2016 Graduate summer school in set theory

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.

Participants

• 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

Schedule

• 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 ω₁, as do certain instances of Chang's conjecture.
  • Afternoon session (2pm-5:30pm). Chris Lambie-Hanson. MM implies almost all points of cofinality ω₂ are good. Short PCF-theoretic proofs for strong "outside guessing of clubs" theorems.
• 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.