In fundamental physics, there exists consensus that the present microscopic theory of elementary particles and their interactions is incomplete and that a new, deeper theory generically called New Physics must replace it. Theorists build models which may grasp a part of New Physics and explore their consequences. One example of such models is the so-called three-Higgs-doublet model. When constructing it, we encounter certain challenges of purely mathematical nature. I will show two examples of these mathematical challenges: one that is solved (classification of finite symmetry groups available of this model) and another which still await its solution (boundedness from below conditions).
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