### Gavril Farkas (Humboldt University of Berlin, Alemanha)

My field of work is algebraic geometry. A significant part of my research is concerned with the study of the geometry and topology of various moduli spaces. Among other things, I am interested in moduli of curves, enumerative geometry, syzygies of algebraic varieties and commutative algebra aspects, vector bundles on curves, abelian varieties and theta functions, Prym varieties, K3 surfaces and Brill-Noether type problems.

## ENSPM 2022 Presentation

**Title: Equations and syzygies of algebraic curves**

**Abstract: **Algebraic curves (Riemann surfaces) are among the most studied
objects in mathematics due to the fact that they can be approached from
the point of view of algebraic geometry, complex analysis or Galois
theory. In 1984, Mark Green put forward a deceptively simple conjecture
concerning the structure of the equations of an algebraic curve in its
canonical embedding, amounting to the statement that the complexity of
each curve of genus g can be recovered in a precise way from the
equations among its canonical forms. I will present an introduction to
this circle of ideas, then explain how ideas coming from topology and
geometric group theory led to a recent solution of Green’s Conjecture
for generic curves in arbitrary characteristics.