---
_id: '12877'
abstract:
- lang: eng
  text: We consider billiards obtained by removing from the plane finitely many strictly
    convex analytic obstacles satisfying the non-eclipse condition. The restriction
    of the dynamics to the set of non-escaping orbits is conjugated to a subshift,
    which provides a natural labeling of periodic orbits. We show that under suitable
    symmetry and genericity assumptions, the Marked Length Spectrum determines the
    geometry of the billiard table.
acknowledgement: 'J.D.S. and M.L. have been partially supported by the NSERC Discovery
  grant, reference number 502617-2017. M.L. was also supported by the ERC project
  692925 NUHGD of Sylvain Crovisier, by the ANR AAPG 2021 PRC CoSyDy: Conformally
  symplectic dynamics, beyond symplectic dynamics (ANR-CE40-0014), and by the ANR
  JCJC PADAWAN: Parabolic dynamics, bifurcations and wandering domains (ANR-21-CE40-0012).
  V.K. acknowledges partial support of the NSF grant DMS-1402164 and ERC Grant # 885707.'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Jacopo
  full_name: De Simoi, Jacopo
  last_name: De Simoi
- first_name: Vadim
  full_name: Kaloshin, Vadim
  id: FE553552-CDE8-11E9-B324-C0EBE5697425
  last_name: Kaloshin
  orcid: 0000-0002-6051-2628
- first_name: Martin
  full_name: Leguil, Martin
  last_name: Leguil
citation:
  ama: De Simoi J, Kaloshin V, Leguil M. Marked Length Spectral determination of analytic
    chaotic billiards with axial symmetries. <i>Inventiones Mathematicae</i>. 2023;233:829-901.
    doi:<a href="https://doi.org/10.1007/s00222-023-01191-8">10.1007/s00222-023-01191-8</a>
  apa: De Simoi, J., Kaloshin, V., &#38; Leguil, M. (2023). Marked Length Spectral
    determination of analytic chaotic billiards with axial symmetries. <i>Inventiones
    Mathematicae</i>. Springer Nature. <a href="https://doi.org/10.1007/s00222-023-01191-8">https://doi.org/10.1007/s00222-023-01191-8</a>
  chicago: De Simoi, Jacopo, Vadim Kaloshin, and Martin Leguil. “Marked Length Spectral
    Determination of Analytic Chaotic Billiards with Axial Symmetries.” <i>Inventiones
    Mathematicae</i>. Springer Nature, 2023. <a href="https://doi.org/10.1007/s00222-023-01191-8">https://doi.org/10.1007/s00222-023-01191-8</a>.
  ieee: J. De Simoi, V. Kaloshin, and M. Leguil, “Marked Length Spectral determination
    of analytic chaotic billiards with axial symmetries,” <i>Inventiones Mathematicae</i>,
    vol. 233. Springer Nature, pp. 829–901, 2023.
  ista: De Simoi J, Kaloshin V, Leguil M. 2023. Marked Length Spectral determination
    of analytic chaotic billiards with axial symmetries. Inventiones Mathematicae.
    233, 829–901.
  mla: De Simoi, Jacopo, et al. “Marked Length Spectral Determination of Analytic
    Chaotic Billiards with Axial Symmetries.” <i>Inventiones Mathematicae</i>, vol.
    233, Springer Nature, 2023, pp. 829–901, doi:<a href="https://doi.org/10.1007/s00222-023-01191-8">10.1007/s00222-023-01191-8</a>.
  short: J. De Simoi, V. Kaloshin, M. Leguil, Inventiones Mathematicae 233 (2023)
    829–901.
date_created: 2023-04-30T22:01:05Z
date_published: 2023-08-01T00:00:00Z
date_updated: 2023-10-04T11:25:37Z
day: '01'
department:
- _id: VaKa
doi: 10.1007/s00222-023-01191-8
ec_funded: 1
external_id:
  arxiv:
  - '1905.00890'
  isi:
  - '000978887600001'
intvolume: '       233'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1905.00890
month: '08'
oa: 1
oa_version: Preprint
page: 829-901
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'
scopus_import: '1'
status: public
title: Marked Length Spectral determination of analytic chaotic billiards with axial
  symmetries
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 233
year: '2023'
...
---
_id: '10704'
abstract:
- lang: eng
  text: We define and study the existence of very stable Higgs bundles on Riemann
    surfaces, how it implies a precise formula for the multiplicity of the very stable
    components of the global nilpotent cone and its relationship to mirror symmetry.
