Dr. Saskia Roos

wissenschaftliche Mitarbeiterin

Kontakt
Raum:
2.09.3.14
Telefon:
+49 331 977-1195
...

Sprechzeiten: Mittwochs 14:00 - 15:00

  • Differentialgeometrie
  • Globale Analysis
  • Spin Geometrie
  • Spektral Geometrie
  • Lorentzgeometrie

2020 | The Chiral Anomaly of the Free Fermion in Functorial Field Theory | Matthias Ludewig, Saskia Roos Zeitschrift: Ann. Henri Poincaré Link zur Publikation, Link zum Preprint

The Chiral Anomaly of the Free Fermion in Functorial Field Theory

Autoren: Matthias Ludewig, Saskia Roos (2020)

When trying to cast the free fermion in the framework of functorial field theory, its chiral anomaly manifests in the fact that it assigns the determinant of the Dirac operator to a top-dimensional closed spin manifold, which is not a number as expected, but an element of a complex line. In functorial field theory language, this means that the theory is twisted, which gives rise to an anomaly theory. In this paper, we give a detailed construction of this anomaly theory, as a functor that sends manifolds to infinite-dimensional Clifford algebras and bordisms to bimodules.

Zeitschrift:
Ann. Henri Poincaré

2019 | The Dirac operator under collapse to a smooth limit space | Saskia Roos Zeitschrift: Ann. Glob. Anal. Geom. Verlag: Springer Link zur Publikation, Link zum Preprint

The Dirac operator under collapse to a smooth limit space

Autoren: Saskia Roos (2019)

Let (Mi,gi)i∈ℕ be a sequence of spin manifolds with uniform bounded curvature and diameter that converges to a lower-dimensional Riemannian manifold (B, h) in the Gromov–Hausdorff topology. Then, it happens that the spectrum of the Dirac operator converges to the spectrum of a certain first-order elliptic differential operator DB on B. We give an explicit description of DB and characterize the special case where DB equals the Dirac operator on B.

Zeitschrift:
Ann. Glob. Anal. Geom.
Verlag:
Springer

2018 | Scalar curvature and the multiconformal class of a direct product Riemannian manifold | Nobuhiko Otoba, Saskia Roos Link zum Preprint

Scalar curvature and the multiconformal class of a direct product Riemannian manifold

Autoren: Nobuhiko Otoba, Saskia Roos (2018)

For a closed, connected direct product Riemannian manifold (M,g)=(M1×⋯×Ml,g1⊕⋯⊕gl), we define its multiconformal class [[g]] as the totality {f12g1⊕⋯⊕fl2gl} of all Riemannian metrics obtained from multiplying the metric gi of each factor Mi by a function fi2>0 on the total space M. A multiconformal class [[g]] contains not only all warped product type deformations of g but also the whole conformal class [g~] of every g~∈[[g]]. In this article, we prove that [[g]] carries a metric of positive scalar curvature if and only if the conformal class of some factor (Mi,gi) does, under the technical assumption dimMi≥2. We also show that, even in the case where every factor (Mi,gi) has positive scalar curvature, [[g]] carries a metric of scalar curvature constantly equal to −1 and with arbitrarily large volume, provided l≥2 and dimM≥3. In this case, such negative scalar curvature metrics within [[g]] for l=2 cannot be of any warped product type.

2018 | Dirac operators with W1,∞ -potential on collapsing sequences losing one dimension in the limit | Saskia Roos Zeitschrift: Manuscripta Mathematica Verlag: Springer Seiten: 1-24 Link zur Publikation, Link zum Preprint

Dirac operators with W1,∞ -potential on collapsing sequences losing one dimension in the limit

Autoren: Saskia Roos (2018)

We study the behavior of the spectrum of the Dirac operator together with a symmetric W1,∞-potential on a collapsing sequence of spin manifolds with bounded sectional curvature and diameter losing one dimension in the limit. If there is an induced spin or pin structure on the limit space N, then there are eigenvalues that converge to the spectrum of a first order differential operator D on N together with a symmetric W1,∞-potential. In the case of an orientable limit space N, D is the spin Dirac operator DN on N if the dimension of the limit space is even and if the dimension of the limit space is odd, then D=DN⊕−DN.

