Aula BCAM-UPV/EHU: The arc-Floer conjecture: When arithmetic encounters Hamiltonian systems

The talk will be in Spanish. 

Abstract:

In the 1980s, Japanese mathematician Jun-ichi Igusa observed a mysterious relationship between arithmetic and the topology of algebraic hypersurfaces. More specifically, he noticed that by taking a polynomial f with integer coefficients and considering the number of solutions to the equation f ≡ 0 modulo n, certain topological invariants of the space {f = 0} ⊂ C^d were obtained. These observations led to what is now known as the Monodromy Conjecture, which quickly gained the interest of the algebraic geometry community by establishing a connection between number theory and singularity theory. Despite being verified in several cases, this conjecture is far from being resolved today. Nevertheless, the problem has spurred advances in other related areas, such as motivic integration, non-Archimedean geometry, and model theory. More recently, mathematicians Budur, F. de Bobadilla, Lê, and Nguyen proposed another approach to address the conjecture by exploiting the symplectic properties of the singularity {f = 0}. In this way, they formulated a second conjecture in the same philosophy as the Monodromy Conjecture: the arc-Floer conjecture. In this case, arithmetic objects would be related not only to the topology of the singularity but also to Floer theories inherent to symplectic topology. This talk will introduce students to the Monodromy and arc-Floer conjectures and explain their current status.