    The main ingredients are the Bialynicki-Birula theory of C∗-actions on semiprojective
    varieties, C∗ characters of indices of C∗-equivariant coherent sheaves, Hecke
    transformation for Higgs bundles, relative Fourier–Mukai transform along the Hitchin
    fibration, hyperholomorphic structures on universal bundles and cominuscule Higgs
    bundles.
acknowledgement: We would like to thank Brian Collier, Davide Gaiotto, Peter Gothen,
  Jochen Heinloth, Daniel Huybrechts, Quoc Ho, Joel Kamnitzer, Gérard Laumon, Luca
  Migliorini, Alexander Minets, Brent Pym, Peng Shan, Carlos Simpson, András Szenes,
  Fernando R. Villegas, Richard Wentworth, Edward Witten and Kōta Yoshioka for interesting
  comments and discussions. Most of all we are grateful for a long list of very helpful
  comments by the referee. We would also like to thank the organizers of the Summer
  School on Higgs bundles in Hamburg in September 2018, where the authors and Richard
  Wentworth were giving lectures and where the work in this paper started by considering
  the mirror of the Lagrangian upward flows W+E investigated in [17]. The second author
  wishes to thank EPSRC and ICMAT for support. 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: Tamás
  full_name: Hausel, Tamás
  id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
  last_name: Hausel
- first_name: Nigel
  full_name: Hitchin, Nigel
  last_name: Hitchin
citation:
  ama: Hausel T, Hitchin N. Very stable Higgs bundles, equivariant multiplicity and
    mirror symmetry. <i>Inventiones Mathematicae</i>. 2022;228:893-989. doi:<a href="https://doi.org/10.1007/s00222-021-01093-7">10.1007/s00222-021-01093-7</a>
  apa: Hausel, T., &#38; Hitchin, N. (2022). Very stable Higgs bundles, equivariant
    multiplicity and mirror symmetry. <i>Inventiones Mathematicae</i>. Springer Nature.
    <a href="https://doi.org/10.1007/s00222-021-01093-7">https://doi.org/10.1007/s00222-021-01093-7</a>
  chicago: Hausel, Tamás, and Nigel Hitchin. “Very Stable Higgs Bundles, Equivariant
    Multiplicity and Mirror Symmetry.” <i>Inventiones Mathematicae</i>. Springer Nature,
    2022. <a href="https://doi.org/10.1007/s00222-021-01093-7">https://doi.org/10.1007/s00222-021-01093-7</a>.
  ieee: T. Hausel and N. Hitchin, “Very stable Higgs bundles, equivariant multiplicity
    and mirror symmetry,” <i>Inventiones Mathematicae</i>, vol. 228. Springer Nature,
    pp. 893–989, 2022.
  ista: Hausel T, Hitchin N. 2022. Very stable Higgs bundles, equivariant multiplicity
    and mirror symmetry. Inventiones Mathematicae. 228, 893–989.
  mla: Hausel, Tamás, and Nigel Hitchin. “Very Stable Higgs Bundles, Equivariant Multiplicity
    and Mirror Symmetry.” <i>Inventiones Mathematicae</i>, vol. 228, Springer Nature,
    2022, pp. 893–989, doi:<a href="https://doi.org/10.1007/s00222-021-01093-7">10.1007/s00222-021-01093-7</a>.
  short: T. Hausel, N. Hitchin, Inventiones Mathematicae 228 (2022) 893–989.
date_created: 2022-01-30T23:01:34Z
date_published: 2022-05-01T00:00:00Z
date_updated: 2023-08-02T14:03:20Z
day: '01'
ddc:
- '510'
department:
- _id: TaHa
doi: 10.1007/s00222-021-01093-7
external_id:
  arxiv:
  - '2101.08583'
  isi:
  - '000745495400001'
file:
- access_level: open_access
  checksum: a382ba75acebc9adfb8fe56247cb410e
  content_type: application/pdf
  creator: dernst
  date_created: 2023-02-27T07:30:47Z
  date_updated: 2023-02-27T07:30:47Z
  file_id: '12687'
  file_name: 2022_InventionesMahtematicae_Hausel.pdf
  file_size: 1069538
  relation: main_file
  success: 1
file_date_updated: 2023-02-27T07:30:47Z
has_accepted_license: '1'
intvolume: '       228'
isi: 1
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
page: 893-989
project:
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
  name: IST Austria Open Access Fund
publication: Inventiones Mathematicae
publication_identifier:
  eissn:
  - 1432-1297
  issn:
  - 0020-9910
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
  link:
  - description: News on the ISTA Website
    relation: press_release
    url: https://ista.ac.at/en/news/the-tip-of-the-mathematical-iceberg/
scopus_import: '1'
status: public
title: Very stable Higgs bundles, equivariant multiplicity and mirror symmetry
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: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 228
year: '2022'
...
