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Octav Cornea

Publications and source records attributed to Octav Cornea.

At least 19 recordsLinked to original sources
arXiv2026

Approximability for Lagrangian submanifolds

This paper introduces a notion of categorical approximability for metric spaces that can be viewed as a categorification of approximability for metric groups, as defined by Turing in 1938. Approximability as introduced here is a property of metric spaces that is more general than precompactness. It is shown that several classes of Lagrangian submanifolds - closed Lagrangian submanifolds in a cotangent disk bundle; equators on the sphere; weakly exact Lagrangians on the torus-endowed with the spectral metric are approximable in this sense. Among other geometric applications, we show that there are such examples of spaces of Lagrangians that are approximable but are not precompact.

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arXiv2023

Persistence K-theory

This paper studies the basic K-theoretic properties of a triangulated persistence category (TPC). This notion was introduced in our earlier papers on triangulation, persistence, and Fukaya categories (arXiv:2304.01785 and arXiv:2104.12258) and it is a type of category that can be viewed as a refinement of a triangulated category in the sense that the morphisms sets of a TPC are persistence modules. We calculate the K-groups in some basic examples and discuss an application to Fukaya categories and to the topology of exact Lagrangian submanifolds. The second version contains some changes in the exposition and the correction of some minor imprecisions.

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arXiv2023

Triangulation, Persistence, and Fukaya categories

This paper introduces a new algebraic notion - triangulated persistence category (TPC) - that refines that of triangulated category in the same sense that a persistence module is a refinement of the notion of a vector space. The spaces of morphisms of such a TPC are persistence modules and this category is endowed with a class of weighted distinguished triangles. Under favourable conditions we show that the derived Fukaya category admits a TPC refinement and this is applied to deduce a global rigidity result for spaces of compact, exact Lagrangians in certain Liouville manifolds: we construct a metric on this space with intrinsic symplectic properties.

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arXiv2021

Triangulation and persistence: Algebra 101

This paper lays the foundations of triangulated persistence categories (TPC), which brings together persistence modules with the theory of triangulated categories. As a result we introduce several measurements and metrics on the set of objects of some triangulated categories. We also provide examples of TPCs coming from algebra, algebraic topology, microlocal sheaf theory and symplectic topology.

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arXiv2020

Bounds on the Lagrangian spectral metric in cotangent bundles

Let $N$ be a closed manifold and $U \subset T^*(N)$ a bounded domain in the cotangent bundle of $N$, containing the zero-section. A conjecture due to Viterbo asserts that the spectral metric for Lagrangian submanifolds that are exact-isotopic to the zero-section is bounded. In this paper we establish an upper bound on the spectral distance between two such Lagrangians $L_0, L_1$, which depends linearly on the boundary depth of the Floer complexes of $(L_0, F)$ and $(L_1, F)$, where $F$ is a fiber of the cotangent bundle.

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arXiv2020

A Lagrangian Pictionary

The purpose of this paper is to describe a dictionary geometry <--> algebra in Lagrangian topology. As a by-product we obtain a tautological (in a sense explained in the body of the paper) proof of a folklore conjecture (sometimes attributed to Kontsevich) claiming that the objects and structure of the derived Fukaya category can be represented through immersed Lagrangians. Our construction is based on certain Lagrangian cobordism categories endowed with a structure called "surgery models".

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arXiv2018

Lagrangian Shadows and Triangulated Categories

Under certain assumptions (such as weak exacteness or monotonicity) we show that splitting Lagrangians through cobordism has an energy cost and, from this cost being smaller than certain explicit bounds, we deduce some strong forms of rigidity of Lagrangian intersections. As a consequence, we construct some new pseudo-metrics and metrics on certain classes of Lagrangian submanifolds. We also fit these constructions in a more general setting, independent of Lagrangian cobordism. As a main technical tool, we develop aspects of the theory of (weakly) filtered A-infinity categories.

