Videos

Recent Developments in Commutative Algebra: "Closure operations and rational singularities"

Presenter
April 18, 2024
Keywords:
  • Commutative rings
  • modules
  • ideals
  • mixed characteristic
  • Frobenius powers
  • test ideals
  • tight closure
  • perfectoid methods
  • singularities
  • birational algebraic geometry
  • multiplier ideals
  • symbolic powers
  • syzygies
  • free resolutions
  • homological methods
  • derived categories
  • polynomials
  • monomial ideals
  • toric varieties
  • Schubert varieties
  • combinatorial commutative algebra
  • equivariant ideals
  • maximal Cohen-Macaulay modules
  • applications of representation theory
  • twisted commutative algebras
  • D-modules
  • local cohomology
  • computational commutative algebra
  • graded rings and projective varieties
MSC:
  • 05Exx - Algebraic combinatorics
  • 11Sxx - Algebraic number theory: local fields
  • 11Txx - Finite fields and commutative rings (number-theoretic aspects)
  • 13-XX - Commutative algebra
  • 14-XX - Algebraic geometry
  • 16Exx - Homological methods in associative algebras {For commutative rings
  • see \newline 13Dxx
  • for general categories
  • see 18Gxx}
  • 18Gxx - Homological algebra in category theory
  • derived categories and functors [See also 13Dxx
  • 16Exx
  • 20Jxx
  • 55Nxx
  • 55Uxx
  • 57Txx]
  • 19Axx - Grothendieck groups and $K_0$K_0 [See also 13D15
  • 18F30]
  • 19Lxx - Topological $K$K-theory [See also 55N15
  • 55R50
  • 55S25]
  • 20Jxx - Connections of group theory with homological algebra and category theory
Abstract
This talk will focus on the ways closure operations can be used to detect rational singularities and its analogues. I will give characteristic-free properties a closure operation needs to have in order to generate something like F-rational singularities (joint with Zhan Jiang). Then I will define a closure operation in equal characteristic 0 that can be used to detect singularities there without reduction to characteristic p (joint with Neil Epstein, Peter McDonald, and Karl Schwede).