can an integer $n$ be represented as a sum of two squares $n=x^2+y^2$? if so, how many different representations are there? we begin with the answers to these classical questions due to fermat and jacobi. we then illustrate hurwitz's class number formula for binary quadratic forms, and put all these classical formulas under the modern perspective of the siegel-weil formula. we explain how the latter perspective led gross-keating to discover a new type of identity between arithmetic intersection numbers on modular surfaces and derivatives of certain eisenstein series. after outlining the influential program of kudla and rapoport for generalization to higher dimensions, we report a recent proof (joint with w. zhang) of the kudla-rapoport conjecture and hint at the usage of the uncertainty principle in the proof.