I will discuss old and new results about the distribution of zeros of modular forms, and relation to Quantum Unique Ergodicity. It is known that a modular form of weight k has about k/12 zeros in the fundamental domain . A classical question in the analytic theory of modular forms is “can we locate the zeros of a distinguished family of modular forms?”. In 1970, F. Rankin and Swinnerton-Dyer proved that the zeros of the Eisenstein series all lie on the circular part of the boundary of the fundamental domain. In the beginning of this century, I discovered that for cuspidal Hecke eigenforms, the picture is very different - the zeros are not localized, and in fact become uniformly distributed in the fundamental domain. Very recently, we have investigated other families of modular forms, such as the Miller basis (ZR 2024, Roei Raveh 2025, Adi Zilka 2026), Poincare series (RA Rankin 1982, Noam Kimmel 2025) and theta functions (Roei Raveh 2026),  finding a variety of possible distributions of the zeroes.

Physical networks are spatially embedded complex networks composed of nodes and links that are tangible objects which cannot overlap. Examples of physical networks range from neural networks and networks of bio-molecules to computer chips and disordered meta-materials. It is hypothesized that the unique features of physical networks, such as the non-trivial shape of nodes and links and volume exclusion affect their network structure and function. However, the traditional tool set of network science cannot capture these properties, calling for a suitable generalization of network theory. Here, I present recent efforts to understand the impact of physicality through tractable models of network formation.

We introduce a graph renormalization procedure based on the coarse-grained Laplacian, which generates reduced-complexity representations across scales. This method retains both dynamics and large-scale topological structures, while reducing redundant information, facilitating the analysis of large graphs by decreasing the number of vertices. Applied to graphs derived from electroencephalogram recordings of human brain activity, our approach reveals collective behavior emerging from neuronal interactions, such as coordinated neuronal activity. Additionally, it shows dynamic reorganization of brain activity across scales, with more generalized patterns during rest and more specialized and scale-invariant activity in the occipital lobe during attention.

Persistent homology (PH) is an operation which, loosely speaking, describes the different holes in a point cloud via a collection of intervals called a barcode. The two most frequently used variants of persistent homology for point clouds are called Čech PH and Vietoris-Rips PH. How much information is lost when we apply these kinds of PH to a point cloud? We investigate this question by studying the subspace of point clouds with the same barcodes under these operations. We establish upper and lower bounds on the dimension of this space, and find that the question of when the persistence map is identifiable has close ties to rigidity theory. For example, we show that a generic point cloud being locally identifiable under Vietoris-Rips persistence is equivalent to a certain graph being rigid on the same point cloud.