At the conference at the University of Toronto, Al-Shbeil​ presented my work on constructing a self-map of \( M_4(\Gamma) \), the space of meromorphic weight 4 modular forms for a subgroup of the modular group. This map is defined using two essential tools: the Schwarz derivative and rational equivariant functions. Interestingly, the fixed points of this map satisfy a specific fourth-order nonlinear differential equation with constant coefficients. In my presentation, I illustrated this concept with four examples of fixed points derived from the Eisenstein series and Jacobi theta functions.

Applications of this work include advancements in number theory and mathematical physics. Specifically, the methods and results have potential implications for understanding special functions that appear in studying automorphic forms, string theory, and conformal field theory. Additionally, these constructions could provide new insights into solving differential equations that naturally arise in various theoretical physics contexts, such as modeling complex dynamical systems and exploring symmetry structures in mathematical models.​