In your opinion what is the most amazing thing in math?
The unexpected interconnectedness of everything!
Let me give some examples:
- You’re beyond a doubt familiar with Fermat’s last theorem, stating the unsolvability of the Diophantine equation xn + yn = zn for n > 2. It took more than 350 years to actually prove this result, but more astonishing is how it was proved: by weaving together elliptic curves and modular forms, two seemingly unrelated objects in mathematics.
- A less known mathematical gem is Monsky’s theorem: it is not possible to dissect a square into an odd number of triangles of equal area. The only known method to prove this geometrical curiosity uses properties of 2-adic integers (algebra) and Sperner’s lemma (combinatorics).
- Another, more technical example is monstrous moonshine, describing a unexpected and deep connection between the Monster group and a specific modular form, the j-invariant.
To me, mathematical objects an sich can be amazing, but the ways these objects are intertwined are even more beautiful.