BrainMap

Mr. Sergiu Moroianu

CS 1 dr. habil.

Professor - INSTITUTUL DE MATEMATICA "SIMION STOILOW" AL ACADEMIEI ROMANE

Researcher

Curriculum Vitae (16/03/2020)

Expertise & keywords

Spectral methods in Hyperbolic Geometry Call name: P 4 - Proiecte de Cercetare Exploratorie, 2020
PN-III-P4-ID-PCE-2020-0794 2021 - 2023

Role in this project:

Coordinating institution: INSTITUTUL DE MATEMATICA "SIMION STOILOW" AL ACADEMIEI ROMANE

Project partners: INSTITUTUL DE MATEMATICA "SIMION STOILOW" AL ACADEMIEI ROMANE (RO)

Affiliation:

Project website: http://www.imar.ro/%7Esergium/0794/0794.html

Abstract:

The moduli space of Riemann surfaces is intimately linked to hyperbolic geometry. The Riemannian metric on this space is defined through the L^2 product with respect to the unique hyperbolic metric in the conformal class. Moreover, the Kähler form on the (regular part of the) moduli space has several Kähler potentials on the orbifold universal cover, the celebrated Teichmüller space. One of them is the determinant of the Laplacian with respect to the hyperbolic metric. Another one is the renormalized volume of a certain hyperbolic metric of infinite volume associated to a hyperbolic cobordism in dimension 2+1. We plan to extend this type of objects - and results - to the boundary of Teichmüller space. We plan to investigate the determinant of the Dirac operator on hyperbolic surfaces, its adiabatic limit (pinching a set of disjoint simple geodesics) and to construct in this way a Kähler potential which extends to some of the bounding Teichmüller spaces of punctured surfaces. The determinant of the Laplacian is known to become singular at every boundary face of the Teichmüller space corresponding to the pinching. We expect the Selberg trace formula to play a role in our analysis, but we also plan to use heavily algebras of pseudodifferential operators adapted to the adiabatic limit of hyperbolic surfaces, in order to control the adiabtic limit of Schwartz kernel of the complex powers of the Dirac operator.

Volumes of hyperbolic and Einstein manifolds Call name: P 4 - Proiecte de Cercetare Exploratorie
PN-III-P4-ID-PCE-2016-0330 2017 - 2019

Role in this project:

Coordinating institution: INSTITUTUL DE MATEMATICA "SIMION STOILOW" AL ACADEMIEI ROMANE

Project partners: INSTITUTUL DE MATEMATICA "SIMION STOILOW" AL ACADEMIEI ROMANE (RO)

Affiliation: INSTITUTUL DE MATEMATICA "SIMION STOILOW" AL ACADEMIEI ROMANE (RO)

Project website: http://www.imar.ro/~sergium/0330/0330.html

Abstract:

The current project aims to study, among other invariants of hyperbolic manifolds, the so-called renormalized volume of geometrically finite 3-manifolds, and in higher dimensions that of asymptotically hyperbolic Einstein manifolds. The renormalized volume depends on a choice of metric in the given conformal class on the ideal boundary Σ. If we choose this metric to be the unique hyperbolic metric (the Fuchsian uniformization) we get a well-defined functional on the Teichmüller space of the ideal boundary. There are two separate problems we want to attack: proving positivity properties of the renormalized volume, and proving that the renormalized volume extends to the boundary of Teichmüller space as a Kähler potential for the Weil-Petersson metric. The first result in this direction was due to Krasnov-Schlenker, who showed that the Hessian of the renormalized volume functional at the Fuchsian locus equals the Weil-Petersson inner product, hence it is positive definite. In a joint paper with C. Ciobotaru, the project leader showed that the renormalized volume is positive on the open set of almost-fuchsian manifolds. The Kähler potential property was proved for compact Σ and arbitrary geometrically finite X without cusps of rank 1 by Guillarmou and the project leader. We are also interested in the geometry and topology of compact hyperbolic manifolds, with a focus on volumes, analytic and Reidemeister torsions, and cohomological invariants. We propose here an in-depth analysis of the relationship between the volume as a geometric invariant, and twisted cohomology as a topological one.

Investigation of quantum and finite type invariants, applications in geometry and topology Call name: Projects for Young Research Teams - TE-2012 call
PN-II-RU-TE-2012-3-0492 2013 - 2016

Role in this project:

Coordinating institution: INSTITUTUL DE MATEMATICA "SIMION STOILOW" AL ACADEMIEI ROMANE

Project partners: INSTITUTUL DE MATEMATICA "SIMION STOILOW" AL ACADEMIEI ROMANE (RO)

Affiliation: INSTITUTUL DE MATEMATICA "SIMION STOILOW" AL ACADEMIEI ROMANE (RO)

Project website: http://www.imar.ro/~dcheptea/grantul_0492.html

Abstract:

The main set of open problems focusses around finding a comprehensive relation between quantum and finite-type invariants of homology cylinders, studying the topological consequences, when via this “translation” methods from one can be used in solving the problems that difficult for the other. Of particular importance to establishing this correspondence in a proper manner is to look not only at quantum topological problems, but also more generally at related quantum geometric problems. To achieve the aim to further the development in the field as much as possible to his forces, the project leader has identified several works by the two other team members that border on some of the open geometric problems needed for the goal, therefore this collaboration. To develop this work qualitatively, at the international research level, we seek to enhance the collaboration between the Geometry and Topology group at IMAR, Bucharest, and CQM in Aarhus, Uppsala, and Mathematics Department at Uppsala, Sweden.

Quantum invariants in hyperbolic geometry Call name: Projects for Young Research Teams - TE-2011 call
PN-II-RU-TE-2011-3-0053 2012 - 2014

Role in this project:

Coordinating institution: Institutul de Matematica Simion Stoilow al Academiei Romane

Project partners: Institutul de Matematica Simion Stoilow al Academiei Romane (RO)

Affiliation: Institutul de Matematica Simion Stoilow al Academiei Romane (RO)

Project website: http://www.imar.ro/~sergium/0053/0053.html

Abstract:

We are interested in the renormalized volume of geometrically finite hyperbolic manifolds of dimension 3 as a functional on the Teichmüller space of the conformal infinity of the funnels and rank-1 cusps of the manifold. We aim to prove that the renormalized volume is a Kähler potential for the Weil-Petersson metric, extending results known for the quasi-fuchsian and Schottky moduli spaces. We also want to define real and holomorphic Chern-Simons invariants in the infinite-volume case, and to provide an explicit isomorphism between the Chern-Simons line bundle and the determinant bundle over the Teichmüller space.

FILE DESCRIPTION	DOCUMENT
List of research grants as project coordinator or partner team leader	Download (30.86 kb) 16/03/2020
Significant R&D projects for enterprises, as project manager
R&D activities in enterprises
Peer-review activity for international programs/projects