Teaching

Information regarding the course Model Theory of differential fields in characteristic zero.

Practical information

The lectures take place on the Fridays from 9.45am to 11.15am and from 11.30am to 1.00 pm with a fifteen minutes break in the middle. The rooms for each lecture are indicated below. Be careful the room change with the lectures.

• Lectures 1/2 on 12/01/2024. Quai 43, s.104 (1er étage)
• Lectures 2/3 on 19/01/2024. Darwin D71 (RDC)
• Lectures 3 on 26/01/2024. Braconnier Séminaire 1
• Lectures 4/5 on 02/02/2024. Darwin C salle Caullery (RDC)
• Lectures 5/6 on 09/02/2024. Darwin C salle Caullery (RDC)
• Lectures 6/7 on 23/02/2024. Darwin C salle Caullery (RDC)
• Lectures 8 on 08/03/2024. Braconnier Séminaire 1
• Lectures 9 on 15/03/2024. Braconnier Séminaire 1

More information about the location of each of the rooms can be found here .

Lecture notes

• Lecture 1. Motivation
• Lecture 2. Ritt-Raudenbush theory
• Lecture 3. Differentially closed fields
• Lecture 4. Seidenberg embedding Theorem
• Lecture 5. Differential forms
• Homework. Homework due on March 8th
• Lecture 6. The Ax-Schanuel Theorem

References

• David Marker, Model Theory of differential fields (second chapter of the book Model Theory of fields) pdf.
• Rosenlicht, On Liouville theory of elementary functions, Pacific Journal of Mathematics, Vol. 65, No. 2, 1976. pdf.
• Bouscaren E., Model Theory and Algebraic Geometry, Lecture notes in Mathematics link .

Oral examination

For the oral examination, you will be given two subjects in the following list from which you will choose one:

• The Ritt-Raudenbush theorems (Lecture 2)
• Differentially closed fields and elimination of quantifiers (Lecture 3)
• Elimination of imaginaries (Lecture 4)
• The Seidenberg embedding Theorem (Lecture 5)
• Algebraic properties of differential forms (Lecture 6)
• The Ax-Schanuel Theorem (Lecture 8)

You will have 20 minutes of preparation (without access to course notes) to prepare a 15 minutes presentation which will be followed by 20 minutes of questions and discussion on the topic you have picked.

The oral examinations will take place on Thursday March 26th from 9am to 13am and on Friday March 27th from 9am to 12.25am in the basement of the Batiment Braconnier. Previsional schedule:

• Thursday March 26th from 9am to 9.55am: Lucas
• Thursday March 26th from 10am to 10.55am: Mathilde
• Thursday March 26th from 11am to 11.55am: Romain

• Thursday March 26th from 12am to 12.55am: Nawzad

• Friday March 27th from 8.45am to 9.40am: Pierre-Louis
• Friday March 27th from 9.40am to 10.35am: Jeremie
• Friday March 27th from 10.35am to 11.30am: Quentin
• Friday March 27th from 11.30am to 12.25am: Antonin

Please come 10 minutes before the start of your oral exam.