One of the leading figures in higher-dimensional algebraic geometry – János Kollár, an external member of the Hungarian Academy of Sciences, receives the János Bolyai International Mathematics Award at a formal ceremony

“For Hungarian mathematics, this celebration is a continuation of an exceptionally rich tradition. It also serves as a reminder that this tradition lives on through people: in families, schools, teachers and research communities, and then through collaborations that transcend national borders,” said Mihály Pósfai, President of the Hungarian Academy of Sciences (MTA), in his ceremonial address. In his view, for the Academy, the Bolyai Prize represents a contribution to the international scientific community’s collective judgement of achievement. As a Hungarian institution, the Academy recognises work of universal significance. Today, we are especially pleased that the award recipient belongs to both the Hungarian mathematical community and our Academy.”

János Kollár and Mihály Pósfai Photo: mta.hu / Tamás Szigeti

“There is something else that gives today’s celebration particular significance: we are honouring a mathematician of flesh and blood. Almost every day, we read news reports about mathematical problems solved by artificial intelligence. We are living in extraordinary times, and mathematics is also going through an extraordinary period. We do not know how the human role in discovery will change, and in a few years, we may be talking about who solved a problem in an entirely different way. That is precisely why we should cherish this moment. The person whose work we are celebrating is here among us. His life is connected to family memories, teachers, colleagues and friendships, and we can express our appreciation to him in person.”

Zsolt Páles, an academician and President of MTA’s Section of Mathematics, described Kollár as one of the most influential figures in contemporary mathematics, particularly in higher-dimensional algebraic geometry. Páles explained: “According to the prize committee, which consists of five full members of the Academy’s Section of Mathematical Sciences and five distinguished foreign mathematicians, although classification problems arise in almost every area of mathematics, algebraic geometry has a distinctive approach: it does not regard a classification problem as complete simply because the objects in question have been classified by the elements of a set. Rather, it seeks to equip these classification sets with a geometric structure. This idea leads to what is known as moduli theory. In this framework, objects are classified by constructing moduli spaces, allowing us to parametrise them.

Zsolt Páles Photo: mta.hu / Tamás Szigeti

“A classic example of this is the construction of the moduli space of smooth curves of genus at least two. This was the culmination of a century-long programme that began with Riemann and culminated in the work of Deligne, Grothendieck and Mumford. Mumford introduced geometric invariant theory (GIT), which uses the quotient of a variety by a suitable reductive algebraic group. The end result is the moduli space of stable curves, which provides a geometric compactification of the moduli space of smooth curves. Gieseker took the first step towards solving the analogous problem in higher dimensions in 1977.

“However, no one had succeeded in applying GIT methods beyond the case of curves. The 1988 paper Threefolds and Deformations of Surface Singularities, by Kollár and N. Shepherd-Barron, marked a breakthrough in the theory of higher-dimensional moduli spaces. This paper proposed the use of methods from birational geometry instead of GIT. It launched a monumental programme that encompassed more than 30 major papers by Kollár and culminated in his book Families of Varieties of General Type.”

János Kollár Photo: mta.hu / Tamás Szigeti

As part of the ceremony, Sándor Kovács, Professor at the University of Washington, gave a lecture on Kollár’s distinctive body of work and its influence on the development of algebraic geometry. Kollár, an external member of MTA and a professor at Princeton University, then gave a lecture entitled ‘Which Surfaces Can Be Constructed from Circles?’

Click on the photo for more images Photo: mta.hu / Tamás Szigeti