In 1922, Jacques Hadamard published his (arguably) most famous book: "Lectures on Cauchy's
Problem in Linear Partial Differential Equations". This book based on a series of lectures given
at Yale University is generally considered as the first `mathematically rigorous' treatise on the
general theory of PDEs and its applications in physics. In this talk, I will discuss one of the most
famous problems posed by Hadamard in his book that is nowadays known as the problem of lacunas
or Hadamard's Conjecture on Huygens' principle. This problem has a fascinating history, and despite
efforts of many mathematicians over the past 100 years still remains widely open. One remarkable
feature of Hadamard's problem is its connections to different parts of mathematics that (at first glance)
have nothing to do with classical analysis or hyperbolic PDEs. I will focus on some of these connections
linking Hamadard's problem to representation theory, algebraic geometry, differential geometry and topology.