# 2015 graduate summer school in set theory

Supported by NSF Grant DMS-1044150

Theme:  Forcing and large cardinals
Dates: July 6 through July 17, 2015
Location:  University of California at Irvine
Instructors:  Brent Cody (VCU), Monroe Eskew (VCU), and Spencer Unger (UCLA).
Organizer:  Sean Cox (Department of Mathematics and Applied Mathematics, Virginia Commonwealth University).  If you are a graduate student (Masters or PhD) with background in set theory, or the adviser of such a student, please feel free to contact Sean (scox9@vcu.edu) about the program.

Program:  The goal of the course is to learn about advanced topics in forcing and large cardinals.  Participants are expected to be familiar with basic theory of forcing and large cardinals; familiarity with the following sources would suffice:
• Kunen’s Set Theory: An Introduction To Independence Proofs
• Some exposure to proper forcing (e.g. from Jech’s Set Theory or Abraham’s chapter in the Handbook of Set Theory).
• Familiarity with large cardinals, especially measurable cardinals and supercompact cardinals.  These are covered, for example, in Jech’s Set Theory and Kanamori’s The Higher Infinite.  Participants should be most familiar with the characterization of these cardinals in terms of elementary embeddings between models of set theory.

Here is a tentative list of topics that each instructor intends to cover.

Spencer Unger:   Intro to the continuum problem. Konig’s theorem. Powers of regular cardinals. The singular cardinals hypothesis. What is known about the singular cardinals problem.  Measurable $\kappa$ with $2^\kappa$ large. Prikry forcing and properties of the Prikry extension. Prikry forcing with collapses for $\aleph_\omega$.  Gitik and Sharon’s theorem about failure of SCH and weak square. A sketch of some recent theorems of Sinapova-Unger.

Brent Cody: Woodin’s surgery argument for forcing GCH to fail at a measurable cardinal from the existence of a $\kappa^{++}$-tall cardinal $\kappa$. Easton’s theorem and supercompactness.  Baumgartner’s theorem that from a weakly compact cardinal, there is a forcing extension in which every stationary subset of $\omega_2\cap\text{cof}(\omega)$ reflects at a point of cofinality $\omega_1$. Magidor’s proof for consistency of every stationary subset of $\omega_{\omega+1}$ reflects from $\omega$-many supercompacts.Sacks forcing at inaccessible cardinals and the Friedman-Magidor result on the number of normal measures.

Monroe Eskew:  Basics of precipitous ideals, disjointing and closure of ultrapowers. Review some basic forcing facts. Proof of the Duality Theorem, with some elaborations (the general version appearing in Eskew’s thesis). Applications of Duality from a measurable cardinal: Precipitous ideals on successor cardinals, saturated ideals on inaccessibles, precipitous (non-prime) ideals on inaccessible $\kappa$ that adds no subsets of $\kappa$.
Real valued measurable cardinals. Prove some general facts about random real forcing, and get a nice approach to RVM’s through Duality. Further applications. Possible topics: precipitousness of $\text{NS}_{\kappa^+} \restriction \text{cof}(\kappa)$ from a measurable, using iterated club shooting. Indestructibility of “$\omega_1$ is minimally generically supercompact” under countably closed forcing, while destructibility of this notion under typical ccc forcing.

Financial Support:  The school is able to provide full support to US citizens and permanent residents. Foreign students are welcomed to participate, but would need to obtain funding from their universities or advisers.

Sc
hedule and location:  We will meet Monday through Friday of both weeks.  All meetings will be in Rowland Hall 306 (RH306), except on Thursdays; Thursday meetings will take place in room 1201 of Natural Sciences II  (NS2 1201).   A campus map is here.

Here is the rough daily schedule of activities:

09:00 – 10:30 Lecture
10:30 – 11:00 Break
11:00 – 12:30 Problem session
12:30 – 02:00 Lunch break
02:00 – 03:30 Lecture
03:30 – 04:00 Break
04:00 – 05:30 Problem session

Participants:
William Chan (CalTech)
Bill Chen (UCLA)
Jin Du (U. of Illinois at Chicago)
Miha Habic (CUNY)
Curtis Koch (UC Irvine)
Kameran Kolahi (UC Irvine)
Maxwell Levine (U. of Illinois at Chicago)
Kaethe Minden (CUNY)
Benny Siskind (UC-Berkeley)
Ryan Sullivant (UC Irvine)
Kameryn Williams (CUNY)

Housing:
Participants coming from outside Irvine will be housed in the on-campus “Middle Earth” housing (http://housing.uci.edu/housingOptions/Middle_Earth.html), located at 529 East Peltason Drive, Irvine, CA 92697-5521.

