M9260 Teorie modelů

Přírodovědecká fakulta
podzim 2017
Rozsah
2/0/0. 2 kr. (příf plus uk k 1 zk 2 plus 1 > 4). Ukončení: zk.
Vyučující
Michael Joseph Lieberman, B.A., Ph.D. (přednášející)
Garance
prof. RNDr. Jiří Rosický, DrSc.
Ústav matematiky a statistiky – Ústavy – Přírodovědecká fakulta
Dodavatelské pracoviště: Ústav matematiky a statistiky – Ústavy – Přírodovědecká fakulta
Rozvrh
Po 18. 9. až Pá 15. 12. Út 16:00–17:50 M4,01024
Předpoklady
Familiarity with predicate logic, basic abstract algebra, and naive set theory would be very helpful, but is not strictly required. The chief prerequisite is a level of mathematical maturity commensurate with an advanced undergraduate/graduate course.
Omezení zápisu do předmětu
Předmět je nabízen i studentům mimo mateřské obory.
Mateřské obory/plány
Cíle předmětu
At the end of the course, students will be able to understand and apply the tools and techniques of modern model theory, including:
Logical compactness
Quantifier elimination
Saturated models
Fraisse limits
Omitting types, stability
Abstract independence, bases, and dimension
Osnova
  • The main emphasis of the course will be on the following topics:
  • Similarity types, structures.
  • Completeness and compactness theorems, Lowenheim-Skolem theorem.
  • Construction of models from constants, Henkin's omitting types theorem, prime and atomic models.
  • Elementary chains of models, saturated models.
  • Quantifier elimination, model completeness.
  • Results on countable models, including Ryll-Nardzewski's theorem.
  • Stability (especially omega-stability), rank functions.
  • Indiscernible sequences, Ehrenfeucht-Mostowski models.
  • Morley's categoricity theorem.
  • Applications: algebraic geometry, number theory, analysis.
  • Additional topics may include:
  • Combinatorial set theory
  • Fraisse limits
  • Related categorical structures
Literatura
    doporučená literatura
  • MARKER, David. Model theory : an introduction. New York: Springer, 2002, viii, 342. ISBN 0387987606. info
    neurčeno
  • HODGES, Wilfrid. A shorter model theory. Cambridge: Cambridge University Press, 1997, x, 310. ISBN 0521587131. info
  • BUECHLER, Steven. Essential stability theory. Berlin: Springer-Verlag, 1996, xiv, 355. ISBN 3540610111. info
Výukové metody
The course will consist of two hours of lecture each week, with optional biweekly problem sets to be completed outside of class.
Metody hodnocení
Students will be evaluated on the basis of one written exam, given during the middle of the term and completed outside of class, and a final oral exam, which will consist of a small number of problems, most of which will be selected at random from a list circulated to students before the beginning of the exam period.
Attendance will not be required for completion of the course, but is essential to the achievement of the course's objectives. Optional assignments will be handed out regularly but will not count towards a student's grade, although they will be corrected and returned promptly.
Informace učitele
http://www.math.muni.cz/~lieberman/M5920P17.html
Dr. Michael Lieberman, PhD.
Výzkumný a vývojový pracovník
Ústav matematiky a statistiky
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