This is the course website for a lecture and accompanying exercise sessions to be held at MPI-MiS Leipzig from November 4 to Christmas 2024. The course will take place twice a week, Mon 13:30-15:00 and Thu 11:00-12:30, usually in room G3 10. A precise schedule can be found below. To register for the lecture, please follow these instructions. In case you are interested in attending the course, but are not able to make it to Leipzig in person, please reach out to me.
Lecturer: Anna-Laura Sattelberger
Teaching assistant: Carlos Rodriguez (MPI-MiS Leipzig)
Course content
Algebraic analysis investigates linear PDEs and their solution functions by algebraic methods. The main actor is the Weyl algebra, denoted D. It is a non-commutative ring that gathers linear differential operators with polynomial coefficients. The theory of D-modules provides deep classification results of linear PDEs, structural insights into problems in the sciences, as well as new computational tools. Functions that can be encoded by an annihilating D-ideal are called holonomic, and these are ubiquitous in the sciences.
The course is hands-on: the focus lies on the introduction of concepts from algebraic analysis and utilizing them for solving problems arising in applications, such as a systematic study of Feynman integrals from particle physics. For software, we are going to employ Macaulay2 and Mathematica exclusively.
Lecture notes
The course was taught in a similar format at KTH in autumn 2023. A current version of the lecture notes is provided here. They might be changed and updated in the course of the lecture. Please report mistakes and typos to anna-laura.sattelberger@mis.mpg.de .
(Tentative) layout*
1. An algebraic counterpart of linear PDEs
1.1 The Weyl algebra
1.2 Properties and D-modules
2. Gröbner deformations of D-ideals
2.1 Initial ideals
2.2 Indicial ideals
7. Encoding D-ideals
7.1 Gröbner bases
7.2 Pfaffian systems
3. The characteristic variety
3.1 Holonomicity
3.2 Singular locus
4. Solutions and their singularities
4.1 Solution space
4.2 Regular vs. irregular singularities
5. Operations on D-modules
5.1 Integral transforms
5.2 Restricting and integrating
6. Holonomic functions
6.1 Weyl closure of D-ideals
6.2 Closure properties of holonomic functions
10. Relations among Mellin integrals
10.1 Bernstein-Sato ideals
10.2 Feynman integrals
*The numbering corresponds to the chapters in the lecture notes.
Schedule:Mon, Nov 4, 13h30: Lecture in G3 10 (Chapter 1)Thu, Nov 7, 11h00: Lecture in E2 10 (Chapter 2)Mon, Nov 11, 13h30: Lecture in G3 10 (Chapter 7)Thu, Nov 14, 11h00: Exercise session in G3 10Mon, Nov 18, 13h30: Exercise session in G3 10Thu, Nov 21, 11h00: Lecture in G3 10 (Chapter 3 and 4.1)Mon, Nov 25: NO LECTURE (instead: Macaulay2 workshop)Thu, Nov 28, 15h15: Lecture in G3 10 (Chapter 4.2)Mon, Dec 2, 13h30: Lecture in G3 10 (Chapter 5)
Thu, Dec 5, 11h00: Exercise session in G3 10Mon, Dec 9, 13h30: G3 10 Lecture in G3 10 (Chapter 6)Thu, Dec 12, 11h00: G3 10 Lecture in G3 10 (Chapter 10.1)
Mon, Dec 16, 13h30: G3 10 Lecture in G3 10Thu, Dec 19, 11h00: Q&A in G3 10
The lectures ended. The course website is going to stay active.
Anna-Laura Sattelberger 