Check-in is from 12pm to 8pm on Sunday, July 5
.  Please let the organizer know if you plan to arrive after 8pm.  Upon arrival on Sunday, participants should park or be dropped off at the Anteater Parking Structure (http://www.parking.uci.edu/maps/imap.cfm), and then walk to the Middle Earth Conference Office (right in front of Pippin Commons; more info here: http://housing.uci.edu/about/Maps.html#me). There are some signs posted in Middle Earth indicating where the conference office is located. Once the participants arrive, they will be greeted by a Conference Assistant who will help check them in. The participants will receive room keys, meal cards (for those that have a board plan), and parking permits (for those that we order parking permits for).  If you unexpectedly arrive after 8pm on Sunday, please call the Middle Earth Conference Office phone number at 949-824-2956. Someone will answer the phone, and meet the participant at the conference office to check him/her in.

Check-out is from 8am to 12pm on Saturday, July 18th.

Local dining:  Participants who opted for a meal card will be provided breakfast and dinner each day near the residence hall.  There are several on-campus dining options, and also many restaurants at the University Center (http://www.shopirvinecompany.com/centers/irvine/university-center), about a 10-15 minute walk from Rowland Hall.

Internet access:

On campus:
The easiest way to access wi-fi on campus is through EduRoam.  See if your home university offers the EduRoam service (e.g. UC Berkeley does; see https://ist.berkeley.edu/airbears/).  If they do, then you can use EduRoam to log in with your home university’s username and password.  It is strongly recommended that you configure your device to your own university’s EduRoam system before travelling.  If your home university does not offer EduRoam, you may access the UCI wi-fi as a guest, but only for 7 days.  Beyond that you will have to register your computer with UCI to gain wi-fi access.

In the residence halls:
There is NO wi-fi in the residence halls.  You will need to bring an ethernet cord in order to access the internet in the residence halls.  You will also need a temporary Summer ID number, which will be available on Monday July 6.

Save all receipts (taxi/shuttle, airline, meals, etc.) in order to be reimbursed.  Food receipts must be itemized (i.e. show what was ordered); alcoholic beverages will not be reimbursed, according to NSF rules.

Daily summaries

WEEK 1

Day 1:  Unger gave historical background.  Forced over a model with a supercompact cardinal to obtain a model with failure of GCH at a measurable.  Proved that SCH holds above a supercompact, via an argument involving simultaneous stationary reflection.  Began discussing Prikry forcing.  Participants presented exercises on:  a) proving Silver’s Theorem via generic ultrapowers; b) Solovay’s stationary splitting theorem via generic ultrapowers; c) Strong Prikry property.

Day 2: Unger spoke briefly about stationary splitting via generic ultrapowers.  Then discussed proof that Prikry forcing extension has very good scale.  Continued Magidor’s consistency proof that SCH can fail at $\aleph_\omega$.  Exercises involving scales, square, Prikry sequences and iterated ultrapowers, stationary reflection.

Day 3:  Finished Magidor’s consistency proof that SCH can fail at $\aleph_\omega$.  More variations of Prikry forcing, including Gitik-Sharon verion.  Mitchell order.  Failure of weak square at singulars above a supercompact (via no good scales).  Exercises involving square, special Aronszajn trees, stationary reflection, and existence of very good scales.

Day 4:  Unger finished discussing Gitik-Sharon forcing.  Discussed when Gitik-Sharon poset adds weak square (due to Sinapova-Unger).  Abstract PCF and Shelah’s Dichotomy Theorem.  Exercises on collapsing behavior of Gitik-Sharon forcing, collapsing behavior of adding Cohen sets under GCH, problems about exact and least upper bounds in PCF theory.  Cody began speaking on the number of normal measures.  Exercises on liftings of ultrapower embeddings, and forcing lemmas to be used later.

Day 5:  Sacks forcing, generalized Sacks forcing, Friedman-Thompson theorem.  Began proof of Friedman-Magidor theorem (2009) about forcing a prescribed number of normal measures.  (Cody)

WEEK 2

Day 1: Friedman-Magidor theorem cont.  Iteration with nonstationary supports.  (Cody)

Day 2: Woodin’s surgery argument for failure of GCH at measurable from optimal hypothesis.  Cody-Magidor proof about Easton’s theorem for partially supercompact cardinals.  (Cody)

Day 3:  Introduction to precipitous ideals and related concepts.  Foreman’s Duality Theorem and applications.  (Eskew)

Day 4:  Ideals on inaccessible cardinals with varying degrees of saturation.  (Eskew)

Day 5:  Measure algebras and real-valued measurable cardinals.  Foreman’s indestructibility of generic supercompactness of $\omega_1$, and related results of Eskew.