Banca de DEFESA: Nowras Naufel Ali Mahamoud Otmen

Uma banca de DEFESA de MESTRADO foi cadastrada pelo programa.
STUDENT : Nowras Naufel Ali Mahamoud Otmen
DATE: 11/07/2023
TIME: 14:00
LOCAL: PPGMAT
TITLE:

Galois groups of function fields with prescribed 
ramification

KEY WORDS:

Riemann surfaces; ramification; differential forms; sheaf 
cohomology; Riemann-Roch; Riemann-Hurwitz; function fields; Galois 
theory of valuations; profinite groups; Galois groups.

PAGES: 115
BIG AREA: Ciências Exatas e da Terra
AREA: Matemática
SUMMARY:

he purpose of this dissertation is to understand the phenomenon of 
ramification.

On one hand, we investigate what happens in the more classical and 
‘geometrical’ case of Riemann surfaces, exploring their basic 
properties, what it means for a holomorphic function between surfaces 
to have ramification and branch points, the definitions of divisors 
and of the genus of a compact Riemann surface X and the theorems of 
Riemann-Roch and Riemann-Hurwitz. We aim to exemplify these concepts 
via a few examples and calculations.

On the other hand, we talk about the concept of function fields and, 
using the language of valuations, places and valuation rings, we 
define for function fields concepts which are, in some sense, very 
similar to the ones we study in the Riemann surfaces case. It is the 
intention to highlight the similarity between both cases.

Finally, in the last chapter, we explore how the genus of functions 
fields can be used to prove results regarding their Galois groups; 
specifically, that the genus and the ramification of certain prime 
divisors profoundly influence the structure of these profinite groups.

COMMITTEE MEMBERS:
Externo à Instituição - AMILCAR PACHECO - UFRJ
Externo à Instituição - MARCO BOGGI - UFF
Interno - 2255154 - MARTINO GARONZI
Presidente - 2554004 - THEO ALLAN DARN ZAPATA