Geometric structures on manifolds

• Christian El Emam (Torino) - Complex minimal surfaces and rank-2 Hitchin components.

 Abstract: Let S be a closed oriented surface. Classic Teichmüller theory builds a bridge between different worlds - complex structures on S, hyperbolic Riemannian metrics on S, and Fuchsian representations of the fundamental group of S into PSL(2,R) - with a deep interplay among these perspectives. In recent decades, an active line of research aims to generalize this picture to higher-rank Lie groups G, in particular by replacing Fuchsian representations with the Hitchin component Hit(G), a distinguished connected component of the character variety defined for G simple and split (e.g. SL(n,R), SO(n,n), ...). In this setting, the interplay with the study of minimal immersions into locally symmetric spaces has proved to be very successful. Hitchin constructed a correspondence between equivariant minimal immersions into the symmetric space of G with holonomy in Hit(G), and some holomorphic forms on Riemann surfaces. Moreover, for G of rank 2 (e.g. SL(3,R)), Labourie proved that the holonomy map determines a diffeomorphism between such minimal immersions (up to isotopy) and Hit(G). Together, these results give Hit(G) a complex structure.

In this talk, we show that this complex structure on Hit(G) is compatible with the Goldman symplectic form - a natural symplectic form on character varieties - defining a pseudo-Kähler structure. This compatibility is closely connected to a generalization of Bers' simultaneous uniformization Theorem for Hit(G), which will also be discussed. The main tool is a notion of "complex minimal immersion" in holomorphic symmetric spaces, whose role in this framework will be explained during the talk. This is joint work with Nathaniel Sagman.

• Suzanne Schlich (Torino) - Bowditch representations in Gromov-hyperbolic spaces. 

Abstract: Bowditch, followed by Tan-Wong-Zhang, introduced in 1998 a class of representations of the once-punctured torus group into PSL(2,C). Using trace relations in PSL(2,C), they give a condition on a representation which ensures that the translation lengths of the images of simple closed curves grow linearly with respect to the word length. 

In this talk, I will explain how to define and study a generalization of these conditions in the context of Gromov-hyperbolic spaces, where no trace relations hold, and give several characterizations of this set. I will also discuss the dynamics of the mapping class group on the space of Bowditch representations. 

 Abstract:

• Andrea Tamburelli (Pisa) - On the topology of the domain of discontinuity for SL(n,R)-Hitchin representations acting on flag manifolds.

Abstract: By work of Guichard-Wienhard and Kapovich-Leeb-Porti, an SL(n,R)-Hitchin representation of a surface group acts properly discontinuously on a domain \Omega in the partial flag manifold of line-hyperplane pairs, which turns out to be a fiber bundle over the universal cover of the surface.  In a joint work in progress with Parker Evans, we use recent techniques developed by Colin Davalo and Parker Evans to establish some topological properties of the fiber of this bundle. This leads to a complete description up to (almost)-diffeomorphisms if n is even and up to connected sums with rational homology spheres if n is odd.

• Sara Maloni (University of Virginia) - Geometric Structures in Higher Teichmüller spaces.

Abstract: The Teichmüller space of a surface S is the deformation space of marked hyperbolic structures. This can be viewed as a component of representations of the fundamental group π1(S) into the isometry group of hyperbolic space. Higher Teichmüller Theory generalizes this idea by studying representations of surface groups into Lie groups of higher rank.

In the first part of the talk, we will introduce key ideas in the theory of deformations of geometric structures, (higher) Teichmüller theory,  and Anosov representations. In the second part of the talk, we will present the work of Guichard-Wienhard and Kapovich-Leeb-Porti that helps relating Anosov representations to deformations of geometric structures. We'll also discuss recent work with Alessandrini, Tholozan, and Wienhard (and independently obtained also by Davalo) and of my PhD student Mason Hart, on the topology of such geometric structures.

How to get there: Bologna has both an international airport ("Guglielmo Marconi") and a (fast) railway station. The airport is connected to the station through a (fast but expensive: 12.80€ single and 23.30€ return) monorail that runs every ~10 minutes from 5:40 am to 12 am; you can pay contactless.

From the station to the department (Piazza di Porta San Donato, 5) you can either walk (~25 minutes), hire a bike, or take a bus (e.g.  bus n. 32; you can pay contactless).

Aula Pincherle is on the 2nd floor. Once you enter the department, the elevators are in front of you, stairs are right before on your right.

Where to eat: There is a canteen ("ristorante universitario") close to the department. Other quick options include pizza al taglio ("I Gemelli") and a Sicilian rotisserie on via Zamboni, or a self-service restaurant ("Le Salentine") on via San Donato. Some restaurants (e.g. Baracca e Burattini and Sartoria Gastronomica) offer a lunch menu (main + water + coffee = 13-15€). Vegan restaurants include Malerba and Estravagario). There are several other options on via Petroni and Piazza Aldrovandi.

Typical food (is quite heavy and based on meat, mostly pork) include tigelle e crescentine with fresh cheese (squacquerone) and cold cuts (mortadella, prosciutto crudo, etc.), tagliatelle al ragù, lasagne verdi, tortellini in brodo or alla crema di parmigiano, gramigna alla salsiccia, cotoletta petroniana. There are several delicious ice cream shops.

What to do: Ask the locals! 

• Visit the historical city centre (Piazza Maggiore, le due torri, Piazza Santo Stefano, Piazza Malpighi). 

• Take a walk to the sanctuary at San Luca. 

• Explore the gardens and parks: orto botanico, giardino del Guasto, Palazzo Hercolani, giardini Margherita, Montagnola, parco di Villa Angeletti.