---
_id: '7901'
abstract:
- lang: eng
  text: We derive rigorously the leading order of the correlation energy of a Fermi
    gas in a scaling regime of high density and weak interaction. The result verifies
    the prediction of the random-phase approximation. Our proof refines the method
    of collective bosonization in three dimensions. We approximately diagonalize an
    effective Hamiltonian describing approximately bosonic collective excitations
    around the Hartree–Fock state, while showing that gapless and non-collective excitations
    have only a negligible effect on the ground state energy.
acknowledgement: We thank Christian Hainzl for helpful discussions and a referee for
  very careful reading of the paper and many helpful suggestions. NB and RS were supported
  by the European Research Council (ERC) under the European Union’s Horizon 2020 research
  and innovation programme (grant agreement No. 694227). Part of the research of NB
  was conducted on the RZD18 Nice–Milan–Vienna–Moscow. NB thanks Elliott H. Lieb and
  Peter Otte for explanations about the Luttinger model. PTN has received funding
  from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
  Germany’s Excellence Strategy (EXC-2111-390814868). MP acknowledges financial support
  from the European Research Council (ERC) under the European Union’s Horizon 2020
  research and innovation programme (ERC StG MaMBoQ, grant agreement No. 802901).
  BS gratefully acknowledges financial support from the NCCR SwissMAP, from the Swiss
  National Science Foundation through the Grant “Dynamical and energetic properties
  of Bose-Einstein condensates” and from the European Research Council through the
  ERC-AdG CLaQS (grant agreement No. 834782). All authors acknowledge support for
  workshop participation from Mathematisches Forschungsinstitut Oberwolfach (Leibniz
  Association). NB, PTN, BS, and RS acknowledge support for workshop participation
  from Fondation des Treilles.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Niels P
  full_name: Benedikter, Niels P
  id: 3DE6C32A-F248-11E8-B48F-1D18A9856A87
  last_name: Benedikter
  orcid: 0000-0002-1071-6091
- first_name: Phan Thành
  full_name: Nam, Phan Thành
  last_name: Nam
- first_name: Marcello
  full_name: Porta, Marcello
  last_name: Porta
- first_name: Benjamin
  full_name: Schlein, Benjamin
  last_name: Schlein
- first_name: Robert
  full_name: Seiringer, Robert
  id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
  last_name: Seiringer
  orcid: 0000-0002-6781-0521
citation:
  ama: Benedikter NP, Nam PT, Porta M, Schlein B, Seiringer R. Correlation energy
    of a weakly interacting Fermi gas. <i>Inventiones Mathematicae</i>. 2021;225:885-979.
    doi:<a href="https://doi.org/10.1007/s00222-021-01041-5">10.1007/s00222-021-01041-5</a>
  apa: Benedikter, N. P., Nam, P. T., Porta, M., Schlein, B., &#38; Seiringer, R.
    (2021). Correlation energy of a weakly interacting Fermi gas. <i>Inventiones Mathematicae</i>.
    Springer. <a href="https://doi.org/10.1007/s00222-021-01041-5">https://doi.org/10.1007/s00222-021-01041-5</a>
  chicago: Benedikter, Niels P, Phan Thành Nam, Marcello Porta, Benjamin Schlein,
    and Robert Seiringer. “Correlation Energy of a Weakly Interacting Fermi Gas.”
    <i>Inventiones Mathematicae</i>. Springer, 2021. <a href="https://doi.org/10.1007/s00222-021-01041-5">https://doi.org/10.1007/s00222-021-01041-5</a>.
  ieee: N. P. Benedikter, P. T. Nam, M. Porta, B. Schlein, and R. Seiringer, “Correlation
    energy of a weakly interacting Fermi gas,” <i>Inventiones Mathematicae</i>, vol.
