# Program

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.
Bibliography

**Keywords and Mathematics Subject Classification (MSC)**

**Tags/Keywords**

Blowup algebras

combinatorial commutative algebra

computational aspects

differential and characteristic p methods

D-modules

elimination theory

equisingularity theory

F-modules

Frobenius

homological algebra

K-theory

linkage and residual intersection

multiplicity theory

Noetherianity properties

singularity theory

syzygies

valuation theory

**Primary Mathematics Subject Classification**

**Secondary Mathematics Subject Classification**

January 18, 2024 - January 19, 2024 | Connections Workshop: Commutative Algebra |

January 22, 2024 - January 26, 2024 | Introductory Workshop: Commutative Algebra |

April 15, 2024 - April 19, 2024 | Recent Developments in Commutative Algebra |