## Commutative Algebra

**Date:**
16 January - 24 May 2024

**Location: ** Simons Laufer Mathematical Research Institute

**Event type:**
Extended Format

**Organisers: ** Aldo Conca (Genova),
Steven Cutkosky (Missouri), Claudia Polini (Notre Dame),
Claudiu Raicu (Notre Dame),
Steven Sam (UCSD), Kevin Tucker (Illinois at Chicago), Claire Voisin (Collège de France; Jussieu)

Commutative algebra is, in its essence, the study of algebraic objects, such as rings and modules over them, arising from polynomials and integral numbers. It has numerous connections to other fields of mathematics including algebraic geometry, algebraic number theory, algebraic topology and algebraic combinatorics. Commutative Algebra has witnessed a number of spectacular developments in recent years, including the resolution of long-standing problems, with new techniques and perspectives leading to an extraordinary transformation in the field. The main focus of the program will be on these developments. These include the recent solution of Hochster’s direct summand conjecture in mixed characteristic that employs the theory of perfectoid spaces, a new approach to the Buchsbaum–Eisenbud–Horrocks conjecture on the Betti numbers of modules of finite length, recent progress on the study of Castelnuovo–Mumford regularity, the proof of Stillman’s conjecture and ongoing work on its effectiveness, a novel strategy to Green’s conjecture on the syzygies of canonical curves based on the study of Koszul modules and their generalizations, new developments in the study of various types of multiplicities, theoretical and computational aspects of Gröbner bases, and the implicitization problem for Rees algebras and its applications.

Professor Bernd Ulrich (Purdue) has been appointed as a Clay Senior Scholar to participate in this program.

CMI Enhancement and Partnership Program