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. 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.
Mod 2 index theorem for non-orientable pin^- manifolds.
problem Analyzing real vector bundles over non-orientable pin^- manifolds.
method Using twisted Dirac operators and KO-theory.
result Extends Atiyah-Singer index theorem to non-orientable manifolds.
Study real structures and Pin−(2)-monopoles in symplectic and Kähler manifolds.
problem Investigate Pin−(2)-monopole invariants in real symplectic and Kähler manifolds. method Prove nonvanishing theorem and give Kobayashi-Hitchin type correspondence.
result Analogous nonvanishing theorem for real symplectic 4-manifolds.
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…
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…
Computes Pin(2)-equivariant monopole Floer homology for certain plumbed 3-manifolds.
problem Computing Pin(2)-equivariant monopole Floer homology for specific 3-manifolds.
method Uses Heegaard Floer/monopole Floer lattice complex and ranks of monopole Floer homology groups.
result Pin(2)-equivariant monopole Floer homology can be determined by ranks of monopole Floer homology groups.
Study non-formality in Pin(2)-monopole Floer homology.
problem Understanding non-formality in Pin(2)-monopole Floer homology.
method Introduced a natural A-infinity structure and computed Massey products.
result Explicit descriptions and computations of Pin(2)-monopole Floer homology.
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.
New method computes Yamabe invariants for specific 4D manifolds.
problem Computing Yamabe invariants for new 4D manifolds.
method Using twisted Seiberg-Witten equations, Pin(2)-monopole equations.
result Computed Yamabe invariants for a new class of 4D manifolds.
New methods prove weak homotopy inequivalence for manifold symmetries.
problem Understanding symmetries of smooth 4-manifolds.
method Using Pin(-2)-monopole equations.
result New examples of 4-manifolds where symmetries are not fully represented.
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.
Proves exact triangle for Pin(2)-monopole Floer homology.
problem Existence of exact triangle in Pin(2)-monopole Floer homology.
method Using specific Dehn surgeries and elementary cobordism.
result Describes invariants of homology spheres.
We introduce a diffeomorphism invariant of 4-manifolds, the Pin−(2)-monopole invariant, defined by using the Pin−(2)-monopole equations. We compute the invariants of several 4-manifolds, and prove gluing formulae. By using the invariants, we construct exotic smooth structures on the connecte…
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.
Study of Pin(2)-monopole Floer homology under connected sums.
problem Understanding Pin(2)-monopole Floer homology behavior under connected sums.
method Constructing a partially defined A-infinity module structure and using Eilenberg-Moore spectral sequences.
result Identify the Floer chain complex of a connected sum via an A-infinity tensor product of modules.
Floer theory applied to manifold surgery and triangulation.
problem Understanding manifold surgeries and triangulations.
method Floer homology and Pin(2)-equivariant Seiberg-Witten Floer homology.
result Floer stable homotopy types and Pin(2)-equivariant Seiberg-Witten Floer homology provide insights into the triangulation conjecture.
Proof outlines existence of non-triangulable manifolds in high dimensions.
problem Existence of non-triangulable manifolds in dimensions >4.
method Homology cobordism invariants from Pin(2) symmetry of Seiberg-Witten equations.
result Existence of non-triangulable manifolds proven in dimensions >4.
New classification for some unorientable 4-manifolds using modified surgery theory.
problem Classifying stable diffeomorphism classes of unorientable 4-manifolds.
method Modified surgery theory applied to unorientable 4-manifolds with specific fundamental groups.
result Found nine stable diffeomorphism classes for pin+ manifolds, one for pin−, and four for neither, under certain conditions. 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.
We introduce a variant of the Seiberg-Witten equations, Pin^-(2)-monopole equations, and give its applications to intersection forms with local coefficients of 4-manifolds. The first application is an analogue of Froyshov's results on 4-manifolds which have definite forms with local coefficients. The second is a local …
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\…
Remark on Pin(2)-equivariant Floer homology and its connections.
problem Understanding the relationship between Pin(2)-equivariant Floer homology and other invariants.
method Analyzing the monopole Frøyshov invariant and Involutive Heegaard Floer correction terms.
result Identifying the only interesting correction terms as coming from subgroups Z/4, S^1, and Pin(2).
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. Paper generalizes properties of oriented 3d TFTs to unoriented case.
problem Generalize properties of oriented 3d TFTs to unoriented case.
method Show how Turaev-Viro construction can be generalized to unoriented 3d TFTs and Pin^+ TFTs.
result Pin^+ TFTs can be constructed from unoriented TFTs with a mixed anomaly.
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…
Research examines triangulations and homology cobordism groups.
problem Understanding the three-dimensional homology cobordism group.
method Local equivalence methods from Pin(2)-equivariant Seiberg-Witten Floer spectra and involutive Heegaard Floer homology.
result Review of known results and new insights into the group.
This note records two results which were inexplicably omitted from our paper on Pin structures on low dimensional manifolds, [KT]. Kirby chose not to be listed as a coauthor. A Pin^- structure on a surface F induces a quadratic enhancement of the mod 2 intersection form, q: H_1(F;Z/2Z) -> Z/4Z Theorem 1.1 says that q v…
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.
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.
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.
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.
Monopole Floer homology links Rokhlin invariants in 3-manifolds.
problem Understanding the relationship between Pin(2)-monopole Floer homology and Rokhlin invariants. method Using mod 2 index theory, the bar version of Pin(2)-monopole Floer homology is determined by triple cup products and Rokhlin invariants. result The Pin(2)-monopole Floer homology of a 3-manifold is determined by its Rokhlin invariants. Two Pin groups fit fermions in gravity.
problem Matching fermion behavior with gravity.
method Examined eight Pin groups; identified two compatible with gravity.
result Only two Pin groups fit fermions in gravity.
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.
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.
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.
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…
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.
Proves M-theory anomaly cancellation on nonorientable manifolds.
problem Anomaly cancellation in M-theory on nonorientable manifolds.
method Computational techniques for eta-invariants, algebraic theory of cubic forms, Adams spectral sequence techniques.
result No parity anomaly in M-theory in the low-energy field theory approximation.
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…
Paper shows how Non-Abelian T-duality solves pure spinor equations in supersymmetric vacua.
problem Preserving N=1 supersymmetry in Type II supergravity requires specific pure spinor equations. method Demonstrates that Non-Abelian T-duality (NATD) is a solution generating transformation for these pure spinor equations, showing covariance under Pin(d,d) transformations. result Non-Abelian T-duality (NATD) generates a flux that matches the geometric flux associated with the isometry group.
Geometrically interprets cup products and defines combinatorial Pin structures.
problem Understanding Steenrod's cup products and their geometric interpretation.
method Constructs vector fields and combinatorial frames to interpret cochain-level formulas.
result Geometrically interprets cup products and defines Pin structures combinatorially.
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.
We show that if M is an orientable 4-dimensional infrasolvmanifold and either β=β1(M;Q)≥2 or M is a Sol04- or a Solm,n4-manifold (with m=n) then M is parallelizable. There are non-parallelizable examples with β=1 for each of the other solvable Lie geometries $\ma…
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.
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.