Maximum likelihood estimation is a ubiquitous task in statistics and its applications. The task is: given some observations of a random variable, find the distribution(s) in your statistical model which best explains these observations. A modern perspective on this classical problem is to study the "likelihood geometry" of a statistical model. By focusing on models which have a polynomial parametrization, i.e., lie on an algebraic variety, this perspective brings in tools, algorithms and invariants from algebraic geometry and combinatorics.

In this talk, I will explain some of the key ideas in likelihood geometry and discuss its recent application to the study of likelihood asymptotics, i.e., understanding likelihood estimation for very large or very small observation counts. Agostini et al. showed that these asymptotics can be modeled and understood using tools from tropical geometry, and they used this to completely describe the asymptotics for linear models. In our work, we use the same approach to treat the class of log-linear models (also known as Gibbs distributions or maximum entropy models) and give a complete and combinatorial description of the likelihood asymptotics under some conditions.

This talk is based on joint work with Emma Boniface (UC Berkeley) and Serkan Hoşten (San Francisco SU), available at: https://epubs.siam.org/doi/full/10.1137/24M1656839

Wave energy has the theoretical potential to meet global electricity demand, but it remains less mature and less cost-competitive than wind or solar power. A key barrier is the absence of engineering convergence on an optimal wave energy converter (WEC) design. In this work, I demonstrate how geometry optimisation can deliver step-change improvements in WEC performance. I present methodology and results from optimisations of two types of WECs: an axisymmetric point-absorber WEC and a top-hinged WEC. I show how the two types need different optimisation frameworks due to the differing physics of how they make waves. For axisymmetric WECs, optimisation achieves a 69% reduction in surface area (a cost proxy) while preserving power capture and motion constraints. For top-hinged WECs, optimisation reduces the reaction moment (another cost proxy) by 35% with only a 12% decrease in power. These result show that geometry optimisation can substantially improve performance and reduce costs of WECs.