Generalized Lagrangian spectral-distance conjecture

The generalized Viterbo conjecture predicts bounded spectral diameter for Lagrangians Hamiltonian isotopic to the zero section inside a compact neighborhood of that section in T*M, for every closed manifold M. The retained counterexample setting has a closed connected negatively curved base, nonzero H¹(M;R), odd Euler characteristic and Z₂ coefficients; later broader hypotheses and coefficient extensions are separate.

Why it matters and what counts as progress

Why it matters: It tests how spectral-distance boundedness depends on the base manifold and coefficient ring.

What counts as progress: A closed exact Lagrangian family with unbounded spectral distance for the specified negatively curved base and Z₂ coefficients, or a valid restricted boundedness theorem.

How the problem developed

2007 · Conjecture [1]
Claude Viterbo Conjectured a uniform bound for the spectral norm of exact Lagrangians Hamiltonian isotopic to the zero section in a cotangent disk bundle over a torus.
15 Oct 2026 · Counterexample [2]
Filip Broćić, Baptiste Serraille, and Ibrahim Trifa Proved unbounded spectral distance for Lagrangians in a disk cotangent bundle of a closed manifold satisfying stated topological assumptions, disproving a generalized Viterbo conjecture; the paper also classifies the surface cases over Z₂.

AI contribution

A counterexample to the generalized Viterbo conjecture on the spectral distance in a disc cotangent bundle

Filip Broćić, Baptiste Serraille, Ibrahim Trifa

The initial negatively curved, Z2-coefficient counterexample to the generalized Lagrangian spectral-distance assertion, as identified in the AI disclosure; exclude the later human extensions.

AI involvementAI-centered

ImportanceModest

Technical verificationReported

Human understandingFull