Symmetries assumptions arise naturally when modeling natural phenomena. For instance, these are commonly obtained assuming the existence of some group of symmetry on a certain Riemannian manifold. The relation with physics, and hence, to the best of our current knowledge, to nature, is reinforced by the classical Noether Theorem.It is classical nowadays that geometric analytic problems modeled on manifolds equipped with symmetries coming from group actions are easier to deal with given both the existence of better compactness embeddings of Sobolev spaces in Lebesgue spaces and the classical Principle of Symmetric Criticality due to Palais. In this talk, we aim to show how symmetries can lead to easier solvability to plenty of elliptic partial differential equations related not only to the pure geometric ground but plasma equations, Kirchhoff critical problem -- a model for the motion of electrorheological fluids, characterized by their ability to change thl properties drastically when influenced by an external electromagnetic field; amoeir mechanicang others. A huge advantage coming from our approach is that symmetries allow us to circumvent critical exponents in elliptic partial differential equations. I'll present more problems to be approached by mixing statistics and numerical analysis.Target audience: Professors, postdocs, and Students of Mathematics, Applied Mathematics interested in mixing their capabilities in different contexts.Pre-requisites: The talk is thought to introduce the minimum required for understanding it during the presentation. Deep comments shall be made, but the lecturer intends that everyone could take some aspect of it.

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