Sara Maad: The implicit function theorem made explicit

Given a system of $n$ equations and $m$ unknowns $f_1(x_1,\dots x_m)=c_1$, ... $f_n(x_1,\dots,x_m)=c_n$, we intuitively expect (or at least hope) to be able to solve the system if $m\ge n$, and express $n$ of the variables as functions of the remaining $m-n$ ones. This is of course not always the case, since there may be redundancies in the system (for instance if we have listed one of the equations more than once). The implicit function theorem, or rather theorems, gives sufficient conditions for when the intuitive picture is correct. The simple theorem from advanced calculus has counterparts in Banach spaces, making it useful for proving existence and uniqueness of solutions in differential equations, and also smoothness with respect to parameters and initial conditions.

The speaker will talk about different versions of this theorem, mention something about its history, and show some applications in my research area, differential equations.