Geodesics in the Condition Metric

Abstract: 

Homotopy methods is the name for a set of algorithms designed to find approximate solutions in a broad class of very important problems such as polynomial system solving or the eigenvalue problem. Roughly speaking, these methods consider continuous deformations of problems with a known solution to problems we want to solve. Despite homotopy methods exhibiting an excelent experimental performance, we still don't know optimal bounds for its complexity. The works of Shub and Smale, among others, suggesst to study the behaviour of geodesics in the so-called "condition metric”.

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There are many situations where one imposes certain conditions on a function and its Fourier transform and then wants to optimize a certain quantity. I will describe how these types of Fourier optimization problems can arise in the context of the explicit formula, which relates the primes to the zeros of the Riemann zeta-function. These ideas lead to the strongest known estimates in the classical problem of bounding the maximum gap between consecutive primes assuming the Riemann hypothesis. Our answer depends on the size of the constant in the Brun-Titchmarsh inequality. Using the explicit formula in the other direction, we can also use Fourier optimization to prove the strongest known conditional estimates for the number of zeta zeros in an interval on the critical line. This is based on joint works with E. Carneiro, V. Chandee, and K. Soundararajan.

There are many situations where one imposes certain conditions on a function and its Fourier transform and then wants to optimize a certain quantity. I will describe how these types of Fourier optimization problems can arise in the context of the explicit formula, which relates the primes to the zeros of the Riemann zeta-function. These ideas lead to the strongest known estimates in the classical problem of bounding the maximum gap between consecutive primes assuming the Riemann hypothesis. Our answer depends on the size of the constant in the Brun-Titchmarsh inequality. Using the explicit formula in the other direction, we can also use Fourier optimization to prove the strongest known conditional estimates for the number of zeta zeros in an interval on the critical line. This is based on joint works with E. Carneiro, V. Chandee, and K. Soundararajan.