Graduate Math Meet

Title: The Implicit Function Theorem

Abstract: Given an equation F(x,y)=0, can we solve for one variable (say y) as a nice function of the others (x), at least locally? Surprisingly often, the answer is yes, and the Implicit Function Theorem provides the precise condition under which this magic happens. This is one of the most elegant theorems and has far-reaching tools in multivariable calculus and analysis. This talk begins with motivating examples, followed by the precise statement of the theorem. We will conclude by discussing several applications that demonstrate the theorem's reach and utility in modern mathematics.