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 have to choose and present one of the following subject (out of three randomly picked subjects) in the following list

• Prime differential ideals of differential polynomials (Lecture 2)
• Radical differential ideals of differential polynomials (Lecture 2)
• Elimination of quantifiers in the theory of differentially closed fields and consequences (Lecture 3)
• Elimination of imaginaries in the theory of differentially closed fields (Lecture 3)
• The Seidenberg embedding Theorem (Lecture 4)
• Algebraic properties of differential forms (Lecture 5)
• The Galois-theoretic proof of the functional Lindermann-Weierstrass Theorem (Lecture 1)
• The Liouville-Rosenlicht Theorem (Lecture 6)
• The Ax-Schanuel Theorem (Lecture 6)

The oral examinations will take place on Tuesday March 26th from 1pm to 6.30pm in room Berthollet 126 (1er étage) based on the following schedule.

• 1.15pm to 2pm: Youssef
• 2pm to 2.45pm: Alexandre
• 2.45pm to 3.30pm: Yoann
• 3.30pm to 4.15pm: Enzo
• 4.15pm to 5pm: Quang-Khai

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