Mathematical Sciences Research Institute

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Commutative Algebra January 16, 2024 to May 24, 2024
Organizers Aldo Conca (Università di Genova), Steven Cutkosky (University of Missouri), LEAD Claudia Polini (University of Notre Dame), Claudiu Raicu (University of Notre Dame), Steven Sam (University of California, San Diego), Kevin Tucker (University of Illinois at Chicago), Claire Voisin (Collège de France; Institut de Mathématiques de Jussieu)
9 points theorem
Image for theorem about 9 point on cubic curve, the special case of Cayley–Bacharach theorem.
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.
Keywords and Mathematics Subject Classification (MSC)
  • 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
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Programmatic Workshops
April 15, 2024 - April 19, 2024 Recent Developments in Commutative Algebra