TITLE: What Result Is Valuable in Mathematics?
QUESTION [5 upvotes]: I'm a physics student, and I know that, in physics, value is considered for compliance with experiment. No theory, no matter how beautiful, is taken serious without experimental proof. People publish papers in physics, when they find new applications of physics or when they give new meanings to some physical entities through their new theories. What result is valuable in Mathematics? Do you call it new, for example, when someone presents a fast way to compute number of primes, or when someone presents a complicated formula but a precise one, or just the beauty of the approach is valued?

REPLY [3 votes]: One way to be valuable is to be aesthetically pleasing: $$e^{i\pi}=-1\quad \text {(Euler)}$$ 
A second way is to  solve a problem which had baffled great mathematicians: $$x^n+y^n=z^n \;\text {has no non-trivial solutions in integers for }  n\gt2    \quad    \text {(Wiles) }   $$ 
A third way is to prove  a difficult  and useful result in some branch of mathematics: $$  \text {Every algebraic variety in characteristic zero  has a resolution of singularities }     \text {(Hironaka) } $$   A fourth way is to introduce a powerful new concept:    $$\text {Logarithms } \:\text {(Napier) } \quad\text {Flatness } \:\text {(Serre) }  \quad \text {Distributions }  \;   \text {(Sobolev, Schwartz) }                $$ 
...and there are many, many other ways.