Related papers: Kuranishi structure, Pseudo-holomorphic curve, and…

Let $L$ be a Lagrangian submanifold in a symplectic vector space which is closed, oriented and spin. Using virtual fundamental chains of moduli spaces of nonconstant pseudo-holomorphic disks with boundaries on $L$, one can define a…

We construct global Kuranishi charts for moduli spaces of pseudo-holomorphic maps of arbitrary genus with boundary on an embedded Lagrangian submanifold. We then build the geometric foundations required for obtaining compatible chain-level…

Consider a space $X$, such as a compact space of $J$-holomorphic stable maps, that is the zero set of a Kuranishi atlas. This note explains how to define the virtual fundamental class of $X$ by representing $X$ via the zero set of a map…

Kuranishi structures were introduced to symplectic topology by Fukaya and Ono and recently refined by Joyce, in order to extract homological data from compactified moduli spaces of holomorphic maps in cases where geometric regularization…

The notion of linear K-system is introduced by the present authors as an abstract model arising from the structure of compactified moduli spaces of solutions to Floer's equation in the book [FOOO14]. The purpose of the present article is to…

Motivated by the definition of homotopy $L_\infty$ spaces, we develop a new theory of Kuranishi manifolds, closely related to Joyce's recent theory. We prove that Kuranishi manifolds form a $2$-category with invertible $2$-morphisms, and…

We give a construction of contact homology in the sense of Eliashberg--Givental--Hofer. Specifically, we construct coherent virtual fundamental cycles on the relevant compactified moduli spaces of pseudo-holomorphic curves.

Using homological perturbation theory, we develop a formal version of the miniversal deformation associated with a deformation problem controlled by a differential graded Lie algebra over a field of characteristic zero. Our approach…

We construct algebraic virtual cycles that give us the cohomological field theories of Fan-Jarvis-Ruan invariants by integral transformations.

In this paper I give a major update of quantum gravity framework project. The heuristic conceptual framework proposed in previous versions is expanded to include structure formation and consciousness in the universe. A Path Integral version…

This is the first part of the lecture notes that grew out of the special course given during the 2021-2022 academic year. In these lecture notes we present an approach to the fundamental structures of differential geometry that uses the…

We show that the deformation space of complex parallelisable nilmanifolds can be described by polynomial equations but is almost never smooth. This is remarkable since these manifolds have trivial canonical bundle and are holomorphic…

This paper proposes a basic theory on physical reality and a new foundation for quantum mechanics and classical mechanics. It presents a scenario not only to solve the problem of the arbitrariness on the operator ordering for the…

We study real and integral structures in the space of solutions to the quantum differential equations. First we show that, under mild conditions, any real structure in orbifold quantum cohomology yields a pure and polarized tt^*-geometry…

Quantum physics is a linear theory, so it is somewhat puzzling that it can underlie very complex systems such as digital computers and life. This paper investigates how this is possible. Physically, such complex systems are necessarily…

On a threefold with trivial canonical bundle, Kuranishi theory gives an algebro-geometry construction of the (local analytic) Hilbert scheme of curves at a smooth holomorphic curve as a gradient scheme, that is, the zero-scheme of the…

We describe deformations of vector bundles on surfaces that are a product of two smooth projective curves. We explicitly describe the Kuranishi map around unstable vector bundles and compare the homologies of the Kuranishi spaces of stable…

The motivation for this thesis was to recast quantum self-testing [MY98,MY04] in operational terms. The result is a category-theoretic framework for discussing the following general question: How do different implementations of the same…

This is a survey of the author's paper arXiv:1409.6908 and in-progress book. 'Kuranishi spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic geometry (see e.g. arXiv:1503.07631), as the geometric structure on moduli…

This is a work in progress, far from being in its final form whose purpose is to investigate thoroughly the structure of Berkovich analytic curves and its relation with the semi-stable reduction theorem (of which a new proof is given here,…