A graduate summer school in set theory was held July 25 - August 5 at the University of California Irvine. The subject of the summer school was singular cardinal combinatorics, with a focus on PCF theory.

The instructors were James Cummings (CMU), Christopher Lambie-Hanson (Hebrew University of Jerusalem) and Bill Chen (UCLA).

The summer school was supported by the National Science Foundation grant DMS-1044150 as part of the program EMSW21-RTG: Logic in Southern California.

- Shehzad Ahmed, graduate student student, Ohio University
- David Casey, undergraduate, University of Chicago
- Jin Du, graduate student, University of Illinois at Chicago
- Thomas Gilton, graduate student, University of California, Los Angeles
- Jared Hilliard, graduate student, University of California, Irvine
- Jared Holshouser, graduate student, University of North Texas
- Kameran Kolahi, graduate student, University of California, Irvine
- Hossein Lamei Ramandi, graduate student, Cornell University
- Jeffrey Robert Schatz, graduate student, University of California, Irvine
- Benjamin Siskind, graduate student, University of California, Berkeley
- Ryan Sullivant, graduate student, University of California, Irvine
- Trevor White, graduate student, University of California, Irvine
- Jing Zhang, graduate student, Carnegie Mellon University

- Monday 25 July:
- Morning session (9am-12:30pm). James Cummings. Filters and ideals, ultrafilters, reduced products, least upper bounds and exact upper bounds, obstacles to the existence of exact upper bounds.
- Afternoon session (2pm-5:30pm). James Cummings. Statement and proof of the Trichotomy theorem, good points and strongly increasing sequences, using good points to ensure the existence of exact upper bounds.

- Tuesday 26 July:
- Morning session (9am-12:30pm). James Cummings. Club guessing, building sequences with many good points, some examples of scales.
- Afternoon session (2pm-5:30pm). James Cummings. Progressive sets, PCF of a progressive set, basic facts about PCF and the associated ideals, PCF as a closure operator, universal sequences.

- Wednesday 27 July:
- Morning session (9am-12:30pm). Bill Chen. PCF generators and the associated scales, cofinality of the product of a progressive set, no holes theorem, elementary applications to cardinal arithmetic.
- Afternoon session (2pm-5:30pm). Bill Chen. Internally approaching chains, characteristic functions, reconstruction of an internally approachable structure from its characteristic function, minimally obedient sequences.

- Thursday 28 July:
- Morning session (9am-12:30pm). Bill Chen. Minimal obedience, computing characteristic functions of an internally approachable structure, cofinal families of small subsets of sup(A) for A a progressive interval.
- Afternoon session (2pm-5:30pm). Bill Chen. Pseudopowers, sigma-complete pcf, covering numbers, covering vs pp theorem, statement of the revised GCH theorem.

- Friday 29 July:
- Morning session (9am-12:30pm). Bill Chen. Proof of the revised GCH theorem.
- Afternoon session part 1 (2pm-4pm) Bill Chen. An application (by Kojman) of the revised GCH to graph colouring numbers.
- Afternoon session part 2 (4:30pm-5:30pm) James Cummings. Covering by finite unions of generators, PCF structure for progressive A is invariant under min(A)-cc forcing, a PCF theoretic version of Silver's theorem.

- Monday 1 Aug:
- Morning session (9am-12:30pm). Chris Lambie-Hanson. Good, better and very good scales, approachability, the ideal I[λ], (weak) squares, constructing better and very good scales from weak squares.
- Afternoon session (2pm-5:30pm). Chris Lambie-Hanson. Interaction between large cardinals and scales, supercompact cardinals imply bad scales, supercompact cardinals imply existence of non-reflecting stationary sets. Analysis of of the least singular cardinal where the PCF-theoretic version of SCH fails.

- Tuesday 2 Aug:
- Morning session (9am-12:30pm). Chris Lambie-Hanson. MM implies a stationary set of bad points
in cofinality ω
_{1}, as do certain instances of Chang's conjecture. - Afternoon session (2pm-5:30pm). Chris Lambie-Hanson. MM implies almost all points of cofinality
ω
_{2}are good. Short PCF-theoretic proofs for strong "outside guessing of clubs" theorems.

- Morning session (9am-12:30pm). Chris Lambie-Hanson. MM implies a stationary set of bad points
in cofinality ω
- Wednesday 3 Aug:
- Morning session (9am-12:30pm). Chris Lambie-Hanson. Weak squares in outer models, Jónsson cardinals.
- Afternoon session part 1 (2pm-4pm). Chris Lambie-Hanson. Instances of incompactness: ADS, NPT, almost free Abelian groups, and almost metrisable first countable topological spaces.
- Afternoon session part 2 (4:30pm-5:30pm) James Cummings. Boolean algebras, Stone spaces and Stone duality.

- Thursday 4 Aug:
- Morning session (9am-12:30pm). James Cummings. The transitive generators theorem.
- Afternoon session (2pm-5:30pm). James Cummings. The localisation theorem, bound on the size of the PCF of a progressive interval.

- Friday 5 Aug:
- Morning session (9am-12:30pm). James Cummings. Transitive closed generators, PCF topology as a Stone space, superatomic Boolean algebras, PCF structures.