Large value estimates in number theory and computer science

A large value estimate for a matrix M is a simple type of estimate in quantitative linear algebra.  Estimates of this type appear in many parts of math, both pure and applied.  One example is the large value problem for Dirichlet polynomials from analytic number theory, which is related to estimates about the zeroes of the Riemann zeta function.  We will also give some examples from computer science.  Many large value problems are difficult.  On the pure math side, the sharp conjecture about large values of Dirichlet polynomials has been open for a long time and is out of reach of current methods.  On the computer science side, we don’t know any efficient algorithm to approximately solve the large value problem for a given matrix M.  Many experts think that such an algorithm does not exist.  In this talk we will survey how large value estimates come up, the known methods for working on them, and some of the obstacles to fully understanding them.