## Grant 0053/2011 "Quantum invariants in
hyperbolic geometry"

Identifier: PN-II-RU-TE-2011-3-0053

Period: 2012-2014.

Financed by: ANCS-UEFISCDI contract 7/05.10.2011.

Budget: 705.000 lei

Grant director: Sergiu
Moroianu

Team members:
Vicentiu Pasol (Experienced Researcher), since August 2013.
Alexandra Otiman (Master student, SNSB), since November 2012.
Nicolina Istrati (Master student, SNSB), since November 2012.

Abstract.
Hyperbolic manifolds are Riemannian manifolds of constant sectional curvature equal to -1.
In dimensions 2 and 3 they play a fundamental role in the classification of smooth manifolds,
as well as in complex analysis. Roughly speaking, most manifolds in these dimensions
admit hyperbolic metrics. The moduli spaces of these metrics are some of the most
intriguing objects in mathematics,
and the current project aims to contribute to their understanding.

Objectives:

Computing the variation of the eta invariant and of the analytic torsion for geometrically finite hyperbolic 3-manifolds.

Proving that the renormalized volume of geometrically finite hyperbolic 3-manifolds is a Kahler potential
for the Weil-Petersson symplectic form on Teichmuller space.

Studying holomorphic bundles like the Chern-Simons and the determinant bundle over the Teichmuller space.

Analyzing the degeneration of these invariants towards the boundary of Teichmuller space.

Activity 2012

Raport 2012

*Publications*

Colin Guillarmou, Sergiu Moroianu, and Jinsung Park, *Bergman and Calderon projectors for Dirac
operators*, Journal of Geometric Analysis 24, no. 1 (2014), 298-336.

*Invited talks
*

S Moroianu, IHP Paris, Mar. 2012.

*Talks at conferences*
S Moroianu, Analysis and Geometric Singularities, Oberwolfach, May 2012.
S Moroianu, Analysis, Geometry and Surfaces, Autrans, Jan. 2012.

Activity 2013

Raport 2013

*Publications*

Bernd Ammann, Andrei Moroianu, Sergiu Moroianu, *The Cauchy problems for Einstein metrics and parallel spinors*,
Commun. Math. Phys. 320, 173-198 (2013).
Florin P. Boca, Vicentiu Pasol, Alexandru A. Popa, Alexandru Zaharescu,
*Pair corellation of angles between reciprocal geodesics on the modular surface*,
Algebra and Number Theory 8 (2014), no. 4, 999-1035.
Colin Guillarmou, Sergiu Moroianu, *Chern-Simons line bundle on Teichmuller space*, Geometry & Topology 18 (2014), 327-377.

*Invited talks
*

S Moroianu, IMAR Bucharest, Feb. 2013.
S Moroianu, University of Bucharest, Jun. 2013.

*Talks at conferences*
S Moroianu, Conference in honor of Cabiria Andreian-Cazacu, Bucharest, Feb. 2013.
S Moroianu, Workshop for Young Researchers in Mathematics, Constanta, May 2013.
S Moroianu, Bucharest Faculty of Mathematics 150 years conference, Aug. 2013
S Moroianu, CAIM 2013, University of Bucharest, Sept. 2013

*Participation in conferences*

A Otiman and N Istrati, Young Topologists Meeting, 8-12 July 2013, EPFL, Ecublens, Switzerland.

*Invited positions abroad*

V Pasol was an invited researcher during the month of October 2013 at the Max Plank Institute in Bonn, Germany.

Activity 2014

Raport sintetic 2014 (Romanian),
Final scientific report (english)

*Publications*

Alexandra Otiman, *Currents on locally conformally Kahler manifolds*, Journal of Geometry and Physics 86 (2014), 564-570
Corina Ciobotaru, Sergiu Moroianu, *Positivity of the renormalized volume of almost-Fuchsian hyperbolic 3-manifolds*, preprint arXiv:1402.2555, to appear in the Proceedings of the American Mathematical Society.

*Conferences organized
*

S Moroianu, Workshop for Young Researchers in Mathematics, Constanta, 22-23 mai 2014.
S Moroianu, Real and Complex Differential Geometry, University of Bucharest, 8-12 september 2014.

*Talks at conferences*
V Pasol, Workshop for Young Researchers in Mathematics, Constanta, 22-23 mai 2014.
V Pasol, p-adic Variation in Number Theory, Boston, June 2-6, 2014.
S Moroianu, Geometrie spinorielle et analyse sur les varietes, Marsilia, October 6-10, 2014.

*Invited talks
*

S Moroianu, IMAR Bucharest, March 2014.
N Istrati, IMAR Bucharest, June 2014.
V Pasol, Alfred Renyi Institute of Mathematics of the Hungarian Academy of
Sciences, Sept. 2, 2014: "Multiple Dirichlet series".

*Participation in conferences*

A Otiman, Workshop for Young Researchers in Mathematics, Constanta, 22-23 mai 2014.
A Otiman, Cortona Summer School "An introduction to Calabi extremal Kahler metric", July 21 - Aug. 2, 2014.
A Otiman and N Istrati, Young Topologists Meeting, June 30 - July 4, 2014, Copenhagen, Denmark.
A Otiman and N Istrati, Real and Complex Differential Geometry, University of Bucharest, 8-12 september 2014.

*Modificat: 28 noiembrie 2014.*