PhD course: D-Modules and Holonomic Functions

Course content:
The lecture introduces concepts from algebraic analysis and demonstrates its utility in problems in the sciences. Algebraic analysis investigates linear PDEs 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 PDEs, structural insights into problems in the sciences as well as new computational tools. This course focuses on the applied aspects of D-modules. The applications are ranging from maximum likelihood estimation in statistics through Feynman integrals in high energy physics to the computation of volumes of basic semi-algebraic sets to arbitrary precision.
D-ideals encode systems of linear PDEs with polynomial coefficients. The ideals encode crucial properties of the solutions of the associated system of PDEs, such as their singularities. These occur in two different kinds, namely as regular and irregular singularities. Series solutions of a regular holonomic D-ideal can be computed purely algebraically in terms of Gröbner deformations of the D-ideal. Due to Gröbner basis theories for the Weyl algebra, various software systems are available to compute with holonomic D-ideals and their solutions, which are called holonomic functions. Holonomic functions are ubiquitous in the sciences and their function values can be computed via the holonomic gradient method, a numerical evaluation scheme which makes use of an annihilating D-ideal of the function.

Tentative schedule: (adjustments might be made in the course of the lecture)
Lecture 1: An algebraic counterpart of linear PDEs
Lecture 2: Gröbner deformations of D-ideals
Lecture 3: The characteristic variety
Lecture 4: Solutions and their singularities
Lecture 5: Operations on D-modules
Lecture 6: Holonomic functions
Lecture 7: Encoding D-ideals
Lecture 8: Evaluating holonomic functions
Lecture 9: Computing solutions of D-ideals
Lecture 10: Volumes or Feynman Integrals

Main references:

• M. Saito, B. Sturmfels, and N. Takayama. Gröbner Deformations of Hypergeometric Differential Equations, volume 6 of Algorithms and Computation in Mathematics. Springer Berlin, Heidelberg, 2000.
• A.-L. Sattelberger and B. Sturmfels. D-Modules and Holonomic Functions. Preprint arXiv:1910.01395. To appear in the volume Varieties, polyhedra, computation of EMS Series of Congress Reports. 

Required prerequisites:
Commutative algebra; basic knowledge of complex analysis is helpful.

Intended learning outcome:
The students shall understand and be able to apply the concepts from algebraic analysis that are introduced in the course. For instance, this includes the Fundamental Theorem of Algebraic Analysis and the theorem of Cauchy-Kovalevskaya-Kashiwara. The course participants shall be able to take advantage of computer algebra systems to compute with D-ideals and to apply learned concepts such as the computation of solutions to D-ideals arising in applications.

Exam:
The exam takes place during the two lectures in the last week of study period 2, i.e., Dec 11-15, 2023.
Format: Short presentations by the students + written summary of their presentation. The precise topic will be allocated to the students in due time. The registration procedure is going to be communicated in the course.
Credits: 7.5 hp