The Kakeya Needle: What a Fields Medal Can Tell the Haitian Economy

On 23 July, in Philadelphia, the mathematician Hong Wang received the Fields Medal, notably for work that led, with Joshua Zahl, to the resolution of the Kakeya conjecture in three-dimensional space. Beneath its puzzle-like appearance, this result carries an idea that bears directly on small economies: you can compress the size of a system, never its structure.

An old economists’ question

Since its beginnings, political economy has been dragging along a question of size. Adam Smith already observed that the division of labor is limited by the extent of the market. The development theorists of the 1940s and 1950s, from Paul Rosenstein-Rodan to Ragnar Nurkse, built on this idea: a narrow domestic market would hold back industrialization for lack of sufficient outlets, and it was with the West Indies in mind that W. Arthur Lewis, a future Nobel laureate, raised as early as 1950 the question of industrializing very small economies. Haitian economic analysis still bears its trace: in L’économie haïtienne et sa voie de développement, Gérard Pierre-Charles devoted pages to the narrowness of the local market. Globalization supplied the classic rejoinder: in an open economy, a country does not produce for its own market alone, it produces for the world, and domestic size ceases to be destiny. The objection is strong. Yet it shifts the question more than it settles it: if size is not the decisive constraint, what is? It is precisely a question of this form, posed in the language of geometry, that has just earned a young woman the highest honor in mathematics.

A needle, a century, a woman

On 23 July 2026, at the International Congress of Mathematicians in Philadelphia, Hong Wang, a Chinese mathematician aged thirty-five, received the Fields Medal, awarded every four years to researchers under the age of forty. She is the third woman to receive it, after Maryam Mirzakhani in 2014 and Maryna Viazovska in 2022. Her path runs through France: born in Guilin, educated at Peking University and then at the École polytechnique, where she arrived without speaking a word of French, she earned a master’s degree at Paris-Saclay University before a doctorate at MIT. She now teaches in New York and at the Institut des hautes études scientifiques, on the Paris-Saclay campus.

It all starts with a problem that fits in a single sentence. In 1917, the Japanese mathematician Sōichi Kakeya asked what the smallest area is within which a needle can be turned completely around. The answer, found in the 1920s by Abram Besicovitch, defies intuition: that area can be made as small as one likes. There are even sets of zero area, specks of dust with no measurable weight, that nonetheless contain a segment in every possible direction. Volume, in other words, offers no floor.

What resists is something else: structure. A thread, a sheet of paper, and a box are not built the same way, and mathematicians measure this difference with a number, the dimension: one for the thread, two for the sheet, three for the box. The Kakeya conjecture asserted that a set containing every direction can lose almost all of its matter, but never its structure. Think of a sea urchin, the chadwon of our beaches. Set it on the table in any position: there will always be spines standing straight up, because its spines point in every direction. That is why you cannot flatten it between the pages of a book without breaking something. The Kakeya set is a sea urchin taken to the extreme: almost no body, but a needle in every possible direction. You can strip away almost all of its matter; the needles remain, and needles pointing in every direction will never lie flat within a sheet. The intuition is simple; turning it into a rigorous proof is another matter entirely. In February 2025, together with the Canadian mathematician Joshua Zahl, Hong Wang proved it for three-dimensional space, settling, for that space, a conjecture that grew out of the 1917 problem; the question remains open in higher dimensions. The result consolidates a field of mathematical analysis closely tied to the study of waves and to signal processing; any technological spin-offs remain indirect and long-term.

Volume can be compressed; structure resists

What geometry has just established can be carried over, with caution, into an economic thought experiment. No theorem travels unchanged from a mathematical space to a society. But the shape of this one is worth the detour.

Let us replace geometric directions with productive ones: generating energy, processing a harvest, writing software, insuring a risk, training an engineer. Let us replace area with the size of the economy. Kakeya’s question then becomes: what is the minimal structure of a productive system capable of covering every direction, that is, of sustaining effective supply and competition in each of them?

