Handwritten notes
- May 20 Benstein-Sato polynomial, differntial direct summands, BS poly on direct summands
- May 18 General construction of the D-module R_f[s] f^s, Benstein-Sato functional equation
- May 13 Local cohomology modules of infinite length and infinite associated primes, the D-module R_f[s] f^s
- May 11 Finite length and finite associated primes for local cohomology of polynomial rings in char 0
- May 6 Bernstein's inequality, holonomic D-modules
- May 4 Bernstein filtration, dimension and multiplicity for D-modules over char 0 poly rings
- Apr 29 Symmetry of differential operators on hypersurface, D-algebra simplicity of finite group invariants, good filtrations
- Apr 27 Symmetry of differential operatorson poly ring, symmetry on hypersurface
- Apr 22 Right Noetherianity of differential operators on node
- Apr 20 NonNoetherianity of differential operators on char p poly ring, cubic cone, assoc graded of node
- Apr 3 Filtrations, Noetherianity of differential operators on char 0 polynomial ring and finite group invariants
- Apr 1 D-algebra simplicity, faithfulness of local cohomology, Cohen-Macaulayness
- Mar 30 Frobenius, D-module simplicity, D-algebra simplicity
- Mar 27 D-ideals, D-module simplicity, Frobenius
- Mar 25 D-modules, local cohomology, differential equations
- Mar 23 Differential operators in positive characteristic, D-modules
- Mar 11 Differential operators and primary components, Stanley Reisner rings, cubic cone, positive characteristic
- Mar 4 Differential operators on invariants of finite groups 2
- Mar 2 Differential operators on invariants of finite groups 1
- Feb 26 Differential operators on cusp
- Feb 24 Operators into residue field in quasicoefficient field case, Zariski-Nagata theorems
- Feb 19 Zariski-Nagata easy containment, quasicoefficient fields
- Feb 17 Operators on general poly rings and quotients of poly rings, localization
- Feb 12 Modules of principal parts
- Feb 10 General definition of differential operators
- Feb 5 More on differential operators on poly rings in char 0
- Jan 22 Invariants of SL2
- Jan 20 Intro, differential operators on poly rings in char 0
Time and place:
Lunes y Miercoles, 9:30–10:50 AM, online
Zoom meeting ID: 973 5286 9532
Course information:
The course is a class on differential operators and applications to commutative algebra. A rough outline for the course is
- Basics of differential operators
- Principal parts
- Zariski-Nagata theorems on symbolic powers
- Examples of rings of differential operators
- D-modules and local cohomology
- D-algebra simplicity, D-module simplicity, and singularities
- Noetherianity of differential operator rings
- Holonomicity
- Finiteness properties of local cohomology
- Bernstein-Sato polynomials
Recommended texts:
- Countinho's A Primer of Algebraic D-modules, for differential operators on polynomial rings of characteristic zero
- Weyl's The Classical Groups, for the use of differential operators in invariant theory
- Grothendieck and Dieudonné's EGA IV Chapter 16 Section 8, for basics of differential operators
- McConnell and Robson's Noncommutative Noetherian Rings last chapter, for basics of differential operators
- Dao, De Stefani, Grifo, Huneke, Núñez-Betancourt's survey on symbolic powers, for Zariski-Nagata
- Cid-Ruiz's Noetherian operators, primary submodules and Zariski-Nagata theorem, for Zariski-Nagata
- Bernstein IN, Gelfand IM, Gelfand SI. Differential operators on a cubic cone. Uspehi Mat. Nauk. 1972;27(1):163, for operators on the cubic
- Tripp, J. "Differential operators on Stanley-Reisner rings." Transactions of the American Mathematical Society 349.6 (1997): 2507-2523.
- Twenty-four hours of local cohomology, by Iyengar et al, for local cohomology