M@X — Math @ Xavier

Many problems in science lead to a differential equation that we cannot solve with a formula. A natural question is then simpler but still deep. Does a solution exist at all? In this talk we study one such problem, an equation on a region whose behavior is shaped by what happens on its boundary. The main idea is to find one function that stays below a solution and one that stays above, and then to build the solution step by step from them. Each step improves the approximation, and the process settles down to an answer. We will see how ideas from calculus and from the study of functions come together to make this work. The talk assumes only calculus and a curiosity about how different parts of mathematics connect. Along the way I will share how a love of functional analysis as an undergraduate led me, without my expecting it, to partial differential equations.