---
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:
- access_level: open_access
  checksum: 487fa9113e1bbf32a6c70e6d1e8f63bc
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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_size: 2256345
  relation: main_file
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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'
...
---
_id: '17475'
abstract:
- lang: eng
  text: "As a discrete analogue of Kac’s celebrated question on ‘hearing the shape
    of a drum’ and towards a practical\r\ngraph isomorphism test, it is of interest
    to understand which graphs are determined up to isomorphism by\r\ntheir spectrum
    (of their adjacency matrix). A striking conjecture in this area, due to van Dam
    and Haemers,\r\nis that ‘almost all graphs are determined by their spectrum’,
    meaning that the fraction of unlabelled n-vertex\r\ngraphs which are determined
    by their spectrum converges to 1 as n → ∞.\r\nIn this paper, we make a step towards
    this conjecture, showing that there are exponentially many n-vertex\r\ngraphs
    which are determined by their spectrum. This improves on previous bounds (of shape
    e\r\nc\r\n√\r\nn\r\n). We also\r\npropose a number of further directions of research.\r\n"
acknowledgement: Matthew Kwan was supported by ERC Starting Grant ‘RANDSTRUCT’ No.
  101076777.
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
- first_name: Matthew Alan
  full_name: Kwan, Matthew Alan
  id: 5fca0887-a1db-11eb-95d1-ca9d5e0453b3
  last_name: Kwan
  orcid: 0000-0002-4003-7567
citation:
  ama: Koval I, Kwan MA. Exponentially many graphs are determined by their spectrum.
    <i>Quarterly Journal of Mathematics</i>. 2024;75(3):869-899. doi:<a href="https://doi.org/10.1093/qmath/haae030">10.1093/qmath/haae030</a>
  apa: Koval, I., &#38; Kwan, M. A. (2024). Exponentially many graphs are determined
    by their spectrum. <i>Quarterly Journal of Mathematics</i>. Oxford University
    Press. <a href="https://doi.org/10.1093/qmath/haae030">https://doi.org/10.1093/qmath/haae030</a>
  chicago: Koval, Illya, and Matthew Alan Kwan. “Exponentially Many Graphs Are Determined
    by Their Spectrum.” <i>Quarterly Journal of Mathematics</i>. Oxford University
    Press, 2024. <a href="https://doi.org/10.1093/qmath/haae030">https://doi.org/10.1093/qmath/haae030</a>.
  ieee: I. Koval and M. A. Kwan, “Exponentially many graphs are determined by their
    spectrum,” <i>Quarterly Journal of Mathematics</i>, vol. 75, no. 3. Oxford University
    Press, pp. 869–899, 2024.
  ista: Koval I, Kwan MA. 2024. Exponentially many graphs are determined by their
    spectrum. Quarterly Journal of Mathematics. 75(3), 869–899.
  mla: Koval, Illya, and Matthew Alan Kwan. “Exponentially Many Graphs Are Determined
    by Their Spectrum.” <i>Quarterly Journal of Mathematics</i>, vol. 75, no. 3, Oxford
    University Press, 2024, pp. 869–99, doi:<a href="https://doi.org/10.1093/qmath/haae030">10.1093/qmath/haae030</a>.
  short: I. Koval, M.A. Kwan, Quarterly Journal of Mathematics 75 (2024) 869–899.
corr_author: '1'
das_tickbox: '0'
date_created: 2024-09-01T22:01:07Z
date_published: 2024-06-19T00:00:00Z
date_updated: 2026-10-16T10:14:59Z
day: '19'
ddc:
- '500'
department:
- _id: MaKw
- _id: VaKa
doi: 10.1093/qmath/haae030
external_id:
  arxiv:
  - '2309.09788'
  isi:
  - '001249741500001'
file:
- access_level: open_access
  checksum: abf200d37ad69e6f2c0750a30296ad97
  content_type: application/pdf
  creator: cchlebak
  date_created: 2024-09-06T12:23:57Z
  date_updated: 2024-09-06T12:23:57Z
  file_id: '17851'
  file_name: 2024_QuJofMath_Koval.pdf
  file_size: 946411
  relation: main_file
  success: 1
file_date_updated: 2024-09-06T12:23:57Z
fulldoi: https://doi.org/10.1093/qmath/haae030
has_accepted_license: '1'
intvolume: '        75'
isi: 1
issue: '3'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 869-899
project:
- _id: bd95085b-d553-11ed-ba76-e55d3349be45
  grant_number: '101076777'
  name: Randomness and structure in combinatorics
publication: Quarterly Journal of Mathematics
publication_identifier:
  eissn:
  - 1464-3847
  issn:
  - 0033-5606
publication_status: published
publisher: Oxford University Press
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Exponentially many graphs are determined by their spectrum
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: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 75
year: '2024'
...