Séminaire de Mathématiques Supérieures 2025: An Introduction to Recent Trends in Commutative Algebra

Overview 

Commutative algebra, the study of commutative rings and their modules, is a central field in mathematics. Commutative algebra provides the algebraic language used within algebraic geometry, and via the Stanley-Reisner correspondence, commutative algebraic techniques can be brought to bear upon problems in combinatorics. The implementation of Buchberger’s work on Gröbner bases and modern computing power has enabled the commutative algebra community to experiment and generate examples and conjectures that would have been unattainable to founders of the field, like Hilbert and Noether. In recent years, commutative algebra has also found numerous applications beyond pure mathematics, to areas such as algebraic statistics, computer vision, cryptography, and coding theory. In light of all of these connections, commutative algebra is an important and active area of research. 

The 2025 SMS will allow graduate students to learn about a number of recent trends and advances in the field of commutative algebra. The aim of the SMS is to provide an “on-ramp” for graduate students interested in algebra, combinatorics, and/or algebraic combinatorics to learn more about commutative algebra’s interaction with these fields. The introductory courses will introduce fundamental skills in commutative algebra, the more intermediate courses will expose students to cutting-edge research in the field. The school will focus on four topics within commutative algebra: Combinatorial Methods, Homological Methods, Computational Methods, and Characteristic p Methods. The SMS will provide both a series of introductory lectures and intermediate/advanced lectures from leaders in one of the four areas. The lectures will include a series of problem sessions that will allow participants to develop and hone their skills in these areas, which will be especially helpful for new people to the field. Participants will be encouraged to work collaboratively, both to enhance their own mathematical networks as well as to promote future collaborations beyond the school. 

This summer school will take place from June 2-13 2025 and is part of the Fields Institute thematic program in Commutative Algebra and Applications.

School structure 

Each course will have a problem session, with problems provided by the instructors. The organizers and other instructors will work with the students during the problem session to create a more collaborative experience. The second week will additionally include lightning talks to give students an opportunity to present their work and also a professional development panel. In both the first and second weeks, we will reserve Wednesday afternoon as a free block for informal discussions, collaboration, or restorative down time and socializing.

Prerequisites 

This school is primarily aimed at graduate students, especially those who are at the beginning of their program and have an interest in algebra, algebraic combinatorics, or algebraic geometry. We will expect students to have completed at least the first year of their graduate program, including the material covered in a typical first-year graduate algebra sequence that many schools use as preparation for a qualifying/prelim exam. To accommodate differing levels of background, students will self-select into either a beginning group or an advanced group for the problem session component of each course. Advanced students will also be encouraged to help the beginning students. The organizers and speakers will also be on hand to help provide any additional background participants may require.

Basic courses 

The first week of the proposed SMS will consist of four introductory courses, which introduce the participants to four main themes of current research in commutative algebra. 

1. Combinatorial Methods in Commutative Algebra. The course will cover the basics about the connection between simplicial complexes and monomial ideals via the Stanley-Reisner and facet ideal constructions. Students will learn how to use this dictionary between combinatorial algebraic topology and commutative algebra. This course will be taught by Sara Faridi. 
2. Homological Methods in Commutative Algebra. The graded minimal free resolution of a graded module was first introduced by Hilbert. Resolutions continue to be the source of many interesting research questions. This course will introduce the basic concepts in the area, along with important invariants like Hilbert functions, Betti numbers and the Castelnuovo-Mumford regularity. This course will be taught by Claudia Miller. 
3. Computational Methods in Commutative Algebra. Gröbner bases are the underlining tool used to perform computations in commutative algebra and algebraic geometry. This course will introduce the basic results of Gröbner bases, and explain their importance in computational commutative algebra. Students will also learn how to use computer algebra systems to compute these bases. This course will be taught by Federico Galetto. 
4. Characteristic p Methods in Commutative Algebra. This course will introduce key positive characteristic methods, including a suite of techniques used to study problems in commutative algebra and algebraic geometry that makes use of the Frobenius morphism. This course will be taught by Jack Jeffries. 

Advanced courses (details subject to change) 

The basic courses will be followed up in the second week by four intermediate to advanced level courses. Possible titles and topics for these courses include: 

1. Gröbner Geometry and Applications. This course will explore the use of Gröbner bases, degenerations, and related combinatorics to study problems in commutative algebra and algebraic geometry. This course will be taught by Sergio Da Silva and Patricia Klein. 
2. Multigraded Modules. In this follow up course on resolutions, we will discuss recent techniques and progress in the study of multigraded modules. In this context, we get “finer” invariants, like a multi-graded version of the Castelnuovo-Mumford regularity. This course will be taught by Christine Berkesch. 
3. Homological Invariants of Points in Projective Space. This course will focus on the study of homological questions and invariants of points in projective space (e.g., Hilbert functions, resolutions, regularity). Students will see the interplay between classical algebraic geometry and commutative algebra, e.g., how to use the Hilbert function to deduce geometric information about the set of points. This course will be taught by Adam Van Tuyl and Elena Guardo. 
4. New Developments in Positive Characteristic Commutative Algebra. This course will survey recent results on F-singularity theory, such as new results about F-regularity, F-pure thresholds, and test ideals. This course will be taught by Daniel Hernández.