Two Pin(2)-equivariant Floer homologies are shown to be equivalent.
problem Equivalence of Pin(2)-equivariant Floer homologies.
method Comparison of definitions by Lin and Manolescu.
result Pin(2)-equivariant monopole Floer homology is isomorphic to Seiberg-Witten Floer homology.
Paper solves a 4D topology problem using Pin(2)-equivariant Mahowald invariants.
problem Existence of Pin(2)-equivariant stable maps between certain representation spheres.
method Analysis of Pin(2)-equivariant Mahowald invariants and use of cell diagrams, Atiyah-Hirzebruch spectral sequence.
result Proves a '10/8+4'-Theorem and solves the 11/8-Conjecture.
In this remark, we show how the monopole Frøyshov invariant, as well as the analogues of the Involutive Heegaard Floer correction terms d,d, are related to the Pin(2)-equivariant Floer homology SWFH∗G. We show that the only interesting correction terms of a $\mathrm{Pin}…
We compute the Pin(2)-equivariant monopole Floer homology for the class of plumbed 3-manifolds with at most one "bad" vertex (in the sense of Ozsvath and Szabo). We show that for these manifolds, the Pin(2)-equivariant monopole Floer homology can be calculated in terms of the Heegaard Floer/monopole Floer lattice compl…
Using Seiberg-Witten Floer spectrum and Pin(2)-equivariant KO-theory, we prove new Furuta-type inequalities on the intersection forms of spin cobordisms between homology 3-spheres. As an application, we give explicit constrains on the intersection forms of spin 4-manifolds bounded by Brieskorn spheres $\pmΣ(2,3,6k\…
We compute the Pin(2)-equivariant Seiberg-Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu's conjecture that β=−μˉ for Seifert integral homology three-spheres. We show that the Manolescu invari…
We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov's correction term in this setting is an integer-valued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that there…
New equations use Pin(2) symmetry to study spinor and connection solutions.
problem Study of spinor and connection solutions on 4-manifolds.
method Define and analyze Seiberg-Witten-like equations with Rarita-Schwinger operators.
result Moduli space of solutions is non-compact and non-empty under certain conditions.
Let M be a manifold homotopy equivalent to the complex projective space $\C P^m$. Petrie conjectured that M has standard total Pontrjagin class if M admits a non-trivial action by S1. We prove the conjecture for m<12 under the assumption that the action extends to a nice Pin(2)-action with fixed point. The…
Exotic diffeomorphism survives stabilizations on a contractible 4-manifold.
problem Finding exotic diffeomorphisms on contractible 4-manifolds.
method Developed a Pin(2) × Z_2-equivariant refinement for computing Seiberg-Witten Floer homotopy types.
result Constructed an exotic diffeomorphism that survives two stabilizations.
Authors prove existence of exotic surfaces and invariants not detecting self-diffeomorphisms.
problem Existence and properties of exotic surfaces and diffeomorphisms.
method Vanishing theorem of family Bauer--Furuta invariant for diffeomorphisms on spin 4-manifolds.
result Family Bauer--Furuta invariants do not detect exotic self-diffeomorphisms on S4 or S2imesS2. New invariant shows Dehn twist on connected sum of homology tori is not isotopic to identity.
problem Determining when Dehn twists on connected sums of homology tori are isotopic to identity.
method Generalized Pin(2)-equivariant family Bauer-Furuta invariant to nonsimply connected manifolds and constructed a refinement.
result Dehn twist on X1#X2 is not isotopic to identity if determinants r1,r2 are odd. New example of non-smooth isotopy after stabilization in 4-manifolds.
problem Non-smooth isotopy of Dehn twist after stabilization.
method Pin(2)-equivariant Bauer-Furuta invariant.
result Dehn twist is not smoothly isotopic even after stabilization.
This study analyzes a non-orientable spacetime model in 1+1D quantum gravity.
problem Analyzing a non-orientable spacetime model in 1+1D quantum gravity.
method Formulated a Jackiw-Teitelboim gravity toy model on the Möbius band, computed Stiefel-Whitney classes, and analyzed the Dirac operator.
result Half-integer momentum quantization, spectral symmetry, vanishing mod-2 index, and η_D(0) = 0 follow.
The study of triangulations on manifolds is closely related to understanding the three-dimensional homology cobordism group. We review here what is known about this group, with an emphasis on the local equivalence methods coming from Pin(2)- equivariant Seiberg-Witten Floer spectra and involutive Heegaard Floer homolog…
Researchers compute mod 2 Seiberg-Witten invariants for spin structures and families.
problem Computing mod 2 Seiberg-Witten invariants for spin structures and families.
method Using Pin(2)-symmetry and localisation in equivariant cohomology, the researchers enhance and compute the invariants.
result The mod 2 Seiberg-Witten invariants are computed for spin structures and families, confirming the simple type conjecture mod 2.
