Séminaire TNGA : Vaidehee Thatte

Oratrice : Vaidehee Thatte (King’s College London)

Titre : Ramification Theory for Henselian Valued Fields

Résumé : Ramification theory serves the dual purpose of a diagnostic tool and treatment by helping us locate, measure, and treat the anomalous behaviour of mathematical objects. In the classical setup, the degree of a finite Galois extension of « nice » fields splits up neatly into the product of two well-understood numbers (ramification index and inertia degree) that encode how the base field changes.

In the general case, however, a third factor called the defect (or ramification deficiency) can pop up. The defect is a mysterious phenomenon and the main obstruction to several long-standing open problems, such as obtaining resolution of singularities. The primary reason is, roughly speaking, that the classical strategy of « objects become nicer after finitely many adjustments » fails when the defect is non-trivial. I will discuss my previous and ongoing work in ramification theory in this setting; in particular, it allows us to understand and treat the defect. Background in ramification theory or valuation theory is not assumed.