Complex projective structures, or PSL( $2, \mathbb{C})$-opers, play a central role in the theory of uniformization of Riemann surfaces. A very natural generalization of this notion is to consider complex projective structures with ramification points. This gives rise to the notion of branched projective structure, which is much more flexible in many aspects. For example, any representation of a surface group with values in $\operatorname{PSL}(2, \mathbb{C})$ is obtained as the holonomy of a branched projective structure. We will show that one of the central properties of complex projective structures, namely the complex analytic structure of their moduli spaces, extends to the branched case.[-]

Starting with the Gauss-Bonnet formula : rigidity phenomena on bounded symmetric domains Ngaiming MOK (The University of Hong Kong, Hong Kong) Let $E$ be a compact Riemann surface of genus 1, and $Z$ be a compact Riemann surface of genus $≥ 2$. Then, every holomorphic map $f : E → Z$ is constant, as can be proven by contradiction by pulling back a nontrivial holomorphic differential on $Z$ which necessarily vanishes at some point. A metric version of the proof using the Gauss-Bonnet formula is more flexible, and a variation of the proof based on a Chern integral gives a Hermitian metric rigidity theorem, first established by the author in 1987 in the case of compact quotients $X\left\lceil := \Omega/\right\lceil$ of irreducible bounded symmetric domains $\mathrm{X}_{Γ} := \Omega/Γ$ of rank $≥ 2$ and then extended in the finite-volume case by To in 1989, which gives rigidity results on holomorphic maps from $X\lceil$ to Kähler manifolds of nonpositive holomorphic bisectional curvature, and geometric superrigidity results in the special cases of $Γ\G/K$ for $G/K$ of Hermitian type and of rank $≥ 2$ and for cocompact lattices $Γ ⊂ G$ via the use of harmonic maps and the $∂∂$-Bochner-Kodaira formula of Siu's in 1980. The Hermitian metric rigidity theorem was the starting point of the author's investigation on rigidityphenomena mostly on bounded symmetric domains $\Omega$ irreducible of rank $≥ 2$, but also, in the presence of irreducible lattices Γ ⊂ G := Aut0(Ω), on reducible $Omega$, and, for certain problems also on the rank-1 cases of n-dimensional complex unit balls Bn. The proof of Hermitian metric rigidity serves both (I)as a prototype for metric rigidity theorems and (II) as a source for proving rigidity results or making conjectures on rigidity phenomena for holomorphic maps. For type-I results the author will explain (1) the finiteness theorem on Mordell-Weil groups of universal polarized Abelian varieties over functionfields of Shimura varieties, established by Mok (1991) and by Mok-To (1993), (2) a Finsler metric rigidity theorem of the author's (2004) for quotients $XΓ := Ω/Γ$ of bounded symmetric domains Ω of rank $\ge2$ by irreducible lattices and a recent application by He-Liu-Mok (2024) proving the triviality of the spectral base when $XΓ$ is compact, (3) a rigidity result of Clozel-Ullmo (2003) characterizing commutants of certain Hecke correspondences on irreducible bounded symmetric domains Ω of rank $\ge 2$ via a reduction to a characterization of holomorphic isometries and the proof of Hermitian metric rigidity. For type-II results the author will focus on irreducible bounded symmetric domains Ω of rank $\ge2$ and explain (4) the rigidity results of Mok-Tsai (1992) on the characterization of realizations of Ω as convex domains in Euclidean spaces, (5) its ramification to a rigidity result of Tsai's (1994) on proper holomorphic maps in the equal rank case, (6) a theorem of Mok-Wong (2023) characterizing Γ-equivariant holomorphic maps into arbitrary bounded domains inducing isomorphisms on fundamental groups, and (7) a semi-rigidity theorem of Kim-Mok-Seo (2025) on proper holomorphic maps between irreducible bounded symmetric domains of rank $\ge2$ in the non-equirank case. Through Hermitian metric rigidity the author wishes to highlight the fact that complex differential geometry links up with many research areas of mathematics, as illustrated for instance by the aforementioned results (6) of Mok-Wong in which harmonic analysis meets ergodic theory and Kähler geometry, and (7) of Kim-Mok-Seo on proper holomorphic maps in which techniques of several complex variables cross-fertilize with those in $CR$ geometry and the geometric theory of varieties of minimal rational tangents ($VMRTs$).[-]

The mean curvature flow (MCF) describes the evolution of a hypersurface in time, where the velocity at each point is given by its mean curvature vector (i.e., the unit normal vector multiplied by the mean curvature). When initiated with a sphere in Rn, the MCF will shrink it homothetically to a point in finite time. In this talk, we introduce an adaptation of the mean convex MCF within the Heisenberg group setting. Our initial objective was to explore potential connections between this flow and the Heisenberg isoperimetric problem. Wewill discuss the existence and uniqueness of solutions and prove that the Pansu sphere does not evolve homothetically under the MCF. This work is based on joint research with Gaia Bombardieri and Mattia Fogagnolo.[-]