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arXiv2015

Lagrangian cobordism and metric invariants

We introduce new pseudo-metrics on spaces of Lagrangian submanifolds of a symplectic manifold $(M,\omega)$ by considering areas associated to projecting Lagrangian cobordisms in $\mathbb{C} \times M$ to the "time-energy plane" $\mathbb{C}$. We investigate the non-degeneracy properties of these pseudo-metrics, reflecting the rigidity and flexibility aspects of Lagrangian cobordisms.

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arXiv2015

Lagrangian cobordism in Lefschetz fibrations

Given a symplectic manifold $(M^{2n},\omega)$ we study Lagrangian cobordisms $V\subset E$ where $E$ is the total space of a Lefschetz fibration having $M$ as generic fiber. We prove a generation result for these cobordisms in the appropriate derived Fukaya category. As a corollary, we analyze the relations among the Lagrangian submanifolds $L\subset M$ that are induced by these cobordisms. This leads to a unified treatment - and a generalization - of the two types of relations among Lagrangian submanifolds of $M$ that were previously identified in the literature: those associated to Dehn twists that were discovered by Seidel and the relations induced by cobordisms in trivial symplectic fibrations described in our previous work.

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arXiv2015

Categorification of Seidel's representation

Two natural symplectic constructions, the Lagrangian suspension and Seidel's quantum representation of the fundamental group of the group of Hamiltonian diffeomorphisms, Ham(M), with (M,ω) a monotone symplectic manifold, admit categorifications as actions of the fundamental groupoid Π(Ham(M)) on a cobordism category recently introduced in \cite{Bi-Co:cob2} and, respectively, on a monotone variant of the derived Fukaya category. We show that the functor constructed in \cite{Bi-Co:cob2} that maps the cobordism category to the derived Fukaya category is equivariant with respect to these actions.

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arXiv2013

Lagrangian Cobordism and Fukaya Categories

Given a symplectic manifold M, we consider a category with objects finite ordered families of Lagrangian submanifolds of M (subject to certain additional constraints) and with morphisms Lagrangian cobordisms relating them. We construct a functor that maps this category to a variant of the derived Fukaya category of M in a way that takes into account the triangulated structure of the latter.

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arXiv2012

Lagrangian Cobordism I

This paper discusses the cobordism of Lagrangian submanifolds (in the monotone setting) and structures it as a category that is related in a functorial way to an appropriate (derived) Fukaya category. Are also discussed obstructions to cobordism based on properties of Lagrangian quantum homology, relations to Lagrangian surgery, as well as examples of non-isotopic but cobordant Lagrangians. This is a revision of our earlier preprint from September 2011.

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arXiv2008

Lagrangian Quantum Homology

The present paper is mainly a survey of our work arXiv:0708.4221 and arXiv:0808.2440 but it also contains the announcement of some new results. Its main purpose is to present an accessible introduction to a technique allowing efficient calculations in Lagrangian Floer theory.

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arXiv2008

Rigidity and uniruling for Lagrangian submanifolds

This paper explores the topology of monotone Lagrangian submanifolds $L$ inside a symplectic manifold $M$ by exploiting the relationships between the quantum homology of $M$ and various quantum structures associated to the Lagrangian $L$.

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arXiv2007

Quantization of the Serre spectral sequence

The present paper explores how the spectral sequence introduced in a previous work (and obtained by taking moduli spaces of any dimension into account in the Floer construction), interacts with the presence of bubbling. As consequences are obtained some relations between binary Gromov-Witten invariants and relative Ganea-Hopf invariants, a criterion for detecting the monodromy of bubbling as well as algebraic criteria for the detection of periodic orbits.

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arXiv2005

Cluster Homology

We assign, to a Langrangian submanifold $L$, a new homology which manages the bubbling of disks by means of auxiliary Morse data. This invariant of the Hamiltonian isotopy class of $L$ has many applications and naturally leads to a universal Floer theory for Lagrangian intersections.

math.SG↗