Zeitschrift:
Manuscripta Mathematica
Verlag:
Springer
Seiten:
1-24

2017 | A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter | Saskia Roos Zeitschrift: Journal of Geometric Analysis Verlag: Springer Seiten: 2707-2724 Band: 28, no. 3 Link zur Publikation, Link zum Preprint

A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter

Autoren: Saskia Roos (2017)

 Let M(n,D) be the space of closed n-dimensional Riemannian manifolds (M,g) with diam(M)≤D and |secM|≤1. In this paper we consider sequences (Mi,gi) in M(n,D) converging in the Gromov–Hausdorff topology to a compact metric space Y. We show, on the one hand, that the limit space of this sequence has at most codimension one if there is a positive number r such that the quotient vol(BMir(x))/injMi(x) can be uniformly bounded from below by a positive constant C(n, r, Y) for all points x∈Mi. On the other hand, we show that if the limit space has at most codimension one then for all positive r there is a positive constant C(n, r, Y) bounding the quotient vol(BMir(x))/injMi(x) uniformly from below for all x∈Mi. As a conclusion, we derive a uniform lower bound on the volume and a bound on the essential supremum of the sectional curvature for the closure of the space consisting of all manifolds in M(n,D) with C≤vol(M)/inj(M).

Zeitschrift:
Journal of Geometric Analysis
Verlag:
Springer
Seiten:
2707-2724
Band:
28, no. 3

2017 | Eigenvalue pinching on spin-c manifolds | Saskia Roos Zeitschrift: J. Geom. Phys. Verlag: Elsevier Seiten: 59-73 Band: 112 Link zur Publikation, Link zum Preprint

Eigenvalue pinching on spin-c manifolds

Autoren: Saskia Roos (2017)

We derive various pinching results for small Dirac eigenvalues using the classification of spinc and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for spinc manifolds which involves a general study on convergence of Riemannian manifolds with a principal S1-bundle. We also analyze the relation between the regularity of the Riemannian metric and the regularity of the curvature of the associated principal S1-bundle on spinc manifolds with Killing spinors.

Zeitschrift:
J. Geom. Phys.
Verlag:
Elsevier
Seiten:
59-73
Band:
112

2015 | Estimates for eigensections of Riemannian vector bundles | Saskia Roos Link zum Preprint

Estimates for eigensections of Riemannian vector bundles

Autoren: Saskia Roos (2015)

We derive a bound on the L-norm of the covariant derivative of Laplace eigensections on general Riemannian vector bundles depending on the diameter, the dimension, the Ricci curvature of the underlying manifold, and the curvature of the Riemannian vector bundle. Our result implies that eigensections with small eigenvalues are almost parallel.

Konferenzen

Seminare

  • 4. Februar 2020, The Dirac operator under collapse to a smooth manifold, Forschungsseminar, Radboud University, Nijmegen, Niederlande.
  • 11. Juni 2018, Characterization of codimension one collapse, Forschungsseminar für Geometrie,  Augsburg.
  • 6. Februar 2018, Dirac eigenvalues under codimension one collapse, Forschungsseminar für Geometrie und Topologie, Stuttgart.
  • 22. Januar 2018, Dirac eigenvalues under codimension one collapse, Forschugnsseminar für Geometrie, Münster.
  • 13. Dezember 2017, Dirac eigenvalues under codimension one collapse, Seminar für Differentialgeometrie, KIT, Karlsruhe.
  • 26. Oktober 2017, Characterization of codimension one collapse, Kolloquium der Graduiertenschule, Regensburg.
  • 19. Oktober 2017, Dirac eigenvalues under codimension one collapse, Forschungsseminar für Geometrie, Potsdam.
  • 9. Mai 2017, Collapse under bounded curvature, Geometrie Seminar,  Köln.
  • 25. April 2017, Collapse under bounded curvature, Oberseminar für Analysis und Zahlentheorie, Tübingen.