Examples of o-minimality in algebraic geometry
In this introductory talk, we will define o-minimality (a way of augmenting algebraic geometry with functions like $e^x$, $\sin$, $\cos$, etc.), and show:
(1) The number of solutions to a system of polynomials equations is bounded by a function of the sizes of the supports of the equations, independent of the sizes of the exponents.
(2) For an irreducible polynomial $f(x,y)$ not of the form $ax^iy^j+bx^ky^l$ there are only finitely many solutions to $f(x,y)=0$ with $x$, $y$ roots of unity.