## PhD research topic - Gabriele Mondello

#### Research area

Differential geometry, algebraic geometry

#### Keywords

Riemann surfaces, geometric structures, moduli spaces

#### Research topics

Riemann surfaces are at the crossroad of several topics: differential geometry, complex algebraic geometry, low-dimensional topology, sometimes non-linear analysis.

A geometric structure (say, on a surface) is an atlas with values in a homogeneous space G/H, with transition functions with values in the group G.

Examples of such geometric structures are: metrics of constant curvature, affine structures, translation structures, foliations, complex structures, complex-projective structures, real-projective structures and so on. These structures often occur "in nature".

Broadly speaking, the main focus of my current research is to study families of such geometric structures: their local invariants and their global moduli spaces. In particular, I am also interested in topological, differential-geometric and algebro-geometric properties of such moduli spaces.

For example, the topology is such moduli spaces can be studied by the aid of geometrically meaningful cellularizations, vanishing results in cohomology can be obtained through Morse theory by using geometrically meaningful functions, volumes of moduli spaces can be recursively computed by chopping them in (geometrically meaningful) pieces.

#### Bibliography

[1] William Goldman, Two papers which changed my life: Milnor's seminal work on flat manifolds and flat bundles, "Frontiers in Complex Dynamics" in celebration of John Milnor's 80th birthday (Banff International Research Station, 24 February 2011)

[2] William Goldman, Locally homogeneous geometric manifolds, Proceedings of the 2010 International Congress of Mathematicians, Hyderabad, India (2010), 717--744, Hindustan Book Agency, New Delhi, India

[3] William Goldman, Higgs Bundles and Geometric Structures on Surfaces, "The Many Facets of Geometry: a Tribute to Nigel Hitchin," Bourgignon, Garcia-Prada & Salamon, eds, Oxford Univ. Press

[4] François Labourie, Lectures on representation of surface groups, EMS Zurich Lectures in Advanced Mathematics (2013), Vol.17

[5] Gabriele Mondello, Stratifications of the moduli space of curves and related questions, Rendiconti di Matematica, Serie VII, 35 (2014), pp. 131-158