ProofAtlas

Symplectic topology

Nearby Lagrangian Conjecture

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For a closed connected smooth manifold Q, must every closed exact Lagrangian inside T*Q be deformable to the zero section by a Hamiltonian isotopy?

Qclosed, connected, smooth,L⊂T*Qclosed exact Lagrangian⇒L∼HQ
Known results and sources
For a closed connected smooth base manifold Q, a schematic cotangent bundle T*Q shows a closed exact Lagrangian above the zero section with an open isotopy question between them.
The card depicts the objects over a closed connected smooth base Q and leaves the Hamiltonian-isotopy relation explicitly open; it does not depict a proof route.

Research problem

Exact mathematical statement

Let QnQ^n be a closed connected smooth manifold. Let

ι=(q,p):Ln↪T*Qn,ι*λQ=df,\iota=(q,p):L^n\hookrightarrow T^*Q^n,\qquad \iota^*\lambda_Q=df,

be a closed exact Lagrangian embedding. The conjecture states:

Every closed exact LagrangianL⊂T*Qis Hamiltonian-isotopic to the zero section.\text{Every closed exact Lagrangian }L\subset T^*Q\text{ is Hamiltonian-isotopic to the zero section.}

The retained source explicitly says that the conjecture is not solved in the source.

Problem infographic

Problem at a glance

For a closed connected smooth manifold Q, a three-panel explainer defines a closed exact Lagrangian in T*Q and asks whether it is Hamiltonian-isotopic to the zero section.
The plate restores the closed, connected, smooth hypothesis on Q, states the exact open problem without a closeness hypothesis, and marks the Hamiltonian isotopy as unresolved.

Current mathematical picture

Where work on Nearby Lagrangian Conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failurePromoting the local phase to a global solution

The source separately lists those three global blockers and says none is solved by the local phase. Keep the local phase as a conditional geometric block while proving the global and ordered-incidence inputs independently.

Route status · Narrowed route
Main reductionCurrent reduction

The source separates the program into product topology and formal-vanishing inputs, an ordered framed homotopy-pullback nullbordism, and a coherent exact time-mixed polarization; none of the global inputs is supplied by the local phase.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeDefine the precise relative ordered-incidence invariant and prove its exact-homotopy invariance with complete conventions.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Nearby Lagrangian Conjecture in numbers

5.1kretained lines of mathematical investigation5,133 in the current working snapshot
Argument development
4,142 · 81%
Explored or eliminated routes
282 · 5%
Computational analysis
99 · 2%
Open obligations
284 · 6%
Definitions and setup
326 · 6%
5selected mapped statements1routes investigated6open questions6contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byVladimir Arnol'dStatus checkedAug 15, 2026
Source recordedAug 15, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

12 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

12 selected steps

Scroll horizontally to explore the route

Working route overview for Nearby Lagrangian ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.For closed connected smooth Q, is every closed exact Lagrangian in T*Q Hamiltonian-isotopic to the zero section? — Depends on missing premiseFor closed connected smoothQ, is every closed exactLagrangian…Current reduction — Depends on missing premiseCurrent reductionClosing target — Depends on missing premiseClosing targetPath extraction from full pullback data — Depends on missing premisePath extraction from fullpullback dataRelative cutoff under explicit margins — Depends on missing premiseRelative cutoff underexplicit marginsPromoting the local phase to a global solution — stoppedPromoting the local phase toa global solutionDefine the precise relative ordered-incidence invariant and prove its exact-homotopy invariance with complete conventions. — OpenDefine the precise relativeordered-incidence invariantand…Prove the product-topology and fixed-formal-vanishing inputs required before the local geometric phase can apply. — OpenProve the product-topologyand fixed-formal-vanishinginputs…Construct a coherent exact time-mixed polarization on the full corridor with fixed endpoint germs and controlled projected incidence. — OpenConstruct a coherent exacttime-mixed polarization onthe…Hamiltonian isotopy to the zero section — OpenHamiltonian isotopy to thezero sectionProduct topology — OpenProduct topologyFixed formal vanishing — OpenFixed formal vanishing
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded
Narrowed routePromoting the local phase to a global solution

The source separately lists those three global blockers and says none is solved by the local phase. Keep the local phase as a conditional geometric block while proving the global and ordered-incidence inputs independently.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

6 featured tasks
01
Define the precise relative ordered-incidence invariant and prove its exact-homotopy invariance with complete conventions.Suggested move: Write the homotopy-pullback normal-bordism target, boundary conditions, ordering conventions, and induced stable-framing comparison in one source-auditable statement.
Ready to work on
02
Prove the product-topology and fixed-formal-vanishing inputs required before the local geometric phase can apply.Suggested move: Separate the smooth product-topology theorem from every component of the formal obstruction and audit the external inputs for each dimension.
Ready to work on
03
Hamiltonian isotopy to the zero section

For a closed connected smooth manifold Q, prove that every closed exact Lagrangian in T*Q is Hamiltonian-isotopic to the zero section.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Product topology

Prove that the projection is homotopic to a diffeomorphism in the required smooth category.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Fixed formal vanishing

Prove vanishing of the fixed formal obstruction, including dimension-three winding and all higher or torsion components.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
06
Construct a coherent exact time-mixed polarization on the full corridor with fixed endpoint germs and controlled projected incidence.Suggested move: Formulate a relative polarized-tubularization lemma whose hypotheses include the one-root, center-gap, radial-barrier, and external-separation margins.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 15, 2026
Current statusOpen conjecture

The general conjecture remains open: every closed exact Lagrangian in a cotangent bundle is conjectured to be Hamiltonian-isotopic to the zero section. Current primary results impose homotopy and stable-Gauss-map constraints but do not establish the isotopy conclusion.

