arXiv research

The paper studies $ext{Pin}^{\pm}$-structures on non-oriented 4-manifolds via Lefschetz fibrations.

The study explores psc-metrics on non-spin manifolds with pin^\pm or spin^c structures.

The paper explores properties of Pin structures on surfaces and their cobordism.

The paper localizes an index and provides a condition for the orientability of a monopole moduli space.

This study analyzes a non-orientable spacetime model in 1+1D quantum gravity.

Geometrically interprets cup products and defines combinatorial Pin structures.

The present, partly expository, monograph consists of three parts. The first part treats Spin- and Pin-structures from three different perspectives and shows them to be suitably equivalent. It also introduces an intrinsic perspective on the relative Spin- and Pin-structures of Fukaya-Oh-Ohta-Ono and Solomon, establishe…

We give necessary and sufficient conditions for the existence of pin+, pin- and spin structures on Riemannian manifolds with holonomy group $Z_2^k$. 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 $pin^c$ structure on the product of two Riemannian manifolds is derived and applied to construct examples of manifolds having the weaker Lipschitz structure, but no $pin^c$ structure. An example of a five-dimensional manifold with this property is given; it is pointed ou…

Two Pin(2)-equivariant Floer homologies are shown to be equivalent.

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.

We investigate the $\mathrm{Pin}^-(2)$-monopole invariants of symplectic $4$-manifolds and Kähler surfaces with real structures. We prove the nonvanishing theorem for real symplectic $4$-manifolds which is an analogue of Taubes' nonvanishing theorem of the Seiberg-Witten invariants for symplectic $4$-manifolds. Further…

In previous work, we introduced a natural $\mathcal{A}_{\infty}$-structure on the $\mathrm{Pin}(2)$-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 $S^3$. In this paper, we provide explicit descriptions of seve…

The paper introduces discrete Dirac structures for mechanics, simplifying dynamics.

PIN models feature interactions using a neural network that mimics decision trees.

Unified Pin-SVM improves accuracy over existing Pin-SVM model.

We show that the bar version of the $\mathrm{Pin}(2)$-monopole Floer homology of a three-manifold $Y$ equipped with a self-conjugate spin$^c$ structure $\mathfrak{s}$ is determined by the triple cup product of $Y$ together with the Rokhlin invariants of the spin structures inducing $\mathfrak{s}$. This is a manifestati…

We study the behavior of $\mathrm{Pin}(2)$-monopole Floer homology under connected sums. After constructing a (partially defined) $\mathcal{A}_{\infty}$-module structure on the $\mathrm{Pin}(2)$-monopole Floer chain complex of a three manifold (in the spirit of Baldwin and Bloom's monopole category), we identify up to …

Classifies stability of flat-core $p$-elasticae pinned at boundaries.

Study pin manifolds using Clifford linear Dirac operator and KO-theory.

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…

New system uses wearable bio-signals for easy authentication.

Established a stable cohomotopy refinement for Pin(2) monopole invariants.

We introduce a diffeomorphism invariant of $4$-manifolds, the $\mathrm{Pin}^-(2)$-monopole invariant, defined by using the $\mathrm{Pin}^-(2)$-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…

Study on a pinning model with random walk increments, showing convergence to a critical disordered pinning measure.

The pinning ideal of multiloops is shown to be NP-complete.

Complex pinning problem simplified for simple multiloops.

In this remark, we show how the monopole Frøyshov invariant, as well as the analogues of the Involutive Heegaard Floer correction terms $\underline{d},\overline{d}$, are related to the $\mathrm{Pin}(2)$-equivariant Floer homology $\mathit{SWFH}^G_*$. We show that the only interesting correction terms of a $\mathrm{Pin}…

Real del Pezzo surfaces split real lines into elliptic and hyperbolic types.

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.

New classification for some unorientable 4-manifolds using modified surgery theory.

The paper finds curves minimizing elastic energy pinned at endpoints.

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…

Spaces of positive and tropical points described as Teichmüller and lamination spaces with pinnings.

For a compact almost complex 4-manifold $(M,J)$, we study the subgroups $H^{\pm}_J$ of $H^2(M, \mathbb{R})$ consisting of cohomology classes representable by $J$-invariant, respectively, $J$-anti-invariant 2-forms. If $b^+ =1$, we show that for generic almost complex structures on $M$, the subgroup $H^-_J$ is trivial. …

This report examines the Pinned AUC metric introduced and highlights some of its limitations. Pinned AUC provides a threshold-agnostic measure of unintended bias in a classification model, inspired by the ROC-AUC metric. However, as we highlight in this report, there are ways that the metric can obscure different kinds…

Researchers compute mod 2 Seiberg-Witten invariants for spin structures and families.

In this paper, we introduce an extension of a Brownian bridge with a random length by including uncertainty also in the pinning level of the bridge. The main result of this work is that unlike for deterministic pinning point, the bridge process fails to be Markovian if the pining point distribution is absolutely contin…

We show that if $M$ is an orientable 4-dimensional infrasolvmanifold and either $β=β_1(M;\mathbb{Q})\geq2$ or $M$ is a $\mathbb{S}ol_0^4$- or a $\mathbb{S}ol_{m,n}^4$-manifold (with $m\not=n$) then $M$ is parallelizable. There are non-parallelizable examples with $β=1$ for each of the other solvable Lie geometries $\ma…

New equations use Pin(2) symmetry to study spinor and connection solutions.

Classifies pinned $p$-elasticae and finds unique optimality exponents.

Acute Kidney Injury (AKI) is a common clinical syndrome characterized by the rapid loss of kidney excretory function, which aggravates the clinical severity of other diseases in a large number of hospitalized patients. Accurate early prediction of AKI can enable in-time interventions and treatments. However, AKI is hig…

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…

Let $M$ be any $n$ dimensional smooth manifold and $PM$ be the space of all smooth paths, then we showed that $PM$ is a smooth manifold modelled over a complete normable space. We discussed many geometric structure on Path spaces and its relation to ambient space.

This paper generalizes two facts about oriented 3d TFTs to the unoriented case. On one hand, it is known that oriented 3d TFTs having a topological boundary condition admit a state-sum construction known as the Turaev-Viro construction. This is related to the string-net construction of fermionic phases of matter. We sh…

New methods prove weak homotopy inequivalence for manifold symmetries.

The paper defines a new algebra structure on skein modules of 3-manifolds and shows it's isomorphic to a known algebra.