Martin Burke

AI Research

I have developed a strong interest in AI safety—particularly recursive debate frameworks as a mechanism for scalable oversight of advanced models. I have worked on constructing and analysing structured debate settings inspired by the work of Irving, Christiano, and others, focusing on how complex claims can be decomposed into adversarial sub-claims under constrained verification. My work has explored the “obfuscated arguments” problem: identifying when a dishonest debater can hide a subtle flaw behind computational intractability, layered reasoning, or asymmetric information. I have designed increasingly difficult debate instances across domains—including optimisation theory, physics, combinatorics, and finance—to stress-test whether a bounded judge (human or model) can reliably detect false sub-claims. This line of work reflects my broader interest in formal structure, adversarial reasoning, and the limits of verification in high-dimensional problem spaces—questions that sit at the core of alignment and scalable AI governance.

PhD Research Summary

My doctoral research sat at the intersection of convex optimisation, polynomial algebra, and computational chemistry, with a central objective: to explore whether rigorous optimisation theory—specifically Sum-of-Squares (SOS) techniques and semidefinite programming—could provide new structural insights into molecular conformation problems, and ultimately contribute to aspects of the protein folding challenge.

Research Context

Molecular structure prediction is fundamentally a global optimisation problem. Chemical systems are governed by highly non-convex potential energy landscapes, typically containing vast numbers of local minima separated by complex barriers. Classical approaches in computational chemistry often rely on heuristic search methods, stochastic sampling, or local optimisation techniques. While practically useful, these methods offer limited guarantees regarding global optimality or structural completeness.

My research investigated whether tools from convex relaxation theory—particularly SOS hierarchies—could be used to reformulate certain classes of molecular optimisation problems into tractable semidefinite programs. The overarching motivation was to introduce mathematical rigour, bounding guarantees, and systematic relaxation hierarchies into domains traditionally dominated by heuristic approaches.

Mathematical Framework

The core methodology relied on:

• Polynomial optimisation theory
• Sum-of-Squares (SOS) decompositions
• Semidefinite programming (SDP)
• Moment relaxations and dual representations
• Convex relaxations of non-convex systems

Many molecular energy formulations can be expressed or approximated as multivariate polynomial functions of atomic coordinates (or transformed coordinate representations). This creates an opportunity to apply SOS techniques, which allow a non-convex polynomial minimisation problem to be relaxed into a hierarchy of convex semidefinite programs.

These relaxations provide:

• Lower bounds on the global minimum
• Certificates of non-negativity
• Structured convergence hierarchies
• Systematic trade-offs between computational cost and solution tightness

I implemented and analysed these formulations computationally using SOS toolchains, exploring their scalability, numerical behaviour, and structural interpretability in chemically meaningful systems.

Chemical Structure Prediction

A key component of the work involved mapping chemically relevant constraints—such as bond lengths, angle constraints, and geometric feasibility conditions—into polynomial constraint systems amenable to SOS relaxation. This required careful modelling choices to ensure:

• Algebraic tractability
• Preservation of physical structure
• Numerical stability within semidefinite solvers

The research examined how these convex relaxations behave when applied to simplified molecular systems and structural motifs. Particular attention was given to:

• The tightness of lower bounds
• The hierarchy order required for meaningful structural information
• Computational complexity scaling
• Degeneracy and symmetry considerations in molecular geometry

This work provided insight into when and how convex relaxation techniques meaningfully approximate chemically realistic energy landscapes, and where inherent dimensionality or combinatorial growth limits their applicability.