[1]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. Peer reviewedAbouzaid, Courte, Guillermou, and Kragh proved vanishing results for the stable Lagrangian Gauss map and developed twisted generating functions; this remains short of Hamiltonian isotopy.[3]
  2. Peer reviewedPorcelli states the Nearby Lagrangian Conjecture as a main open problem while studying consequences for C0 Lagrangian monodromy.[1]
  3. PreprintAbouzaid and Kragh proved that the projection of a closed exact Lagrangian to the base is a simple homotopy equivalence.[2]
4 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusNearby Lagrangian Conjecture
Weaker or relaxed formsimple homotopy equivalence of nearby Lagrangians

Simple homotopy equivalence of the projection is a strong topological constraint but is weaker than Hamiltonian isotopy to the zero section.

[2]
Dependency or reductionstable Lagrangian Gauss-map constraints

Twisted generating functions and stable Gauss-map vanishing constrain possible exact Lagrangians without resolving their Hamiltonian-isotopy class.

[3]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetA formal definition of exact Lagrangian embeddings in cotangent bundles and compactly supported Hamiltonian isotopy.
  • Formalization targetA library of symplectic, Floer-theoretic, and smooth-topological constructions supporting the known partial results.
  • Formalization targetA statement-alignment review separating simple homotopy equivalence from the conjectured Hamiltonian-isotopy conclusion.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

3 standing statements2 proposed statements6 open questions1 narrowed routes
Statements by mathematical role5 selected mapped statements
  • theorem candidate1 of 51
  • reduction1 of 51
  • lemma3 of 53
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.18 displayed rows · 1 route included
  • retained route statementFor closed connected smooth Q, is every closed exact Lagrangian in T*Q Hamiltonian-isotopic to the zero section?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementPath extraction from full pullback dataintermediate
  • retained route statementRelative cutoff under explicit marginsintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failurePromoting the local phase to a global solutionreported failure
  • Research targetDefine the precise relative ordered-incidence invariant and prove its exact-homotopy invariance with complete conventions.open
  • Research targetProve the product-topology and fixed-formal-vanishing inputs required before the local geometric phase can apply.open
  • Research targetConstruct a coherent exact time-mixed polarization on the full corridor with fixed endpoint germs and controlled projected incidence.open
  • Research targetHamiltonian isotopy to the zero sectionopen
  • Research targetProduct topologyopen
  • Research targetFixed formal vanishingopen
  • Research targetExact time-mixed polarizationsuperseded
  • Narrowed routePromoting the local phase to a global solutionThe source separately lists those three global blockers and says none is solved by the local phase. Keep the local phase as a conditional geometric block while proving the global and ordered-incidence inputs independently.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeDefine the precise relative ordered-incidence invariant and prove its exact-homotopy invariance with complete conventions.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointDefine the precise relative ordered-incidence invariant and prove its exact-homotopy invariance with complete conventions.

Nearby Lagrangian Conjecture · ready to start

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Research contextPrepared context for any AI agent

For a closed connected smooth manifold Q, must every closed exact Lagrangian inside T*Q be deformable to the zero section by a Hamiltonian isotopy?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references4 cited works · next context review by Nov 15, 2026

The mathematical context was checked on Aug 15, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    C0 Lagrangian monodromypeer reviewed result · Noah Porcelli · Bulletin of the London Mathematical Society · 2025 · DOI 10.1112/blms.70162 · accessed Aug 15, 2026
  2. 2
    Simple Homotopy Equivalence of Nearby Lagrangianspreprint · Mohammed Abouzaid, Thomas Kragh · arXiv · 2016 · ARXIV 1603.05431 · accessed Aug 15, 2026
  3. 3
    Twisted generating functions and the nearby Lagrangian conjecturepeer reviewed result · Mohammed Abouzaid, Sylvain Courte, Stéphane Guillermou, Thomas Kragh · Duke Mathematical Journal · 2025 · ARXIV 2011.13178 · DOI 10.1215/00127094-2024-0052 · accessed Aug 15, 2026
  4. 4
    The nearby Lagrangian conjectureauthoritative webpage · Princeton University Department of Mathematics · 2020-03-05 · accessed Aug 15, 2026

Important qualifications

  • The current open status is supported by a 2025 peer-reviewed paper that states the general conjecture as a main open problem.
  • Simple homotopy equivalence and stable-Gauss-map results are partial constraints and are not promoted to Hamiltonian isotopy.
  • The conventional Arnol'd attribution is supported by an official university seminar page; no exact proposal year is asserted.
  • No packet URL or attachment was fetched, executed, rendered, or treated as external evidence.

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