Black holes satisfy geometric inequalities that relate their shape and size to physical properties like mass, but mathematicians still lack rigorous proofs for many of these relationships—including the Penrose inequality, which links a black hole's area to a lower bound on its mass. This project tackles a fundamental gap in mathematical physics. The Penrose inequality is intimately connected to the Cosmic Censorship Conjecture, the idea that singularities remain hidden inside black holes rather than being visible to distant observers. Proving these inequalities would confirm that the mathematics underpinning black hole theory is sound. The researcher proposes using spinors—mathematical objects originally developed for quantum mechanics—to encode information on a black hole's boundary in a new way, overcoming previous limitations that blocked such proofs. This is fundamental science with no immediate practical application. However, similar mathematical work on black holes has historically underpinned advances in our understanding of gravity, spacetime, and even led to technologies like GPS corrections that rely on general relativity. A deeper mathematical grasp of black hole geometry could eventually inform how we model extreme astrophysical environments or test the limits of Einstein's theory.
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Einstein's theory of General Relativity is the best theory of gravitation we have. It is a geometric theory describing gravity as a manifestation of the curvature of spacetime. One of the central predictions of General Relativity is that of black holes —that is, regions of spacetime with a gravitational field of such magnitude that even light cannot escape. The existence of black holes has been established in recent years by a number of observations and a theorem suggesting that black holes are the unavoidable consequence of gravitational collapse has earned its author, Sir Roger Penrose, the Nobel Prize in Physics. This great achievement is statement of the value and relevance of rigorous mathematical analysis in the understanding of the consequences physical theories. From a mathematical point of view, black holes are the simplest macroscopic objects one can envisage —the only ingredients required for their rigorous definition are our notions of space and time. One of the main realisations of the mathematical theory of black holes is that they satisfy a number of geometric inequalities —that is inequalities relating geometric properties of the black hole and physical observables. Geometric inequalities provide invaluable qualitative information about the generic behaviour of black holes. Usually, geometric inequalities have first come into existence through heuristic arguments, and in many cases rigorous proofs of their validity, or suitable counterexamples, do not exist. One of the most influential examples of geometric inequalities for which a general proof is still lacking is the so-called Penrose inequalities. This relation states that the area of a black hole provides a very specific (and in some circumstances, optimal) lower bound for its mass. The heuristics behind the Penrose inequality are closely tied to one of the most important open problems in General Relativity: the Cosmic Censorship Conjecture —namely, the expectation that singularities in a spacetime are hidden from far away observers. This project is a first step in a project aimed at the rigorous construction of geometric inequalities and the proofs thereof by means of spinors. Spinors are mathematical entities first arising in the description of quantum system. Remarkably, the mathematical theory of spinors has a rich geometric structure which has proven invaluable for the understanding of the physical and mathematical consequences of General Relativity. A celebrated proof of the most basic and fundamental geometric inequality, the statement that the mass of a black hole is positive makes use of spinors in an essential way. This milestone of the mathematical theory of black holes suggests that spinorial arguments may also provide the key to unlock the proofs of other geometric inequalities. This vision has been obstructed by limitations on the amount of information about a spinor that can be encoded in the surface bounding a black hole. In this project we explore an alternative way of encoding spinorial information on the boundary of a black hole which, it is argued, could lead to the proof of a number of conjectured geometric inequalities and to the rigorous construction of new ones —thus providing a deeper insight on the general properties of black holes as a subject of mathematical study.
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