Mini-courses

Nguyen-Thi Dang - Dynamics of the Weyl chamber flow (over a higher rank locally symmetric space)

Qiongling Li - Higgs bundle and minimal surfaces in non-compact symmetric space

A Higgs bundle over a Riemann surface X equipped with a harmonic metric is called a harmonic bundle. Conformal harmonic bundles over a Riemann surface X correspond to equivariant minimal branched immersion from the universal cover of X to the symmetric space associated to GL(n,C).  Our plan of the mini-course is as follows: Part I explains the explicit correspondence between conformal harmonic bundles with minimal surfaces in the symmetric space associated to GL(n,C). Part II provides various interesting examples of minimal surfaces in the product of symmetric space of non-compact type and the Euclidean space in terms of harmonic bundles. Part III discusses further developments on the topics like Labourie conjecture, Morse index, total curvature and so on.

James Farre - Convex pleated surfaces

A quasi-Fuchsian surface group is a discrete, convex co-compact surface subgroup of PSL(2,C), which acts isometrically on hyperbolic 3-space.  The boundary of the convex core of a quasi-Fuschian surface group has the intrinsic structure of a hyperbolic surface.  This hyperbolic surface is bent along a family of geodesic lines that form a geodesic lamination, and the bending angle defines a transverse measure on this lamination.  These convex surfaces, embedded in a complete hyperbolic 3-manifold, are examples of Thurston’s pleated surfaces. 

Assuming a bit of familiarity with basic hyperbolic geometry, will give a crash course on (measured) geodesic laminations on closed hyperbolic surfaces and then build quasi-Fuchsian surface groups by bending a totally geodesic plane in hyperbolic space along a lamination in a group equivariant way.

If time permits, we will also explain how to bend convex projective structures on closed surfaces along a geodesic lamination in 3-dimensional real projective space to obtain certain convex co-compact surface subgroups of SL(4,R). 

Bram Petri - Probabilistic methods in hyperbolic geometry