FOM: arcane, genus, algorithm, etc.

[jshipman@bloomberg.net]

"Number of holes in a surface" is not at all vague or arcane.
The reason Matiyasevich's result can be stated more
"fundamentally" than Faltings's is not that algorithm is a more
basic concept than genus, it is that when you are talking about
polynomial equations it is obvious how algorithms are relevant
but very non-obvious how surfaces can be.  You need to pass
from rational numbers to reals to complex numbers to topology
and so on.  Both the algebraic and the geometric concepts are
fundamental, but Mr. MOTS won't understand the statement of
Faltings's theorem because it involves an "arcane" connection
between them. -- Joe Shipman