lpetrich
Contributor
Today's Google doodle is 
Emmy Noether's 133rd birthday (1882 - 1935). She was a mathematician who made a lot of contributions to abstract algebra. That's generalizing a lot of familiar mathematical operations.
Most of us know about how addition and multiplication have commutativity, associativity, identities, inverses, and one being distributive over the other. One can generalize these properties by studying what happens with operations that have these properties over some set of entities.
Positive integers under addition = semigroup (associative)
Nonnegative integers under addition = monoid (semigroup with identity)
Integers under addition = group (monoid with inverses)
Integers under addition are also a commutative or abelian group
Integers under addition and multiplication = ring (addition = abelian group, multiplication = monoid, distributive)
Rational numbers under addition and multiplication = algebraic field (ring with multiplication minus additive identity being a group)
This is as far as I'll go here, because I don't want to make all your heads spin.
I've written some software to work with some algebraic-algebra constructions: "semisimple Lie algebras" at My Science and Math Stuff (important for elementary particle physics)
She also worked on a theorem that relates continuous symmetries of physical systems to conserved quantities. Like these:
[table=head]Symmetry | Conserved quantity
Time shift | Energy
Position shift | Momentum
Rotation | Angular momentum
Boost | Center-of-mass position
EM gauge | Electric charge
[/table]
Boosts = change in velocity
EM gauge = electromagnetic gauge symmetry, a rather arcane one
Emmy Noether's 133rd birthday (1882 - 1935). She was a mathematician who made a lot of contributions to abstract algebra. That's generalizing a lot of familiar mathematical operations.Most of us know about how addition and multiplication have commutativity, associativity, identities, inverses, and one being distributive over the other. One can generalize these properties by studying what happens with operations that have these properties over some set of entities.
Positive integers under addition = semigroup (associative)
Nonnegative integers under addition = monoid (semigroup with identity)
Integers under addition = group (monoid with inverses)
Integers under addition are also a commutative or abelian group
Integers under addition and multiplication = ring (addition = abelian group, multiplication = monoid, distributive)
Rational numbers under addition and multiplication = algebraic field (ring with multiplication minus additive identity being a group)
This is as far as I'll go here, because I don't want to make all your heads spin.
I've written some software to work with some algebraic-algebra constructions: "semisimple Lie algebras" at My Science and Math Stuff (important for elementary particle physics)
She also worked on a theorem that relates continuous symmetries of physical systems to conserved quantities. Like these:
[table=head]Symmetry | Conserved quantity
Time shift | Energy
Position shift | Momentum
Rotation | Angular momentum
Boost | Center-of-mass position
EM gauge | Electric charge
[/table]
Boosts = change in velocity
EM gauge = electromagnetic gauge symmetry, a rather arcane one
