Elba Garcia-Failde, new RyC of the department, publishes an innovative work in Inventiones Mathematicae

Professor Ramón y Cajal Elba Garcia-Failde has just joined our department in the EPSEB section. She belongs to the GEOMVAP research group of the department, and to the  SYMCREA unit of the UPC. We would like to take advantage of the news of the publication to welcome her to the department.

The work published in Inventiones Mathematicae is entitled Relations on mg,n and the negative r-spin witten conjecture, and authors Nitin Kumar Chidambaram, Elba Garcia-Failde, and Alessandro Giacchetto. 

The article is important because it introduces a new extension of the Witten r-spin class to the negative spin case, with fundamental implications for the theory of cohomological fields (CohFTs), a central topic in algebraic geometry, mathematical physics, and the theory of moduli spaces. This new theory of cohomological fields, the Theta class, is neither semisimple nor has a flat unit. Despite these challenges, the authors develop a semisimple deformation that allows them to apply Teleman’s reconstruction theorem and derive new tautological relations in the moduli space of curves. This result represents a significant extension of the existing framework of CohFTs and deepens our understanding of moduli spaces through previously inaccessible geometric structures. They also show that the Theta class is deeply related to integrable systems, a key area of ​​mathematical physics. In particular, they prove a long-range Norbury conjecture, which links the case r=2 to a well-known function in random matrix theory.

This work not only provides concrete results and conjectures, but also paves the way for future exploration of non-semisimple structures in algebraic geometry and beyond.

Congratulations Elba!