Identifying and resolving pain points is fundamental to launching successful businesses. Customer validation and market analysis are critical steps toward understanding if a problem is worth solving.

In today’s fast-paced business world, innovation isn’t just a luxury—it’s a necessity. Companies that foster fresh thinking, celebrate new ideas, and encourage risk-taking are the ones that thrive.

Alan Veliz-Cuba has received funding from the Simons Foundation and the American Mathematical Society for some of his research. You can probably think of a time when you’ve used math to solve an ...