Given a bounded submanifold M in R^n, how many rational points with common bounded denominator are there in a small thickening of M? How does this counting function behave if we let the size of the denominator go to infinity? The study of the density of rational points near manifolds has seen significant progress in the last couple of years. In this talk I will explain why we might be interested in this question, focusing on applications in Diophantine approximation and the (quantitative) arithmetic of projective varieties.

This colloquium-level talk will survey mathematical contributions of twentieth-century mathematician Hel Braun. If you are wondering "Who's Hel Braun?" this talk is for you. If you already know and instead asked "But why Hel Braun?" it is also for you. Braun's research contributions lie in three areas: classical number theory problems about integers, modular and automorphic forms, and Jordan algebras. I will describe how each of these seemingly distinct topics led Braun naturally to the next, and I will highlight the ongoing impact of some of Braun's work, including in my own research. Along the way, I will briefly mention factors that helped shape her legacy. This talk is intended for a broad audience of mathematicians, and I hope mathematicians from many fields will attend.