Handling degenerations with tropical geometry - Margarida Melo

Abstract:

In algebraic geometry, understanding the geometry of degenerations of fundamental objects as curves is of central importance in the study of the initial objects themselves.

However, dealing with degenerated objects is often hard as one has to handle combinatorial data associated with the degeneration.

In the last few years, a very successful approach to handle degeneration data has been to use tools from tropical geometry.

Indeed, it is often the case that by tropicalizing algebro-geometric objects, one gets a modular way to describe degeneration data, which can then be treated as moduli spaces themselves.

This interplay between degenerated algebro-geometric objects and their tropical counterparts has led to incredible developments in our current understanding of the geometry of both.

In the talk, I will try to explain these ideas in the case of curves and divisors on curves. In particular, I will illustrate tropical versions of the Riemann-Roch theorem and how can these be used to prove results in classical algebraic geometry via their tropical/combinatorial counterparts.

Bio:

Prof. Margarida Melo earned her Maths degree from the University of Coimbra in 2004 and her PhD from University Roma Tre in 2009. She served as a researcher at the University of Coimbra until 2014, when she moved to Roma Tre through the 'Rita Levi Montalcini' program. Since November 2024, she has been a Full Professor of Geometry at University Roma Tre. Her research focuses on moduli spaces of curves, their degenerations, and tropicalizations. She has delivered invited talks and PhD mini-courses at numerous international institutions.

Full programme:

14:30 - 15:30: Colloquium Matematico (A103) 
15:30 - 16:30: Rinfresco (Acquario)   
16:30 - 17:15: Q&A session "Who speaks Mathematics in the world?" (A103)                             

Abstract: The distribution of mathematicians by gender, cultural background, and geographic location is historically uneven. What factors influence why there are more scientists and mathematicians in one society compared to another? Why is it important to increase and broaden scientific culture within a society? How are these data evolving in Italy and globally?