---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '14278'
abstract:
- lang: eng
  text: 'The Birkhoff conjecture says that the boundary of a strictly convex integrable
    billiard table is necessarily an ellipse. In this article, we consider a stronger
    notion of integrability, namely, integrability close to the boundary, and prove
    a local version of this conjecture: a small perturbation of almost every ellipse
    that preserves integrability near the boundary, is itself an ellipse. We apply
    this result to study local spectral uniqueness of ellipses using the connection
    between the wave trace of the Laplacian and the dynamics near the boundary and
    establish local uniqueness for almost all of them.'
acknowledgement: 'The author acknowledges the partial support of the European Research
  Council Grant #885707. He also thanks Vadim Kaloshin for proposing the idea of the
  project and greatly aiding the implementation. The author is also grateful to Hamid
  Hezari, Amir Vig, Steve Zelditch, Comlan E. Koudjinan, Corentin Fierobe, Ngo Nhok
  Tkhai Shon and Roman Sarapin for useful discussions. The author also acknowledges
  partial support of ISTern summer program. The project started in the summer of 2021,
  when the author was an intern at ISTA. Open access funding provided by Institute
  of Science and Technology (IST Austria).'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Illya
  full_name: Koval, Illya
  id: 2eed1f3b-896a-11ed-bdf8-93c7c4bf159e
  last_name: Koval
citation:
  ama: Koval I. Local strong Birkhoff conjecture and local spectral rigidity of almost
    every ellipse. <i>Inventiones Mathematicae</i>. 2026;244:221-298. doi:<a href="https://doi.org/10.1007/s00222-025-01397-y">10.1007/s00222-025-01397-y</a>
  apa: Koval, I. (2026). Local strong Birkhoff conjecture and local spectral rigidity
    of almost every ellipse. <i>Inventiones Mathematicae</i>. Springer Nature. <a
    href="https://doi.org/10.1007/s00222-025-01397-y">https://doi.org/10.1007/s00222-025-01397-y</a>
  chicago: Koval, Illya. “Local Strong Birkhoff Conjecture and Local Spectral Rigidity
    of Almost Every Ellipse.” <i>Inventiones Mathematicae</i>. Springer Nature, 2026.
    <a href="https://doi.org/10.1007/s00222-025-01397-y">https://doi.org/10.1007/s00222-025-01397-y</a>.
  ieee: I. Koval, “Local strong Birkhoff conjecture and local spectral rigidity of
    almost every ellipse,” <i>Inventiones Mathematicae</i>, vol. 244. Springer Nature,
    pp. 221–298, 2026.
  ista: Koval I. 2026. Local strong Birkhoff conjecture and local spectral rigidity
    of almost every ellipse. Inventiones Mathematicae. 244, 221–298.
  mla: Koval, Illya. “Local Strong Birkhoff Conjecture and Local Spectral Rigidity
    of Almost Every Ellipse.” <i>Inventiones Mathematicae</i>, vol. 244, Springer
    Nature, 2026, pp. 221–98, doi:<a href="https://doi.org/10.1007/s00222-025-01397-y">10.1007/s00222-025-01397-y</a>.
  short: I. Koval, Inventiones Mathematicae 244 (2026) 221–298.
corr_author: '1'
das_tickbox: '0'
date_created: 2023-09-06T08:35:43Z
date_published: 2026-04-01T00:00:00Z
date_updated: 2026-07-23T10:58:59Z
day: '01'
ddc:
- '510'
department:
- _id: GradSch
- _id: VaKa
doi: 10.1007/s00222-025-01397-y
ec_funded: 1
external_id:
  arxiv:
  - '2111.12171'
file:
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  creator: dernst
  date_created: 2026-07-23T10:55:24Z
  date_updated: 2026-07-23T10:55:24Z
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file_date_updated: 2026-07-23T10:55:24Z
fulldoi: https://doi.org/10.1007/s00222-025-01397-y
has_accepted_license: '1'
intvolume: '       244'
language:
- iso: eng
mathsc:
- 37C83
- 35J05
- 37J70
- 74J25
month: '04'
oa: 1
oa_version: Published Version
page: 221-298
project:
- _id: 9B8B92DE-BA93-11EA-9121-9846C619BF3A
  call_identifier: H2020
  grant_number: '885707'
  name: Spectral rigidity and integrability for billiards and geodesic flows
publication: Inventiones Mathematicae
publication_identifier:
  eissn:
  - 1432-1297
  issn:
  - 0020-9910
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: yes
title: Local strong Birkhoff conjecture and local spectral rigidity of almost every
  ellipse
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 244
year: '2026'
...