Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas.

Come up with a meaningful definition of symmetry.

Study analytic symmetry and relate it to algebraic symmetry.
Try to express the symmetries of an object, like a polyhedron, in terms of bundle conceptions.
E8 is the symmetry group of itself. What is the symmetry group of?
Relate symmetry and regularity of choice, thus geometry and choice frameworks.

Thoughts about symmetry

Defining symmetry

Symmetry involves a dual point of view: for example, vertices are distinct and yet not distinguishable.
Symmetry: indistinguishable change, thus a lie, a nontruth, what is hidden. Hidden change, the revealing of hidden change.
Noether's theorems: Symmetry yields conversation law. Feynman's argument for that, as related by Sean Caroll. At an extrema of action, perturbations have no impact, to first order. This brings to mind fixed points.
Ockham's razor gets us to focus on the structures which are most basic in that they generate the richest symmetries - the rich symmetries tend solutions towards Ockham's razor.
Love (symmetry) establishes immortality (invariant).

Symmetry group

Symmetry group consists of distinguishable actions which accomplish nothing (leave an object invariant). So they separate the object/environment and its state.

Symmetry group relates:

The mind is augmented through the "symmetric group" which is the system that augments our imagination.

Symmetry and symmetry breaking

Examples

Note that x and y axes are separated by 90 degrees. This is the grounds for the degree four of i, the trigonometric functions, the Cauchy-Riemann equations, etc.
Representations of the symmetric group. Symmetric - homogeneous - bosons - vectors. Antisymmetric - elementary - fermion - covectors. Euclidean space allows reflection to define inside and outside nonproblematically, thus antisymmetricity. Free vector space. Schur functions combine symmetric and antisymmetric in rows and columns.

Readings

Global vs. Local symmetry
Symmetry: changes in one sense, keeps the same in another sense. Like the six qualities of signs: keeps one level fixed, lets one level change.
Symmetry-Type Multiplication Table has a three-cycle. If you multiply a type by itself, then you get the group.
4 Cartan symmetric spaces - spaces of free fermion Hamiltonians
Symmetry: Different causes have the same effect. Ambiguity.

Diffusion symmetry (Norm Wildberger)

Alternative to the usual "transformational" approach to symmetry embodied in group theory.
Related to character theory, strongly regular graphs, von Neumann algebras, Hecke algebras, Lie group representation theory, cyclotomy and conformal field theory.
hypergroup (going back to the 1970's with seminal papers by Dunkl, Jewett and Spector)
fusion rule algebra (appearing in physics and the study of II-1 factors and elsewhere)
the role of character tables.
replace (at least in some situations) non-commutative harmonic analysis with commutative harmonic analysis.
Observing symmetry requires breaking symmetry.