Research

Publications

Finite approximation and continuous change in spectral geometry

AB Stern PhD thesis, Radboud University, 2021.

Schatten classes for Hilbert modules over commutative C⁎-algebras

AB Stern, WD van Suijlekom Journal of Functional Analysis 281 (4), 109042 (2021).

BibLaTeX (click to expand)

@article{Stern.vanSuijlekomSchattenClassesHilbert2021,
      title = {Schatten Classes for {{Hilbert}} Modules over Commutative {{C}}⁎-Algebras},
      author = {Stern, Abel B. and family=Suijlekom, given=Walter D., prefix=van, useprefix=true},
      date = {2021-08-15},
      journaltitle = {Journal of Functional Analysis},
      shortjournal = {J. Funct. Anal.},
      volume = {281},
      pages = {109042},
      issn = {0022-1236},
      doi = {10.1016/j.jfa.2021.109042},
      url = {https://www.sciencedirect.com/science/article/pii/S0022123621001245},
      urldate = {2021-07-15},
      abstract = {We define Schatten classes of adjointable operators on Hilbert modules over abelian C⁎-algebras. Many key features carry over from the Hilbert space case. In particular, the Schatten classes form two-sided ideals of compact operators and are equipped with a Banach norm and a C⁎-valued trace with the expected properties. For trivial Hilbert bundles, we show that our Schatten-class operators can be identified bijectively with Schatten-norm–continuous maps from the base space into the Schatten classes on the Hilbert space fiber, with the fiberwise trace. As applications, we introduce the C⁎-valued Fredholm determinant and operator zeta functions, and propose a notion of p-summable unbounded Kasparov cycles in the commutative setting.},
      keywords = {Hilbert modules,Noncommutative geometry,Schatten classes},
      langid = {english},
      number = {4}
    }

Reconstructing manifolds from truncations of spectral triples

L Glaser, AB Stern Journal of Geometry and Physics 159, 103921 (2021).

BibLaTeX (click to expand)

@article{Glaser.SternReconstructingManifoldsTruncations2021,
      title = {Reconstructing Manifolds from Truncations of Spectral Triples},
      author = {Glaser, Lisa and Stern, Abel B.},
      date = {2021-01-01},
      journaltitle = {Journal of Geometry and Physics},
      shortjournal = {J. Geom. Phys.},
      volume = {159},
      pages = {103921},
      issn = {0393-0440},
      doi = {10.1016/j.geomphys.2020.103921},
      url = {https://www.sciencedirect.com/science/article/pii/S0393044020302138},
      urldate = {2021-07-15},
      abstract = {We explore the geometric implications of introducing a spectral cut-off on compact Riemannian manifolds. This is naturally phrased in the framework of non-commutative geometry, where we work with spectral triples that are truncated by spectral projections of Dirac-type operators. We associate a metric space of ‘localized’ states to each truncation. The Gromov–Hausdorff limit of these spaces is then shown to equal the underlying manifold one started with. This leads us to propose a computational algorithm that allows us to approximate these metric spaces from the finite-dimensional truncated spectral data. We subsequently develop a technique for embedding the resulting metric graphs in Euclidean space to asymptotically recover an isometric embedding of the limit. We test these algorithms on the truncation of sphere and a recently investigated perturbation thereof.},
      keywords = {Connes distance,Graph embedding,Noncommutative geometry,Operator systems,Spectral geometry,Spectral truncation},
      langid = {english}
    }

Understanding truncated non-commutative geometries through computer simulations (arXiv)

L Glaser, AB Stern Journal of Mathematical Physics 61, 033507 (2020).

BibLaTeX (click to expand)

@article{MR4076626,
      ids = {Glaser.SternUnderstandingTruncatedNoncommutative2020},
      title = {Understanding Truncated Non-Commutative Geometries through Computer Simulations},
      author = {Glaser, L. and Stern, A. B.},
      date = {2020},
      journaltitle = {Journal of Mathematical Physics},
      shortjournal = {J. Math. Phys.},
      volume = {61},
      pages = {033507, 11},
      publisher = {{American Institute of Physics}},
      issn = {0022-2488},
      doi = {10.1063/1.5131864},
      url = {https://doi-org.ru.idm.oclc.org/10.1063/1.5131864},
      mrclass = {58B34 (81T75)},
      mrnumber = {4076626},
      number = {3}
    }

Finite-rank approximations of spectral zeta residues

AB Stern Letters in Mathematical Physics 109 (3), 565-577 (2019).

BibLaTeX (click to expand)

@article{MR3910135,
      ids = {SternFiniterankApproximationsSpectral2019a},
      title = {Finite-Rank Approximations of Spectral Zeta Residues},
      author = {Stern, Abel B.},
      date = {2019},
      journaltitle = {Letters in Mathematical Physics},
      shortjournal = {Lett. Math. Phys.},
      volume = {109},
      pages = {565--577},
      issn = {0377-9017},
      doi = {10.1007/s11005-018-1117-5},
      url = {https://doi-org.ru.idm.oclc.org/10.1007/s11005-018-1117-5},
      fjournal = {Letters in Mathematical Physics},
      mrclass = {58J42 (35K08 35P20 58B34)},
      mrnumber = {3910135},
      number = {3}
    }

NSR superstring measures in genus 5 (arXiv)

P Dunin-Barkowski, A Sleptsov, AB Stern Nuclear Physics B 872 (1), 106-126 (2013).

BibLaTeX (click to expand)

@article{Dunin-Barkowski.Sleptsov.eaNSRSuperstringMeasures2013,
      title = {{{NSR}} Superstring Measures in Genus 5},
      author = {Dunin-Barkowski, Petr and Sleptsov, Alexey and Stern, Abel},
      date = {2013-07-01},
      journaltitle = {Nuclear Physics B},
      volume = {872},
      pages = {106--126},
      issn = {0550-3213},
      doi = {10.1016/j.nuclphysb.2013.03.008},
      url = {https://www.sciencedirect.com/science/article/pii/S0550321313001521},
      urldate = {2021-07-15},
      abstract = {Currently there are two proposed ansätze for NSR superstring measures: the Grushevsky ansatz and the OPSMY ansatz, which for genera g⩽4 are known to coincide. However, neither the Grushevsky nor the OPSMY ansatz leads to a vanishing two-point function in genus four, which can be constructed from the genus five expressions for the respective ansätze. This is inconsistent with the known properties of superstring amplitudes. In the present paper we show that the Grushevsky and OPSMY ansätze do not coincide in genus five. Then, by combining these ansätze, we propose a new ansatz for genus five, which now leads to a vanishing two-point function in genus four. We also show that one cannot construct an ansatz from the currently known forms in genus 6 that satisfies all known requirements for superstring measures.},
      keywords = {Lattice theta series,NSR measures,Riemann theta constants,Siegel modular forms,Superstring theory},
      langid = {english},
      number = {1}
    }