The marked length spectrum of a closed, negatively curved Riemannian manifold records the lengths of closed geodesics. We can compare two Riemannian metrics using their marked length spectra, which can be done dynamically via the geodesic flow. This perspective has been extended to other geometric contexts, such as pairs of Anosov representations and actions on metric trees. In this talk, I will discuss a joint work with Stephen Cantrell in which we compare length functions of actions on CAT(0) cube complexes. These are non-positively curved spaces of combinatorial nature that generalize simplicial trees. The role of the geodesic flow is now played by a finite-state automaton, inspired by Calegari-Fujiwara’s work about word metrics on hyperbolic groups.

For a punctured surface S and a split reductive algebraic group G such as SL_n or PGL_n, Fock and Goncharov (and Shen) consider two types of moduli spaces parametrizing G-local systems on S together with certain data at punctures. They show that these spaces have special coordinate charts, hence are birational to cluster varieties. Fock and Goncharov’s duality conjectures predict the existence of a canonical basis of the algebra of regular functions on one of these spaces, enumerated by the tropical integer points of the other space. I will give an introductory overview of this topic, briefly explain recent developments involving quantum topology and mirror symmetry of log Calabi-Yau varieties, and present some open problems if time allows.

The problem of finding the maximal possible multiplicity of the first Laplacian eigenvalues has been studied at least since the 1970’s. I will present a recent work in collaboration with Simon Machado (ETH Zürich) in which we proved, for negatively curved surfaces, the first upper bound which is sublinear in the genus g. Our method also yields an upper bound on the number of eigenvalues in small spectral windows, and this upper bound is shown to be nearly sharp. We also obtain results for higher-dimensional manifolds. Our proof combines a trace argument for the heat kernel and a geometric idea introduced in the context of graphs of bounded degree in a paper by Jiang–Tidor–Yao–Zhang–Zhao (2021). Our work provides new insights on a conjecture by Colin de Verdière and a new way to transfer spectral results from graphs to surfaces.