The lesson would be two-sided. The first side looks like good news: a small volume does not preclude a complete set of directions. Besicovitch’s set weighs nothing and yet points everywhere. The analogy thus invites us to think that an economy could be small and still cover a broad spectrum of capabilities; it would be a matter of architecture, not of mass. The second side is a warning: if a floor exists, it is not volumetric but structural. Below a certain degree of directional richness (the number of capabilities mastered, the density of substitutable suppliers, the thickness of markets), something breaks down: each missing direction becomes a dependence, and each activity deprived of effective competition or regulation can become a rent.

Covering a direction does not, moreover, mean producing everything oneself. No economy lives in autarky, and international specialization remains rational, especially for a small open economy. To cover a direction is to guarantee access to it. Some capabilities must exist on site: energy, basic logistics, human capital, institutions, part of the financing. Others can be obtained through trade, provided there is no dependence on a single supplier, a single corridor, or a single technology. The hole, in this reading, is not the missing activity: it is dependence without a substitute.

This intuition is in line with a well-established literature. The work of Ricardo Hausmann and César Hidalgo indicates that the diversity and sophistication of the capabilities in a productive fabric tell us something about a country’s income level and its growth prospects, beyond its size alone. The detour through Kakeya adds a hypothesis: that of a critical threshold below which efficiency does not degrade gradually but collapses. One could call it, in a spirit of serious play, the economic Kakeya conjecture: any productive system capable of responding to every direction of demand has an irreducible minimum complexity, largely independent of its size. Proving it would require defining the dimension of a productive fabric. It is a research program, not a result. At the very least, it shifts the strategic question from how much to how: for Haiti, the priority would not be to grow bigger first, but to complete its coverage (energy, logistics, financing, skills), on a small scale if need be, but without holes.

A decision for the state

Hong Wang’s path speaks directly to Haitian high school girls. It is not the story of a genius struck by a bolt of grace, but of a trajectory built step by step: teachers, a decade of patience on a single problem, doubts weathered. A problem posed by a Japanese mathematician, turned around by a Russian, settled by a Chinese woman trained in France: geometry asks for no passport. Mathematics, in several of its branches, is among the sciences that require the least equipment: paper, time, access to journals, demanding teachers, and the right to be wrong for a long time. It nonetheless requires stable institutions and protected careers. None of this is beyond the reach of a poor country. What is missing is of another order.

There are signs, nonetheless. The BRH Research and Development Fund, through which the central bank, the Bank of the Republic of Haiti, has been financing scientific research projects since 2020, deserves praise: a monetary institution that invests in the production of knowledge is making a rare choice, and a right one. The efforts of the rector’s office of the State University of Haiti to equip the country with new master’s programs point in the same direction and deserve encouragement. But these initiatives remain what they are: efforts by institutions that are doing, each within the limits of its mandate, what a national policy has not yet taken on.

For the lesson of Kakeya applies here too. Science is one of those directions an economy cannot leave uncovered without paying, sooner or later, the price of dependence. And covering a direction is never the business of a single actor: it is a structural decision, and structural decisions are made at the highest level of the state. A national science policy, with stable funding, built to last, insulated from changes of government, capable of opening complete scientific tracks, from secondary school to the doctorate, to young people, and first of all to girls: that is what would separate today’s signals from a genuine choice.

The chadwon of our beaches is protected neither by its size nor by its weight: it is protected by its spines. An economy, in its own way, obeys the same logic. Haiti can remain small; it cannot afford to be a sea urchin without needles. This country, moreover, is well placed to know it. In 1804, a colony ruined by war and surrounded by slaveholding empires covered the direction that every power of its time left empty: that of universal freedom. It was not a matter of volume; it was a matter of structure. Two centuries later, the same law holds for knowledge. Nations do not carry weight through their mass: they carry weight through the directions they refuse to abandon. Science is one of them, and no one will come and cover it for us. Let the state write it into law, into the budget, and into the long term, and a high school girl in Port-au-Prince, Les Cayes, or Cap-Haïtien will be able to set out, right here, on the path that took Hong Wang from Guilin to Philadelphia. The world will not wait for Haiti. Yet nothing prevents Haiti from pointing, once again, in every direction of the world.