Mathematicians from BCAM solve a 54-year-old conjecture using a connection between algebraic geometry and symplectic geometry

• The conjecture belongs to algebraic geometry, which focuses on solutions of polynomial equations and has a rigid, arithmetic nature. Surprisingly, it has been proven using flexible techniques from symplectic dynamics (based on the laws of motion in classical mechanics).
•  The conjecture, formulated by Zariski, dating back to 1970, is one of the classic problems in Singularity Theory in Algebraic Geometry. The Belarusian mathematician posed eight questions in one of his works, with this being the only one still unanswered 50 years later.
• This work has been published in the Annals of Mathematics, a prestigious scientific journal specializing in mathematics, published by Princeton University and the Institute for Advanced Study.

A team of researchers from BCAM - Basque Center for Applied Mathematics and Ikerbasque, formed by Javier Fernández de Bobadilla - Ikerbasque Research Professor and Group Leader of the Singularity Theory and Algebraic Geometry line at BCAM; and Tomasz Pełka - Postdoctoral Researcher at BCAM, have successfully resolved a problem that had remained unsolved for over five decades.

This work has been published in the Annals of Mathematics, a prestigious scientific journal specializing in mathematics, published by Princeton University and the Institute for Advanced Study.

Using an innovative approach that links algebraic geometry with symplectic geometry, the scientists have proven the conjecture, demonstrating the powerful ability of mathematics to bridge different branches of knowledge.

This conjecture, resolved by Fernández de Bobadilla and Pełka, falls within Singularity Theory in Algebraic Geometry. This theory studies special points in geometric figures defined by algebraic equations, where these figures are not smooth. For instance, in a curve, these singular points can be places where the curve crosses itself or has a sharp point instead of a smooth one. The theory seeks to understand, classify, and describe these singular points to better comprehend the global properties of the figure. By identifying and analyzing singularities, mathematicians can gain a more complete view of the structure and behavior of geometric figures. Singularities frequently appear in natural and social sciences: black holes in physics, fluid mechanics, equilibrium points in optimization problems, bifurcations in dynamical systems...

Oscar Zariski (Poland, 1899 - USA, 1986) was a Belarusian-American mathematician who made essential contributions to the field of algebraic geometry, and whose work and deep influence persist in the development of contemporary mathematics. Zariski founded the Theory of Equisingularity, which aims to study and compare the complexity of different singularities that may appear. In 1970, in one of his pioneering works, he posed eight questions, seven of which have been resolved by different researchers over the years (Fernández de Bobadilla himself solved one of them in 2005). Only the multiplicity conjecture remained open, the most studied case known as "the multiplicity conjecture in families". This case, which is significant due to its relationship with the approach of other schools of Singularity Theory such as those of Arnol'd and Teissier, is the problem solved by Fernández de Bobadilla and Pełka.

The work of Fernández de Bobadilla and Pełka follows the path opened by mathematicians like M. McLean, who establish connections between singularity theory in algebraic geometry and symplectic dynamics theory, pseudoholomorphic curves (by Gromov, Floer, and others), combining them with other very recent techniques from hybrid and tropical geometry, culminating in the solution of Zariski's conjecture in families.

This advance not only solves a complex mathematical problem but also serves as another example of how the combination of techniques from diverse fields and the bridges between different styles of mathematical thinking allow problems to be solved in ways that seem impossible with techniques from a single domain: "This result not only resolves an old question but also strengthens the bridge between different areas of mathematics, showing how seemingly distant ideas can converge within a unified framework, leading to very fertile applications," explains Javier Fernández de Bobadilla.

The connection between algebraic and symplectic geometry has many other manifestations, one of which is mirror symmetry. Motivated by developments in theoretical physics, mirror symmetry is one of the most active and important research topics in mathematics today. This year, the BBVA Foundation awarded the Premios Fronteras del Conocimiento Awards to C. Voisin and Y. Eliashberg precisely for their fundamental contributions that establish connections between algebraic and symplectic geometry, including advances in mirror symmetry.

Future Implications

The work carried out by BCAM mathematicians underscores the importance of mathematics as a discipline that continues to demonstrate its ability to drive discoveries that transcend the boundaries of natural and social sciences, even though many of its advances are discovered through the internal development of mathematics itself. Discoveries like those of Fernández de Bobadilla and Pełka open new doors and illustrate how open problems and conjectures can act as engines for mathematical progress, inspiring the creation of new theories and methods that transform our understanding of the natural and mathematical world around us. In this case, ideas ultimately derived from classical mechanics have shown their connection and effectiveness in problems originating from algebra.

This demonstrates the importance of intellectual curiosity and interdisciplinarity in the pursuit of human knowledge. This interdisciplinarity has been a constant in the historical development of mathematics and its interaction with other sciences. It is the combination of constant dialogue with other sciences, independent development of them, and abstraction of ideas and structures that can be applied in very different contexts, which makes mathematics an extremely effective science.

BRIEF CV OF THE RESEARCHERS

Javier Fernández de Bobadilla is an Ikerbasque Research Professor and Group Leader of Singularity Theory and Algebraic Geometry at BCAM - Basque Center for Applied Mathematics. His research line includes Algebraic Geometry, Topology, and Singularity Theory. In 2011, he demonstrated a conjecture posed by John Nash in the mid-1960s, also related to the study of singularities, together with María Pe Pereira (Professor at the Faculty of Mathematical Sciences and member of the Interdisciplinary Mathematics Institute of the Complutense University of Madrid).

Fernández de Bobadilla obtained an ERC Starting Grant in 2008 and an ERC Consolidator Grant in 2014. He received his PhD from Nijmegen (Netherlands, 2001), and has worked at Utrecht University, UNED, and CSIC. He is also a member of the Institute for Advanced Study (Princeton, USA) and has been a visiting professor at the Renyi Institute of Mathematics (Hungary), IMPA (Brazil), and held a Jean Morlet Chair at CIRM (Marseille).

Tomasz Pełka is an Associate Professor at the University of Warsaw. His research area includes affine algebraic geometry and singularity theory. Together with Karol Palka (Professor at IM PAN, Poland), he completed a conjectural classification of rational cuspidal curves in the plane, a theorem with applications to surface singularities.

His doctoral thesis, defended at IM PAN in 2019, extends this result to an important class of affine surfaces with simple topology. Since then, he has worked at the University of Bern, BCAM, and has been a visiting researcher at CIRM, Marseille, and the Renyi Institute, Budapest.