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.
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.
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.
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 …
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 Yamabe invariants for a new infinite class of closed 4-dimensional manifolds by using a "twisted" version of the Seiberg-Witten equations, the Pin−(2)-monopole equations. The same technique also provides a new obstruction to the existence of Einstein metrics or long-time solutions of the no…
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…
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.
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. 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. 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.
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…
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.
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…
We study the conjugation involution in Seiberg-Witten theory in the context of the Ozsváth-Szabó and Bloom's spectral sequence for the branched double cover of a link L in S3. We prove that there exists a spectral sequence of F[Q]/Q2-modules (where Q has degree −1) which converges to $\widetilde{\m…
We prove the existence of an exact triangle for the Pin(2)-monopole Floer homology groups of three manifolds related by specific Dehn surgeries on a given knot. Unlike the counterpart in usual monopole Floer homology, only two of the three maps are those induced by the corresponding elementary cobordism. We use this tr…
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.
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 …
The paper localizes an index and provides a condition for the orientability of a monopole moduli space.
problem Localization of a KO∗(extpt)-valued index and orientability of the Pin−(2) monopole moduli space. method Introducing G±(n,s+,s−) structures and formulating characteristic submanifolds for these structures. result The KO∗(pt)-valued index is localized to the characteristic submanifold, and a condition for the moduli space to be orientable is provided. 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.
We classify hyperbolic monopoles with continuous symmetries and construct new examples.
problem Classifying and constructing hyperbolic monopoles with continuous symmetries.
method Developed a Structure Theorem and used representation theory to simplify the problem.
result Found constraints on structure groups and constructed novel spherically symmetric Sp(n) hyperbolic monopoles. Unified tau invariants in instanton and monopole Floer theories.
problem Unified tau invariants in instanton and monopole Floer theories.
method Identified τ_{G} with τ_{G}^{\sharp} via cobordism maps.
result Identified τ_{G} with τ_{G}^{\sharp} via cobordism maps.
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…
Proves surgery exact triangle for monopole Floer homology over integers.
problem Proving the surgery exact triangle for monopole Floer homology over integer coefficients.
method Modification of Kronheimer--Mrowka's local system and adaptation of Freeman's computation.
result Obtains a spectral sequence over integer coefficients for an oriented link in S3. Computes conditions for real spinor bundles on pseudo-Riemannian manifolds.
problem Obtaining Dirac operators on real spinor bundles of complex type.
method Using Lipschitz structures and Karoubi Stiefel-Whitney classes, reformulating the problem in terms of Spin^o_α structures.
result Explicit construction and characterization of real spinor bundles of irreducible complex type.
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.
Real Seiberg-Witten and monopole Floer homologies are equivalent for certain 3-manifolds.
problem Equivalence of two Floer homologies for real structures.
method Direct isomorphism proof between real Seiberg-Witten and monopole Floer homologies.
result Real Seiberg-Witten and monopole Floer homologies are isomorphic.
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.
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…
New 4-manifolds found without certain Einstein metrics, using Seiberg-Witten theory.
problem Constructing 4-manifolds without specific Einstein metrics.
method Using Seiberg-Witten theory and constructing solutions on noncompact manifolds.
result Infinitely many examples of 4-manifolds without cusped asymptotically hyperbolic Einstein metrics.
The study examines the behavior of monopoles as their mass increases and finds that they abelianize near singular points.
problem The behavior of monopoles as their mass increases and the formation of singular points.
method Analysis of mass-renormalized energy measures and convergence of fields to a reducible monopole.
result The mass-renormalized energy measures concentrate at singular points, and the fields abelianize near these points.
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. 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.
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. We prove that Witten's Conjecture [arXiv:hep-th/9411102] on the relationship between the Donaldson and Seiberg-Witten series for a four-manifold of Seiberg-Witten simple type with b1=0 and odd b2+≥3 follows from our SO(3)-monopole cobordism formula [arXiv:math/0203047] when the four-manifold has $…
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.
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.
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.
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.