Gianluca Faraco, Ph.D.

Research

Low-dimensional Topology and Geometry

Complex projective structures and Branched projective structures on surfaces, Translation structures on surfaces. Hyperbolic Geometry, hyperbolic structures and branched hyperbolic structures. Relationship between hyperbolic structures and their holonomy representation. Teichmüller theory and higher Teichmüller theory. Moduli spaces of geometric structures in low-dimension. Representation spaces of finite genertated groups into Lie groups and character varieties.

Dynamical Systems

Teichmüller dynamics. Ergodic theory. Mapping class group action on the character varieties and classification of the orbits.

A square-tiled surface along with a system of handle generators

Published papers

Complex projective structures with maximal number of Möbius transformations. Joint work with L. Ruffoni. In: Mathematische Nachrichten, Volume 292, Issue 6, (2019). Available here.
Distances on the Moduli space of Complex Projective Structures. Expositiones Mathematicae, Volume 38, Issue 4, (2020). Available online here.
Geometrization of purely hyperbolic representations in PSL(2,R). Advances in Geometry, Volume 21, Issue 1, (2021). Available online here.
Translation surfaces and periods of meromorphic differentials. Joint work with S. Chenakkod and S. Gupta. Preprint available here. Proc. London Math. Soc. (3) 2022, 124 478-557.
Modular orbits on the representation space of compact abelian Lie groups. Joint work with Y. Bouilly. Groups, Geometry and Dynamics vol 17, no 2, pagg 719-749.
On the distality and expansivity of certain maps on spheres. Joint work with M. Choudhuri and A. K. Yadav. To appear in Dynamical Systems. Preprint available here.
Monodromy of Schwarzian equations with regular singularities. Joint work with S. Gupta. Geometry & Topology (2025) . Available here.
Automorphism groups of certain branched structures on surfaces. Milan J. Math. (2024). https://doi.org/10.1007/s00032-024-00393-w. Preprint avialable here.
Period realization of meromorphic differentials with prescribed invariants. Joint work with Dawei Chen. Forum of Mathematics, Sigma Paper No. e90. Available here . Preprint available here
Counterexamples to the simple loop conjecture in higher-dimension. Mathematics Newletter

Preprints

Isoperiodic foliation of the stratum H(1,1;-2). Joint work with Guillaume Tahar and Yongquan Zhang. Preprint available here.
Tessellation of surfaces. Preprint available here. -- This survey was supposed to be published in Proceedings of The Year-Long Program on Triangle Groups, Belyi Uniformization, and Modularity, Volume 1. This volume appears to be not longer in publications.
Mapping class group orbit closures for Deroin-Tholozan representations. Joint work with Yohann Bouilly and Arnaud Maret. Preprint available here
Relative Period Realization of holomorphic differentials with prescribed invariants. Joint work with Dawei Chen. Preprint available here

A picture from my paper with G. Tahar and Y. Zhang

2057UB_FaracoGupta_Monodromy_of_Schwarzian_equations_with_regular_singularities.pdf

Please, come and look at this poster for a quick overview about my most recent collaboration with Subhojoy Gupta! This poster has been recently selected to appear at Short Communications Satellite 2022. Link here.

RGF_II-2.pdf

Please, come and look at this poster for a quick overview about my collaboration with Dawei Chen!

Collaborators and friends

Yohann Bouilly, University of Strasbourg
Dawei Chen, Boston College, USA
Manoj Choudhuri, IITRAM, India
Soumya Dey, KREA University, India
Camilla Felisetti, University of Modena and Reggio Emilia, Italy
Subhojoy Gupta, IISc Bangalore, India
Krishnendu Gongopadhyay, IISER Mohali, India
Arnaud Maret, University of Strasbourg
Debanjan Nandi, IISc Bangalore, India
Irene Pasquinelli, University of Bristol, UK
Lorenzo Ruffoni, Binghamton University , USA
Gunja Sachdeva, BITS Pilani, India
Alessio Savini, University of Milano-Bicocca, Italy
Maxim Smirnov
Alok Kumar Yadav, Sikkim University

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