# 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.

- 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.
**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.**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.**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:

**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.**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.**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.**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.

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.