The theory of the defect and its application to the problem of local uniformization
Prof. Franz-Viktor Kuhlmann
11Department of Mathematics, University of Szczecin, Poland Email: fvk@math.usask.ca
Timetable: 16 hrs. First lecture on February 19, 2021, 15:00 (dates already fixed, see calendar), the course will be held online.
Some introductory lessons to the course will be held by prof. Giulio Peruginelli starting from February 15, 2021, 09:00.
Course requirements: Basic knowledge of field theory, ordered groups, basic valuation theory Examination and grading: Students will be evaluated by presentations they will give in the seminar that will follow the course.
SSD: MAT/02, MAT/03
Aim: To introduce graduate students to some deep open research problems in valuation theory, providing known partial results and their proofs.
Course contents:
- Meeting 1: Finite extensions of valued fields and their invariants - the fundamental in- equality - the defect - the Lemma of Ostrowski - examples of defect extensions.
- Meeting 2: Ramification theory of normal extensions of valued fields - absolute ramifica- tion theory - tame and separably tame fields.
- Meeting 3: Valued function fields - the Abhyankar inequality and Abhyankar valuations - extensions of valuations to function fields - Zariski spaces of places I.
- Meeting 4: The Generalized Stability Theorem I.
- Meeting 5: The Generalized Stability Theorem II - local uniformization for Abhyankar places.
- Meeting 6: Henselian Rationality I.
- Meeting 7: Henselian rationality II - local uniformization by alteration.
- Meeting 8: Connection with other problems: decidability of the elementary theory of Laurent series fields over finite fields - valued function fields revisited - Zariski spaces of places II - open problems and new approaches.
M-14
