Dr. Mehran Seyed Hosseini

wissenschaftlicher Mitarbeiter

Kontakt
Raum:
2.09.3.15
Telefon:
+49 331 977-1632


CV

 

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Index Theory, Positive Scalar Curvature, Coarse Geometry

2020 | A Variant of Roe Algebras for Spaces with Cylindrical Ends with Applications in Relative Higher Index Theory | Mehran SeyedhosseiniZeitschrift: To appear in Journal of Noncommutative GeometryLink zum Preprint

A Variant of Roe Algebras for Spaces with Cylindrical Ends with Applications in Relative Higher Index Theory

Autoren: Mehran Seyedhosseini (2020)

In this paper we define a variant of Roe algebras for spaces with cylindrical ends and use this to study questions regarding existence and classification of metrics of positive scalar curvature on such manifolds which are collared on the cylindrical end. We discuss how our constructions are related to relative higher index theory as developed by Chang, Weinberger, and Yu and use this relationship to define higher rho-invariants for positive scalar curvature metrics on manifolds with boundary. This paves the way for classification of these metrics. Finally, we use the machinery developed here to give a concise proof of a result of Schick and the author, which relates the relative higher index with indices defined in the presence of positive scalar curvature on the boundary.

Zeitschrift:
To appear in Journal of Noncommutative Geometry

2020 | On an Index Theorem of Chang, Weinberger and Yu | Thomas Schick, Mehran SeyedhosseiniZeitschrift: Münster Journal of MathematicsLink zur Publikation

On an Index Theorem of Chang, Weinberger and Yu

Autoren: Thomas Schick, Mehran Seyedhosseini (2020)

In this paper we prove a strengthening of a theorem of Chang, Weinberger and Yu on obstructions to the existence of positive scalar curvature metrics on compact manifolds with boundary. They construct a relative index for the Dirac operator, which lives in a relative K-theory group, measuring the difference between the fundamental group of the boundary and of the full manifold. Whenever the Riemannian metric has product structure and positive scalar curvature near the boundary, one can define an absolute index of the Dirac operator taking value in the K-theory of the C*-algebra of fundamental group of the full manifold. This index depends on the metric near the boundary. We prove that the relative index of Chang, Weinberger and Yu is the image of this absolute index under the canonical map of K-theory groups. This has the immediate corollary that positive scalar curvature on the whole manifold implies vanishing of the relative index, giving a conceptual and direct proof of the vanishing theorem of Chang, Weinberger, and Yu. To take the fundamental groups of the manifold and its boundary into account requires working with maximal C* completions of the involved *-algebras. A significant part of this paper is devoted to foundational results regarding these completions.

Zeitschrift:
Münster Journal of Mathematics

2020 | Relative torsion and bordism classes of positive scalar curvature metrics on manifolds with boundary | Simone Cecchini, Mehran Seyedhosseini, Vito Felice ZenobiLink zum Preprint

Relative torsion and bordism classes of positive scalar curvature metrics on manifolds with boundary

Autoren: Simone Cecchini, Mehran Seyedhosseini, Vito Felice Zenobi (2020)

In this paper, we define a relative \(L^2\)-\(\rho\)-invariant for Dirac operators on odd-dimensional spin manifolds with boundary and show that they are invariants of the bordism classes of positive scalar curvature metrics which are collared near the boundary. As an application, we show that if a \(4k+3\)-dimensional spin manifold with boundary admits such a metric and if, roughly speaking, there exists a torsion element in the difference of the fundamental groups of the manifold and its boundary, then there are infinitely many bordism classes of such psc metrics on the given manifold. This result in turn implies that the moduli-space of psc metrics on such manifolds has infinitely many path components. We also indicate how to define delocalised \(\eta\)-invariants for odd-dimensional spin manifolds with boundary, which could then be used to obtain similar results for \(4k+1\)-dimensional manifolds.

  • Riemannian Geometry, Vorlesung (Sommer 2022)
  • Einführung in die Differentialgeometrie, Vorlesung (Winter 2021)
  • Schrödinger Operators over Dynamical Systems II, Vorlesung (als Co-Dozent) (Sommer 2021)
  • Riemannian Geometry, Vorlesung (Sommer 2021)
  • Differentialgeometrie I, Vorlesung (Winter 2020)
  • Representation Theory of Lie Algebras, Seminar (Winter 2020)
  • Spectral Geometry, Vorlesung (Sommer 2020)
  • Baum-Connes Conjecture, Seminar (Mitorganisator) (Sommer 2020)
  • Groupoids and Index Theory, Seminar (Mitorganisator) (Winter 2019)
  • Morse Theory, Seminar (Mitorganisator) (Sommer 2019)
  • Geometry and Analysis of the Atiyah-Singer Index Theorem, Seminar (Mitorganisator) (Winter 2018)