In mathematics and science, symmetry, as a powerful tool to understand natural phenomena,  signifies invariance under certain transformations, including rotations, reflections, or translations, which means a system is symmetric if it remains unchanged following these related operations. While symmetries appear at the core of mathematical models of the physical world, they establish a mathematical realization of spatial invariance. This indicates that mathematical modeling concerns creating representations of real-world systems by means of equations, variables and functions, where symmetry can play a pivotal role in these sorts of models. Fractal theory addresses complex geometrical structures that exhibit self-similarity, having its applications in modeling different natural phenomena as well as generating realistic grains, analyzing signals and modeling chaos. Notwithstanding, chaos theory is concerned with the exploration of the behavior of dynamical systems, which are highly sensitive to initial conditions, which is important to understand nonlinear phenomena emerging in population modeling, fluid dynamics, weather prediction, and so on.  Subsequently, graph theory, as a branch of mathematics, offers means to analyze relationships and networks, with its uses ranging from modeling social networks to optimizing transportation systems and from analyzing molecular structures to solving problems in computer science, including database indexing and search algorithms. At these junctures, cryptography involves the study and practice of techniques for secure communication with methods such as encryption, decryption, digital signatures and hashing being of use in different areas and environments like medical record systems, cybersecurity, e-commerce, banking, government communications, and so forth, where data security and sensitivity are highly important.

Differential equations, probability and uncertainty within mathematical modeling are critical in understanding and solving real-world problems across a broad range of disciplines since they display the capability of describing dynamic systems, predicting behavior and providing insights into dynamic and nonlinear phenomena. Given their predictive capability, descriptive power and wide-ranging applications, differential equations particularly relate a function with its derivatives and are employed to model phenomena comprising changes such as growth, decay, motion or waves in mathematics, physics, biology, engineering, applied sciences, economics, finance, environmental sciences, among many others. Thus, graph theory, fractal theory and chaos theory converge in their capabilities of modeling complexity, unpredictability and interconnectedness, which make up the essential features for powerful encryption systems.

Based on these uses, applicability aspects and broad encompassing, our special session aims to contribute to the understanding of symmetry, encryption, differential equations, graph theory, mathematical modeling, and so forth for paving the way of advanced theories and efficient applications in relevant domains.

The topics include, but are not limited to: