critical points and moduli spaces.
result The resulting structure is an 'almost strict' n-category.
Proposes a new category of bundles for Lagrangian reduction in field theory.
problem Lagrangian reduction in field theory.
method Introduces a category of bundles to perform Lagrangian reduction by stages in covariant Field Theory.
result Formulates the Noether theorem in this new theoretical framework.
New stability conditions identified from quadratic differentials on surfaces.
problem Identifying stability conditions from quadratic differentials.
method Comparison of exchange graphs from tilting hearts and flipping mixed angulations.
result Spaces of stability conditions identified with moduli spaces of quadratic differentials.
Generalizes Hasimoto transformation to arbitrary flows on space curves.
problem Modeling and analyzing wave motions on space curves.
method Develops a mapping between curve evolution and scalar equations, transforming general vector fields in the Frenet frame.
result Binormal flows are generally length preserving but bending energy is fragile.
New category of virtual tangles connects to existing theories.
problem Characterize virtual tangles and their invariants.
method Define and study a new category vT of virtual tangles, showing it's universal and equivalent to virtual tangles. result Virtual tangles and their invariants extend to framed oriented virtual links.
Extends Kanai's result to higher dimensions for negatively curved manifolds.
problem Proving ergodic frame flows on negatively curved manifolds.
method Analyzing frame flows over negatively curved manifolds with specific curvature conditions.
result Proves that under certain conditions, the manifold is homothetic to a real hyperbolic manifold.
New 2-representations link spectral enhancements in link homology.
problem Spectral enhancements in link homology.
method Skew Howe duality, frames, and multifunctors.
result Spectral 2-representations of categorified quantum groups.
Frame flows on certain symmetric spaces mix exponentially.
problem Exponential mixing of frame flows in convex cocompact locally symmetric spaces.
method Generalized local non-integrability and non-concentration properties to apply Dolgopyat's method.
result Exponential mixing of frame flows proved for convex cocompact locally symmetric spaces.
The paper proves exponential mixing for hyperbolic manifolds, with applications to geodesic holonomy.
problem Establishing exponential mixing for frame flows on hyperbolic manifolds.
method Using spectral bounds on transfer operators twisted by holonomy, building on Dolgopyat's method.
result Exponential mixing of frame flows for convex cocompact hyperbolic manifolds.
Harmonic maps link Teichmüller spaces to framed representations.
problem Connecting Teichmüller spaces with framed representations.
method Uses harmonic map heat flow to find unique maps.
result Unique harmonic maps exist under specified conditions.
New gauge invariants from framed 3-manifolds match Hopf algebra indicators.
problem Computing indicators for Hopf algebras.
method Using Kuperberg invariants from framed 3-manifolds.
result Kuperberg invariants match higher Frobenius-Schur indicators of Hopf algebras.
Extends Hard Lefschetz Property to isometric flows and shows equivalence.
problem Studying the Hard Lefschetz Property in isometric flows.
method Extending the Hard Lefschetz Property to isometric flows and proving equivalence.
result Equivalence of transverse and non-transverse Hard Lefschetz Properties in isometric flows.
This paper introduces quantum invariants for 3-alterfolds and proves their consistency with topological moves.
problem Quantum invariants for 3-alterfolds and their consistency with topological moves.
method Introduction of 3-alterfolds with embedded separating surfaces and spherical fusion categories.
result Quantum invariants of 3-alterfolds are consistent with topological moves and generalize invariants of 3-manifolds containing framed links.
Paper revisits FRAME model, explaining instability and proposing a new metric.
problem Unstable training energy in FRAME model.
method Theoretical analysis using particle physics, proposing a new Wasserstein distance.
result Proposed Wasserstein distance stabilizes energy dissipation and maintains statistical consistency.
Generative model uses neural flows for next-frame video generation conditioned on labels.
problem Blurriness and instability in video generation models.
method Proposes using Glow, a neural flow model, for next-frame video generation conditioned on labels.
result Glow model produces clearer and more stable videos compared to GANs.
New algorithm calculates Steenrod squares in Khovanov cohomology.
problem Computing Steenrod squares in Khovanov cohomology.
method Flow category simplification techniques to calculate second Steenrod square and Bockstein homomorphisms.
result Observation of new homotopy types and evidence against CP2 summands. Study of Anosov flows using microlocal analysis for ergodicity and mixing properties.
problem Ergodicity and mixing properties of Anosov flows and their isometric extensions.
method Microlocal analysis of Pollicott-Ruelle resonances to study isometric extensions of Anosov flows.
result Ergodicity of frame flow on negatively-curved Riemannian manifolds under specific curvature assumptions.
Geodesic flows between hypersurfaces in Euclidean spaces using Lorentzian geometry.
problem Interpolation between hypersurfaces in Euclidean spaces.
method Lorentzian geodesic flow between tangent spaces of hypersurfaces.
result Geodesic flow is preserved by rigid transformations and homotheties.
Defines a new link invariant for type D webs.
problem Invariants for framed unoriented links in type D.
method Positive state sum for type D webs, derived from quantum so(2N) modules.
result Relates to Reshetikhin-Turaev's invariant.