We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. We then describe Floer stable homotopy types, the related Pin(2)-equivariant Seiberg-Witten Flo…
Formula connects knot invariant to Lefschetz number, proving special case for Seifert solids.
problem Establishing a formula relating Miyazawa's knot invariant to Lefschetz number.
method Using monopole Floer homology with Pin(2)-equivariant perturbations and integer coefficients.
result Proves ∣deg∣=1 for certain 2-knots in S4 with specific Seifert solid properties. Manolescu correction terms are numerical invariants of homology three-spheres arising from Pin(2)-equivariant Seiberg-Witten theory that contain information about homology cobordism. We discuss several constraints on these invariants for homology spheres obtained by Dehn surgery on a knot in the three-sphere…
We prove Furuta-type bounds for the intersection forms of spin cobordisms between homology 3-spheres. The bounds are in terms of a new numerical invariant of homology spheres, obtained from Pin(2)-equivariant Seiberg-Witten Floer K-theory. In the process we introduce the notion of a Floer K_G-split homology sphere; thi…
Steerable E(3) Graph Neural Networks incorporate geometric and physical covariant information.
problem Incorporating covariant information like position, force, velocity, or spin in graph neural networks.
method Steerable E(3) Equivariant Graph Neural Networks (SEGNNs) that use steerable MLPs to incorporate geometric and physical covariant information.
result SEGNNs improve upon classic linear point convolutions and recent equivariant graph networks that send invariant messages.
Unified Pin-SVM improves accuracy over existing Pin-SVM model.
problem Difficulty in Pin-SVM model for −1≤τ<0. method Unified Pin-SVM model that solves a QPP for −1≤τ≤1. result Significant improvement in accuracy over existing Pin-SVM model.
In the present work we generalize the construction of monopole Floer homology due to Kronheimer and Mrowka to the case of a gradient flow with Morse-Bott singularities. Focusing then on the special case of a three-manifold equipped with a spinc structure isomorphic to its conjugate, we define the counterpart in this…
The paper studies extPin±-structures on non-oriented 4-manifolds via Lefschetz fibrations.
problem Understanding extPin±-structures on non-oriented 4-manifolds and Lefschetz fibrations. method Extending work on orientable settings, the paper uses Lefschetz fibrations to analyze extPin±-structures on non-orientable 4-manifolds and vector bundles. result The paper provides existence results of extPin+ and extPin−-structures on closed non-orientable 4-manifolds and Lefschetz fibrations over the 2-sphere. Classifies stability of flat-core p-elasticae pinned at boundaries.
problem Stability of flat-core p-elasticae under pinned boundary conditions. method Classification based on previous work for all p∈(1,∞) and d≥2. result Completes the classification of stable pinned p-elasticae in Rd. Study pin manifolds using Clifford linear Dirac operator and KO-theory.
problem Index theory on Pin manifolds.
method Clifford linear Dirac operator and differential KO-theory.
result Systematic treatment of index theory on Pin manifolds.
Established a stable cohomotopy refinement for Pin(2) monopole invariants.
problem Refining monopole invariants for Pin(2) structures.
method Adapted Bauer-Furuta's Seiberg-Witten refinement method to Pin(2) monopole invariants.
result Connected sum formula for the refined invariants.
The Pinned AUC metric hides unintended bias when class distributions vary.
problem Unintended bias in classification models.
method Examines the Pinned AUC metric and its limitations.
result Pinned AUC can obscure different types of unintended bias.
Study cyclic group actions on spin 4-manifolds with boundary using Seiberg-Witten theory.
problem Understanding cyclic group actions on spin 4-manifolds with boundary.
method Using Seiberg-Witten equations and lattice constructions, define equivariant refinements of the invariant κ.
result Equivariant relative 10/8-ths type inequalities for spin equivariant cobordisms between rational homology spheres.
The paper explores properties of Pin structures on surfaces and their cobordism.
problem Understanding Pin structures on compact surfaces and their cobordism.
method Analyzes Pin structures on surfaces and their cobordism, providing a construction for dimensions up to two.
result Pin structures on surfaces differ by a diffeomorphism if and only if they are cobordant, but this does not extend to higher dimensions.
Study on a pinning model with random walk increments, showing convergence to a critical disordered pinning measure.
problem Understanding the critical behavior of a disordered pinning model.
method Analyzing a disordered pinning model induced by a random walk with specific moment conditions, showing convergence to a limiting measure.
result Convergence of point-to-point partition functions to the critical disordered pinning measure in the critical window.
