Symmetries beyond the Standard Model and elsewhere
The use of symmeties -- transformations of physical quantities which leave the laws of physics unchanged -- is one of the cornerstones of modern physics. The introduction and understanding of such symmetries has lead to our current formulation of the Standard Model, providing an explanation for the strong, weak and electromagnetic interactions. However, despite its immense success, we know the Standard Model is not the Theory of Everything and thus we must consider ways to go beyond the Standard Model.
There are many ways to do this, two of which are (i) keeping the known symmetries but including new particles which follow definite rules of transformation under these symmetries, and (ii) extending the symmetries beyond those that are currently known but reduce to those in the Standard Model under the appropriate conditions. Either of these methods must reproduce all current results but hopefully also explain currently-unexplained phenomena and predict new and testable observations.
The methodology behind any such extension of the Standard Model must include concepts from the mathematical area of abstract algebra: this area provides a natural framework for defining the objects which quantify the symmetries in question and the more concrete realisations of these symmetries and how they act on the physical quantities present in the theory.
Although algebra -- and its subfields group theory and ring theory -- is of vital importance in high-energy physics, it also has applications in many other areas of physics, such as condensed matter physica, general relativity and quantum information theory, so understanding the mathematics behind the Standard Model can prove extremely helpful in a broad range of other subjects in theoretical physics.