Study programme 2020-2021 | Français | ||
Model Theory II Project | |||
Learning Activity |
Code | Lecturer(s) | Associate Lecturer(s) | Subsitute Lecturer(s) et other(s) | Establishment |
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S-MATH-050 |
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Language of instruction | Language of assessment | HT(*) | HTPE(*) | HTPS(*) | HR(*) | HD(*) | Term |
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Français | Français | 15 | 0 | 45 | 0 | 0 | A |
Description of the modifications to the Q3 2020-2021 assessment procedures (Covid-19) |
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No student took the course this year. |
Organisational arrangements for the end of Q1 2020-2021 assessments (Covid-19) online or face-to-face (according to assessment schedule)
Description of the modifications to the Q1 2020-2021 online assessment procedures (Covid-19) online or face-to-face (according to assessment schedule)
The material was the book of Pierre Simon on NIP theories (Lecture Notes in Logic 2015) together with notes of Artem Chernikov.
At the end of the first semester, a first evaluation consisted in an oral presentation by the student of the quantifier elimination result on Presburger arithmetic (not a matrial covered by the course).
A second evaluation consisted of the presentation of aresult of R. Buchi on the decidability of monadic second-order theory of the natural numbers with the successor -written notes (18 pages), together with an oral presenation to the model theory seminar (two lectures and an half)-also a material not covered in class.
A third evaluation consisted in the presentation of a choice of results in the course material together with written notes.
Content of Learning Activity
The aim of the course is to understand the proof of Morley's Theorem on aleph_1-categorical theories.
We begin by Ryll-Nardewski's Theorem on aleph_0-categorical theories. Then we will study the following notions:
-saturation, indiscernible sequences.
-Ramsey theorem and Ehrenfeucht-Mostwski's models.
-Vaught pairs, strongly minimal sets and pregeometries.
Finally of time permits:
- Morley and Cantor-Bendixon's ranks.
- definable types, heirs and co-heirs. Application in theories of modules.
- Fraïssé limits (e.g. the random graph).
Required Learning Resources/Tools
Marker, David Model theory. An introduction. Graduate Texts in Mathematics, 217. Springer-Verlag, New York, 2002.
Tent K., Ziegler M., A course in Model Theory, Lecture Notes in Logic, Cambridge University Press, 2012.
Recommended Learning Resources/Tools
Poizat B., Cours de th\'eorie des mod\`eles, 1985, Nur Al-Mantiq Wal-Ma'rifah. [Version anglaise éditée chez Springer en 2000.]
Hodges, Wilfrid Model theory. Encyclopedia of Mathematics and its Applications, 42. Cambridge University Press, Cambridge, 1993.
Other Recommended Reading
Jacobson, N., Basic Algebra 2, W.H. Freeman and Compagny, San Francisco, 1980.
Pillay A., An introduction to stability theory, Clarendon Press, Oxford, 1983. [Autre édition: Dover].
Mode of delivery
Type of Teaching Activity/Activities
Evaluations
The assessment methods of the Learning Activity (AA) are specified in the course description of the corresponding Educational Component (UE)