PASCA 2.0 @ CIMAT: Lecture Series, May 2026
Typed notes
Soumyadeep Misra typed up notes of my lectures. Typed notes of Schwede's lecture series (typed by Misra) Hand written ntoes can be found below.Exercises
- Exercises for lecture 1. For additional references, see:
- Chapter 1 of Singularities defined by the Frobenius map (Schwede-Smith)
- Frobenius Splitting in Commutative Algebra (Smith-Zhang)
- Exercises for lecture 2. For additional discussion, see:
- A characterization of rational singularities (Kovács).
- Theorem 4.1.3 in Bhargav Bhatt's thesis
- Rational singularities associated to pairs (Schwede-Takagi)
- Section 8 of F-singularities via alterations (Blickle-Schwede-Tucker)
- Multiplier ideals and klt singularities via (derived) splittings (McDonald)
- Chapter 6 of Singularities defined by the Frobenius map (Schwede-Smith)
- Exercises for lectures 3 and 4. For additional discussion see:
- On F-pure thresholds (Takagi-Watanabe).
- F-thresholds and Bernstein-Sato polynomials (Mustață-Takagi-Watanabe)
- BCM-thresholds of hypersurfaces (Rodríguez-Villalobos)
- Plus-pure thresholds of some cusp-like singularities in mixed characteristic (Cai-Pande-(Quinlan-Gallego)-Schwede-Tucker)
- Bounds on the plus-pure thresholds of some hypersurfaces in (ramified) regular rings (Benozzo-Jagathese-Pandey-(Ramírez-Moreno-)Schwede-Sridhar)
- A Criterion for Perfectoid Purity and the Rationality of Thresholds (Yoshikawa)
- Exercises for lecture 5.
For addition discussion see.
- Closure operations induced via resolutions of singularities in characteristic zero (Epstein-McDonald-R.G.-Schwede)
- The Briançon-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings (Ma-McDonald-R.G.-Schwede)
- Integral Closure of Ideals, Rings, and Modules (Swanson-Huneke)
Other notes and references
Much material can be found in the book draft with Karen Smith, Singularities defined by the Frobenius map.Some other material related to to complex RΓ(𝑌 , 𝒪𝑌 ) can be found in these papers: Closure operations induced via resolutions of singularities in characteristic zero (Epstein-McDonald-R.G.-Schwede) and The Briançon-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings (Ma-McDonald-R.G.-Schwede).
- Handwritten notes for Lecture 1
Allhappy families(regular rings) are alike; eachunhappy family(singular ring) is unhappy in its own way. -- Tolstoy - Handwritten notes for Lecture 2
You say you want arevolution(resolution) -- The Beatles - Handwritten notes for Lecture 3
But in waking life, too, we continue to dream beneath the threshold ofconsciousness(singularities) -- Carl Jung - Handwritten notes for Lecture 4
Life is really worth living in aNoetherianring R when all the local rings have the property that every s.o.p. is an R -sequence. -- Mel Hochster - Handwritten notes for Lecture 5
I know that it's over.
I don't needyour(plus) closure. -- Taylor Swift