MSc Pure Mathematics / Course details

Model theory deals with the properties of mathematical structures and theories which can be expressed in a formal language. Model-theoretic tools can be used to classify and analyse various structures and apply this analysis to solve problems in other areas of mathematics such as algebra, geometry, number theory, and analysis. Model theory has applications outside mathematics too, but we will not talk about those in this course.

In this unit you will learn about key concepts in model theory such as elementary embeddings and extensions, definable sets, types, categoricity, quantifier elimination, model completeness, strong minimality, and o-minimality. Some of the major theorems we will prove include the Lowenheim-Skolem theorem, the Ryll-Nardzewski theorem, quantifier elimination for algebraically closed and real closed fields, and categoricity for several important theories. Plenty of examples will be provided to illustrate the introduced ideas and concepts, and the developed tools will be applied to examine properties of important algebraic structures such as fields and orders.

Familiarity with mathematical logic (including basic set theory) and abstract algebra (groups, rings, fields, vector spaces) is required for this course, although the relevant concepts will be briefly recalled when necessary. The lecture notes will contain appendices covering some of the preliminaries. 

Students are not permitted to take, for credit, MATH43051 in an undergraduate programme and then MATH63051 in a postgraduate programme at the University of Manchester, as the courses are identical.

The unit aims to introduce and analyse key concepts, results, and techniques from model theory, and to investigate applications of those to problems both in model theory and in other parts of mathematics such as algebra and geometry.

On successful completion of this course unit students will be able to:

• Define key concepts such as elementary embeddings, types, definable sets, and explain them in various examples of structures.
• State, prove, and apply major theorems, such as Löwenheim-Skolem and Ryll-Nardzewski.
• Construct and describe sentences, formulae, and types expressing specific mathematical properties.
• Describe and analyse definable sets and types in a given structure.
• Define and manipulate key model-theoretic tools to establish properties of structures and theories.
• Construct models of a first-order theory with specific properties and analyse what other attributes they have.
• Classify first-order theories and structures by the properties they have.
• Apply and combine various model-theoretic techniques to solve problems in pure mathematics such as in algebra and geometry.

This is a list of the topics to be covered in lectures. In tutorials we will discuss some deeper/open ended questions related to that week’s material.

1. Review of first-order logic (2 lectures)
2. Theories, compactness (1 lecture)
3. Substructures, morphisms, embeddings, elementary versions (2 lectures)
4. Tarski-Vaught, method of diagrams (2 lectures)
5. Löwenheim-Skolem theorem (1 lecture)
6. Definable sets and their properties (1 lecture)
7. Types (3 lectures)
8. Categoricity, back and forth, Ryll-Nardzewski theorem (2 lectures)
9. Quantifier elimination, model completeness (2 lectures)
10. Algebraically closed fields (2 lectures)
11. Real closed fields (3 lectures)
12. Model-theoretic definable and algebraic closures (1 lecture) 

There will be three contact hours per week, two lectures and one tutorial. Students will be able to submit homework solutions and get feedback on that, and any questions will be answered during tutorials and office hours. Homework reading will also be assigned regularly for independent study. Tutorials will mostly focus on exploring deep and open-ended questions through active learning. Students will also be encouraged to ask, answer, and discuss questions on a suitable discussion forum (that will be set up for them). The lectures will be as interactive as possible to give the students in-depth knowledge of the material. 

• Mid-semester coursework: weighting 20%
• End of semester examination: weighting 80%

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework also provides an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
 

I will provide full course notes but there are quite a few texts on model theory around.  For example, those below, but they are aimed at graduate students so don’t expect to move quickly when reading them.   There are also sets of lecture notes on the web.  So you can browse around and see what you like/what’s helpful.

• David Marker, Model theory. An introduction. Graduate Texts in Mathematics, 217. Springer-Verlag, New York, 2002. viii+342 pp.
• C.C.Chang and H.J.Keisler, Model Theory, various editions (though it goes overboard on ultraproducts).
• Wilfrid Hodges, Model theory. Encyclopedia of Mathematics and its Applications, 42. Cambridge University Press, Cambridge, 1993. xiv+772 pp.  “A Shorter Model Theory” is the cut-down, student, version

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

· You will normally have approximately 75-120 minutes of video content. Normally you would spend approximately 2.5-4 hrs per week studying this content independently
· You will normally have exercise or problem sheets, on which you might spend approximately 2-2.5hrs per week
· There may be other tasks assigned to you on Blackboard, for example short quizzes, short-answer formative exercises or directed reading
·  In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 9 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.

The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.