    225. Springer, pp. 885–979, 2021.
  ista: Benedikter NP, Nam PT, Porta M, Schlein B, Seiringer R. 2021. Correlation
    energy of a weakly interacting Fermi gas. Inventiones Mathematicae. 225, 885–979.
  mla: Benedikter, Niels P., et al. “Correlation Energy of a Weakly Interacting Fermi
    Gas.” <i>Inventiones Mathematicae</i>, vol. 225, Springer, 2021, pp. 885–979,
    doi:<a href="https://doi.org/10.1007/s00222-021-01041-5">10.1007/s00222-021-01041-5</a>.
  short: N.P. Benedikter, P.T. Nam, M. Porta, B. Schlein, R. Seiringer, Inventiones
    Mathematicae 225 (2021) 885–979.
date_created: 2020-05-28T16:48:20Z
date_published: 2021-05-03T00:00:00Z
date_updated: 2023-08-21T06:30:30Z
day: '03'
ddc:
- '510'
department:
- _id: RoSe
doi: 10.1007/s00222-021-01041-5
ec_funded: 1
external_id:
  arxiv:
  - '2005.08933'
  isi:
  - '000646573600001'
file:
- access_level: open_access
  checksum: f38c79dfd828cdc7f49a34b37b83d376
  content_type: application/pdf
  creator: dernst
  date_created: 2022-05-16T12:23:40Z
  date_updated: 2022-05-16T12:23:40Z
  file_id: '11386'
  file_name: 2021_InventMath_Benedikter.pdf
  file_size: 1089319
  relation: main_file
  success: 1
file_date_updated: 2022-05-16T12:23:40Z
has_accepted_license: '1'
intvolume: '       225'
isi: 1
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
page: 885-979
project:
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
  name: IST Austria Open Access Fund
- _id: 25C6DC12-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '694227'
  name: Analysis of quantum many-body systems
publication: Inventiones Mathematicae
publication_identifier:
  eissn:
  - 1432-1297
  issn:
  - 0020-9910
publication_status: published
publisher: Springer
quality_controlled: '1'
scopus_import: '1'
status: public
title: Correlation energy of a weakly interacting Fermi gas
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: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 225
year: '2021'
...
---
_id: '8519'
article_processing_charge: No
article_type: original
author:
- first_name: Vadim
  full_name: Kaloshin, Vadim
  id: FE553552-CDE8-11E9-B324-C0EBE5697425
  last_name: Kaloshin
  orcid: 0000-0002-6051-2628
citation:
  ama: Kaloshin V. The existential Hilbert 16-th problem and an estimate for cyclicity
    of elementary polycycles. <i>Inventiones mathematicae</i>. 2003;151(3):451-512.
    doi:<a href="https://doi.org/10.1007/s00222-002-0244-9">10.1007/s00222-002-0244-9</a>
  apa: Kaloshin, V. (2003). The existential Hilbert 16-th problem and an estimate
    for cyclicity of elementary polycycles. <i>Inventiones Mathematicae</i>. Springer
    Nature. <a href="https://doi.org/10.1007/s00222-002-0244-9">https://doi.org/10.1007/s00222-002-0244-9</a>
  chicago: Kaloshin, Vadim. “The Existential Hilbert 16-Th Problem and an Estimate
    for Cyclicity of Elementary Polycycles.” <i>Inventiones Mathematicae</i>. Springer
    Nature, 2003. <a href="https://doi.org/10.1007/s00222-002-0244-9">https://doi.org/10.1007/s00222-002-0244-9</a>.
  ieee: V. Kaloshin, “The existential Hilbert 16-th problem and an estimate for cyclicity
    of elementary polycycles,” <i>Inventiones mathematicae</i>, vol. 151, no. 3. Springer
    Nature, pp. 451–512, 2003.
  ista: Kaloshin V. 2003. The existential Hilbert 16-th problem and an estimate for
    cyclicity of elementary polycycles. Inventiones mathematicae. 151(3), 451–512.
  mla: Kaloshin, Vadim. “The Existential Hilbert 16-Th Problem and an Estimate for
    Cyclicity of Elementary Polycycles.” <i>Inventiones Mathematicae</i>, vol. 151,
    no. 3, Springer Nature, 2003, pp. 451–512, doi:<a href="https://doi.org/10.1007/s00222-002-0244-9">10.1007/s00222-002-0244-9</a>.
  short: V. Kaloshin, Inventiones Mathematicae 151 (2003) 451–512.
date_created: 2020-09-18T10:49:26Z
date_published: 2003-03-01T00:00:00Z
date_updated: 2021-01-12T08:19:50Z
day: '01'
doi: 10.1007/s00222-002-0244-9
extern: '1'
intvolume: '       151'
issue: '3'
keyword:
- General Mathematics
language:
- iso: eng
month: '03'
oa_version: None
page: 451-512
publication: Inventiones mathematicae
publication_identifier:
  issn:
  - 0020-9910
  - 1432-1297
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
status: public
title: The existential Hilbert 16-th problem and an estimate for cyclicity of elementary
  polycycles
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 151
year: '2003'
...