Applications of model theory to algebra and geometry (MAGIC004) |
GeneralDescription
This will be a 10 hour "version" of what was previously given (Spring 2008, Spring 2010) as a
20 hour course.
However, I plan to be somewhat more ambitious, at the expense of leaving many details of the introductory material to be done in exercises.
Among the new aspects of the new version will be the inclusion of stability theory.
The first 5 hours will introduce model theory and stability theory, including stable groups.
The last 5 hours will concern connections and applications, largely where groups of one form or the other are involved, and will be taken from among: geometric group theory, diophantine geometry over number fields and function fields (Manin-Mumford, Mordell-Lang), approximate subgroups,..
SemesterSpring 2012 (Monday, January 16 to Friday, March 23) Timetable
PrerequisitesSome familiarity with first order logic would be helpful but not essential.
Syllabusbeginitemize
item textbfLectures 1 to 5: BASICS OF MODEL THEORY AND STABILITY THEORY: First order languages, structures and theories, compactness, types, saturation and homogeneity, stability, stable groups.
item textbfLecture 7 to 10: APPLICATIONS: I will cover 4 or 5 topics, explaining why and how material from the earlier lectures can give insights in areas of algebra, geometry, and number theory. More details will be given closer to the start date.
enditemize
Students
Bibliography
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The assignments will be connected with filling in details and solving some problems related to basic material in the first past of the course, and can serve as a means for non-experts to gain some entry into and competence in general model theory.
Assignment 1
Assignment 2
Assignment 3
Assignment 4
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