Mock modular forms, first invented by Ramanujan, provide a beautiful generalization of the usual modular forms. In recent years, it was found that they capture generating functions of the number of microstates of BPS black holes appearing in compactifications of string theory with 8 and 16 supercharges. This review describes these results and their applications which range from the actual computation of these generating functions for both compact and non-compact compactification manifolds (encoding, respectively, Donaldson-Thomas and Vafa-Witten topological invariants) to the construction of new non-commutative structures on moduli spaces of Calabi-Yau threefolds.

Macrophytes are a key component of lagoon ecosystems, and the spatio-temporal variation of their biomass reflects the effect of environmental drivers. Understandingthese dynamics is essential for managing, preserving and anticipating the fate of such fragile ecosystems. First, using both satellite data and field observations,we showcase here a machine learning approach to accurately detect submerged vegetation in a preserved lagoon in South France, the Bagnas lagoon. Second,we investigate the statistical relationship between the surface area of Stuckenia pectinata and abiotic factors known to influence the life history of this aquatic plant.We thus identify and evaluate the factors that most impact Stuckenia pectinata population dynamics within the lagoon. We find that water temperature, salinity, anddepth are significantly related to submerged aquatic vegetation (SAV) coverage. We show that the SAV surface area reaches a peak during summer, which can varyacross years. Furthermore, we examine the lifetime distribution of SAV during each year cycle across the pond. We show that the average annual lifespan differssignificantly from one year to the next, which can be explained by annual differences in exposure time to sufficiently high water levels.

The stacking sequence of two-dimensional hexagonal boron nitride (hBN) is a critical factor that determines its polytypes and its distinct physical properties. Although most hBN layers adopt the thermodynamically stable AA′ stacking sequence, achieving alternative stacking configurations has remained a long-standing challenge. Here we demonstrate the scalable synthesis of hBN featuring unprecedented AA stacking, where atomic monolayers align along the c axis without any translation or rotation. This previously considered thermodynamically unfavourable hBN polytype is achieved through epitaxial growth on a two-inch single-crystalline gallium nitride wafer, using a metal–organic chemical vapour deposition technique. Comprehensive structural and optical characterizations, complemented by theoretical modelling, evidence the formation of AA-stacked multilayer hBN and reveal that hBN nucleation on the vicinal gallium nitride surface drives the unidirectional alignment of layers. Here electron doping plays a central role in stabilizing the AA stacking configuration. Our findings provide further insights into the scalable synthesis of engineered hBN polytypes, characterized by unique properties such as large optical nonlinearity.

Collagen, the primary structural protein in the extracellular matrix, plays a critical role in tissue architecture and mechanical integrity. This study investigates the structural and nonlinear optical properties of atelocollagen (Acol) and telocollagen (Tcol) in response to pH variations using Second Harmonic Scattering (SHS), polarization-resolved SHS (P-SHS), and Atomic Force Microscopy (AFM). AFM imaging revealed distinct morphological differences, with Acol forming wavy, non-cross-linked, and randomly arranged fibers, while Tcol exhibited a more interconnected, mesh-like fibrillar network. SHS and P-SHS measurements further confirmed these differences, indicating significant variations in hyperpolarizability and depolarization ratio (DR) between Acol and Tcol. The results suggest that telopeptides play a crucial role in collagen fibrillogenesis, influencing molecular symmetry and optical responses. The observed differences in nonlinear optical properties highlight the potential of SHS as a powerful tool for characterizing collagen self-assembly mechanisms. These findings contribute to the broader understanding of collagen organization in biological and biomaterial contexts, with implications for tissue engineering, fibrosis research, and regenerative medicine. Future work should explore how external factors such as ionic strength and cross-linking agents further modulate collagen fibril formation and its mechanical properties.

ABSTRACT Collagen is a key protein in mammals that maintains structural integrity within tissues. A failure in fibrillar collagen reorganization can induce cancer or fibrosis formation, such as in spinal cord injury (SCI), where the healing process after the initial trauma leads to the formation of scar tissue, which includes fibrosis. As there is no current treatment targeting the fibrotic process directly, a better understanding of collagen properties can thus help to apprehend malignant states. Characterization of collagen fibers has been widely explored on second‐harmonic generation (SHG) images, due to the label‐free nature of the SHG imaging technique. It has been performed with various fibers extraction methods such as curvelet transform (CT) implemented in the open‐source software CurveAlign. However, when it comes to investigating undergrowth collagen fibers (collagen fibers that are still under reorganization) as observed in SCI, the CT method becomes complex to tune for nonadvanced users in order to properly segment the fibers. To improve collagen detection in the case of undergrowth fibers, we propose a methodology based on the fingerprint enhancement (FP‐E) algorithm that requires fewer user input parameters and is less time‐consuming. Our method was extensively tested on SHG data from injured spinal cord samples. We obtained metrics that depicted changes in collagen organization over time, particularly a significant increase in fiber density, demonstrating the FP‐E algorithm was properly adapted to address the evolution of collagen properties after SCI. Besides the simpler tuning of the method compared to commonly used software, the combination with further characterization of the extracted fibers could lead to consider fibrillar collagen as a biomarker in diseases where fibers are under development. The FP‐E algorithm is provided in the article.