Graduate level course (broad course) at KTH and Stockholm University

Fall term 2023, study period 2 (Oct 30 – Dec 15)**Format and schedule**:

Mondays, 14:00-16:00h and Thursdays, 10:00-12:00h in study period 2 (Oct 30 – Dec 15, 2023)

Seminar room 3418, KTH Campus, D-huset**Lecturer:**

Anna-Laura Sattelberger (Department of Mathematics, KTH Royal Institute of Technology)

Contact: alsat@kth.se

**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 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 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: The Weyl algebra

Lecture 2: Gröbner deformations of D-ideals

Lecture 3: The characteristic variety

Lecture 4: Gröbner bases

Lecture 5: Operations on D-modules

Lecture 6: Solutions and their singularities

Lecture 7: Holonomic functions

Lecture 8: Holonomic gradient method and descent

Lecture 9: Computing volumes

Lecture 10: Computing solutions of D-ideals

**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

**Intended learning outcome:**The students shall understand and be able to compute and to apply the concepts from algebraic analysis that are introduced in the course.

**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:** 5 points**Registration:**

To register for the course, please send an e-mail to alsat@kth.se .