In his lost notebook, Srinivasa Ramanujan documented two important Eisenstein series identities intimately connected with fifth-degree modular equations. Historically, one of these identities played a pivotal role in Ramanujan’s derivation of his celebrated congruence for the partition function modulo 5: p(5n + 4) ≡ 0 (mod 5).

Over the past century, the investigation of Ramanujan’s identities has catalyzed significant progress in both modular forms theory and combinatorial analysis. In this presentation, I will introduce novel parametrizations of these two renowned identities through the framework of elliptic functions.

These parametrizations establish a systematic approach with applications in modular forms theory, proving especially valuable for deriving new identities that connect Eisenstein series with the Dedekind eta function. I will present several new results that illustrate the efficacy and broad applicability of this methodology.

We introduce some recent researches on detection and quantification of quantum entanglement, particle-wave duality, quantum coherence and genuine multipartite entanglement, quantum network steering and quantum uncertainty relations.

Only a few states in high-dimensional systems can be identified as (un)steerable using existing theoretical or experimental methods. We utilize semidefinite programming (SDP) to construct a dataset for steerability detection in qutrit-qutrit systems. For the full-information feature F1, artificial neural networks achieve high classification accuracy and generalization, and perform better than the support vector machine. As feature engineering plays a pivotal role, we introduce a steering ellipsoid-like feature, F2, which significantly enhances the performance of each of our models. To address quantum steering detection in isotropic states, partially entangled states, and random states, we explore, respectively, the most tailored feature and classifier. Given that the SDP method provides only a sufficient condition for steerability detection, we establish the first rigorously constructed, accurately labeled dataset based on theoretical foundations. This dataset enables models to exhibit outstanding accuracy and generalization capabilities, independent of the choice of features. As applications, we investigate the steerability boundaries of isotropic states and partially entangled states, discover new steerable states, and determine their parameter ranges. This work not only advances the application of machine learning for probing quantum steerability in high-dimensional systems but also deepens the theoretical understanding of quantum steerability itself.