Javier de la Bodega will defend his thesis on Tuesday, January 21st

• The defence will take place at Salón de Grados at the Faculty of Science and Technology of the Leioa Campus and online

Javier de la Bodega (Getxo, 1996) is a PhD student in a joint PhD programme between BCAM and the university KU Leuven in Belgium. His research has been carried out under the supervision of Javier Fernández de Bobadilla and Nero Budur respectively, and it lies in the intersection between algebraic geometry and singularity theory. Previously, he did his Bacherlor's Degree in Mathematics at the UPV/EHU and his Master's Degree in Pure Mathematics at the University of Cambridge.

His PhD thesis, The arc-Floer conjecture and the embedded Nash problem in singular spaces,  is under the supervision of Javier Fernández de Bobadilla (BCAM & Ikerbasque)  and Nero Budur (KU Leuven)

On behalf of all BCAM members, we would like to wish Javier the best of luck in his upcoming thesis defense. 

Abstract:

In 2022, Budur, F. de Bobadilla, Lê and Nguyen conjectured a relation between two very different invariants of an isolated hypersurface singularity: contact loci (of algebraic nature), and the Milnor fibration and its symplectic properties (of topological nature). This open problem is now known as the arc-Floer conjecture. However, Budur et al. stated it based on analogies and not on actual evidence. The main result of this thesis is the proof of the arc-Floer conjecture for plane curve singularities, which is a joint work with de Lorenzo Poza and provides the first piece of evidence supporting the conjecture.

In the path towards the proof of the arc-Floer conjecture for plane curves, one has to address the so-called embedded Nash problem. Indeed, the first problem that arises when studying the geometry of contact loci is that they are typically not irreducible, and the embedded Nash problem aims to understand this phenomenon though resolution of singularities. The case of unibranch plane curves was solved completely by Budur, de la Bodega, de Lorenzo Poza, F. de Bobadilla and Pełka in 2024.

The thesis has been written with an ongoing project in mind: generalising the arc-Floer conjecture to allow singularities in the ambient space. The first step towards this goal has already been achieved, for de la Bodega solved the embedded Nash problem when the ambient space is singular surface.