The pinning ideal of multiloops is shown to be NP-complete.
problem Computing the pinning number of multiloops is NP-complete.
method Adapted algorithms from curve intersection and taut loop characterization to prove NP-completeness.
result The pinning number problem is NP-complete for multiloops in the sphere.
Monograph compares curve signs in intrinsic and Pin-dependent definitions.
problem Comparing curve signs in intrinsic and Pin-dependent definitions.
method Intrinsic perspective on Spin- and Pin-structures, detailed constructions of orientations, explicit comparison.
result Explicit comparison between curve signs in intrinsic and Pin-dependent definitions.
Complex pinning problem simplified for simple multiloops.
problem Complexity of pinning simple multiloops in fixed surfaces.
method Analysis of pinning problem in simple multiloops.
result Pinning problem in simple multiloops is P when 3 strands or fewer, NP-complete when 20 strands or more.
We give definitions of moduli spaces of framed, r-Spin and Pin surfaces. We apply earlier work of the author to show that each of these moduli spaces exhibits homological stability, and we identify the stable integral homology with that of certain infinite loop spaces in each case. We further show that these moduli spa…
We give necessary and sufficient conditions for the existence of pin+, pin- and spin structures on Riemannian manifolds with holonomy group Z2k. For any n>3 (resp. n>5) we give examples of pairs of compact manifolds (resp. compact orientable manifolds) M_1, M_2, non homeomorphic to each other, that are Laplace isos…
Paper extends Brownian bridge with random length and pinning point for financial modeling.
problem Modeling financial information flow with uncertainty in pinning point.
method Introduced an extension of Brownian bridge with random length and pinning point, derived formulae for conditional expectations.
result The extended Brownian bridge fails to be Markovian if pinning point distribution is absolutely continuous.
The paper finds curves minimizing elastic energy pinned at endpoints.
problem Finding curves that minimize elastic energy with fixed endpoints.
method Applying the shooting method to identify and classify critical points.
result Critical points consist of wavelike elasticae, and minimizers have no loops or interior inflection points.
The topological condition for the existence of a pinc structure on the product of two Riemannian manifolds is derived and applied to construct examples of manifolds having the weaker Lipschitz structure, but no pinc structure. An example of a five-dimensional manifold with this property is given; it is pointed ou…
Spaces of positive and tropical points described as Teichmüller and lamination spaces with pinnings.
problem Describing spaces of positive and tropical points in moduli space with pinnings.
method Topological description of Teichmüller and lamination spaces with pinnings.
result Formulae relating functions on Teichmüller space with pinnings and topological description of tropicalized amalgamation map.
PIN models feature interactions using a neural network that mimics decision trees.
problem Modeling feature interactions in tabular data for predictive modeling.
method Tree-like Pairwise Interaction Network (PIN) architecture that captures pairwise feature interactions through a shared feed-forward neural network.
result PIN outperforms traditional and modern neural networks benchmarks in predictive accuracy.
A correspondence between different Pin-type structures on a compact surface and quadratic (linear) forms on its homology is constructed. Addition of structures is defined and expressed in terms of these quadratic forms.
In previous work, we introduced a natural A∞-structure on the Pin(2)-monopole Floer chain complex of a closed, oriented three-manifold Y, and showed that it is non-formal in the simplest case in which Y is the three-sphere S3. In this paper, we provide explicit descriptions of seve…
Classifies pinned p-elasticae and finds unique optimality exponents.
problem Classifying and understanding p-elasticae under pinned boundary conditions. method Classification and analysis of p-elasticae, proving uniqueness and existence. result Discovery of a unique exponent p≃1.5728 for full optimality. We study the behavior of Pin(2)-monopole Floer homology under connected sums. After constructing a (partially defined) A∞-module structure on the Pin(2)-monopole Floer chain complex of a three manifold (in the spirit of Baldwin and Bloom's monopole category), we identify up to …
The study explores psc-metrics on non-spin manifolds with pin^\pm or spin^c structures.
problem Existence of psc-metrics on non-spin manifolds with specific structures.
method Analysis of Dirac operators and index theory on pin^\pm or spin^c manifolds.
result The index of a Dirac operator controls the existence of psc-metrics on certain manifolds.
Research provides obstructions to Stein fillings of certain rational homology spheres.
problem Obstructing Stein fillings of specific rational homology spheres.
method Uses Pin(2)-monopole Floer homology.
result Provides obstructions to the intersection forms of Stein fillings.
The paper studies hyperbolic three-manifolds and their geometric constraints.
problem Understanding the interaction between hyperbolic geometry and homology cobordism.
method Derived explicit bounds on relative grading and invariants in monopole Floer homology.
result Explicit bounds on numerical invariants and subgroup structure of homology cobordism.