Theory of Invariants: algebraic expressions that do not change when certain changes of variable are performed.
Number theory: algebraic numbers could be used to solve Diophantine equations, understand reciprocity laws and even to attack Fermat's Last Theorem.
Euclidean group: They are rigid, in the sense that they do not change distances or angles. Quantities that do not change when a transformation from the group is applied.
Kinds of geometry:
LIE group: That of continuos transformation group, it captures the most significant symmetries of the physical universe, and symmetry is a powerful organizing principle.It has both algebraic and topological properties. A set with an operation of composition that satisfies various algebraic identities, most notably the associative law. And topological manifold, a space that locally resembles Euclidian space of some fixed dimension but which may be curved or otherwise distorted on the global level.
Other types of algebraic systems:
Number theory: algebraic numbers could be used to solve Diophantine equations, understand reciprocity laws and even to attack Fermat's Last Theorem.
Euclidean group: They are rigid, in the sense that they do not change distances or angles. Quantities that do not change when a transformation from the group is applied.
- Translation
- Rotation
- Reflection
- Glide reflection
Kinds of geometry:
- Elliptic Geometry
- Hyperbolic Geometry
- Projective Geometry
LIE group: That of continuos transformation group, it captures the most significant symmetries of the physical universe, and symmetry is a powerful organizing principle.It has both algebraic and topological properties. A set with an operation of composition that satisfies various algebraic identities, most notably the associative law. And topological manifold, a space that locally resembles Euclidian space of some fixed dimension but which may be curved or otherwise distorted on the global level.
Other types of algebraic systems:
- Ring
- Field
- Algebra