The CRM-ISM-AMQ 2025 Prize is awarded to Mikhail Karpukhin and Jean Lagacé

We are delighted to announce the winners of the CRM-ISM-AMQ 2025 Prize. It will be awarded to Mikhail Karpukhin (University College London) and Jean Lagacé (King’s College London) for their paper “Flexibility of Steklov eigenvalues via boundary homogenisation”, published in Volume 48 of AMQ.

Shape optimization for eigenvalues is a classical topic in spectral geometry, with connections to various other fields of pure and applied mathematics, including minimal surface theory and inverse problems. In recent years homogenization methods have been introduced in the field, allowing various optimal bounds and stability vs flexibility results to be obtained. The paper that we are proposing is a prime example of this recent approach.

The main result of the paper is easy to state: «the best upper bound for normalised Steklov eigenvalues of surfaces of genus zero and any fixed number of boundary components can always be saturated by planar domains.» In other words, for surfaces of genus 0, even if the natural setting for some maximizer is a curved surface, it is possible to saturate the sharp bound using only planar domains.This result is not only interesting, but also surprising.

The proof is based on recent results on continuity of min-max eigenvalues associated to Radon measures, which is used in combination with Koebe uniformization theorem and conformal invariance properties of the Dirichlet energy functional in dimension 2.

It is noteworthy that homogenization theory is a field stemming from applied and industrial mathematics, which has not been used to its full potential in pure mathematics yet. This paper provides a new step in that direction.

Source: ISM

For more information about this prize.