Algebraic geometry

As the name suggests, algebraic geometry is the study of geometric objects defined by algebraic equations. For example, a parabola, such as all solutions $(x,y)$ of the equation $y-x^{2}=0$, is one such object, whereas the graph of the exponential function---all solutions $(x,y)$ of the equation $y-e^{x}=0$---is not. The key distinction is that the equation defining the first example is a polynomial equation, whereas the second cannot be represented by polynomial equations, even implicitly.