Boban Velickovic at University of Illinois at Chicago on October 15, 2011

Appalachian set theory

Saturday, October 15, 2011

University of Illinois at Chicago

Boban Velickovic : "Proper forcing remastered"

List of participants in this workshop

Lecture notes by Boban Velickovic and Giorgio Venturi (PDF)

Workshop description

The theory of proper forcing, which has been developed by Shelah over the last 30 years, is a powerful tool for proving independence results. However, quite often when we wish to introduce some object by forcing, the obvious partial order is not proper. In the early 1980s Baumgartner introduced the idea of using side conditions which constrain the way in which the forcing conditions are extended. This approach was later reformulated in terms of finite chains of countable elementary submodels of some H(θ) by Todorcevic and has found numerous applications. However, in some situations when one wishes to preserve not only ω₁ but also some larger cardinal countable models as side conditions do not suffice and one is naturally lead to consider side conditions of two or more different types. The first such result is due to Mitchell and Friedman who generalized Baumgartner's poset for adding a club in ω₁ to larger cardinals. Recently, the theory of generalized side conditions has been studied with great success by Neeman who used it to obtain versions of the Proper Forcing Axiom for larger cardinals. In this workshop we present the basic theory of generalized side conditions and use it to obtain several applications such as the Friedman-Mitchell forcing for adding a thin club to ω₂, Kosmzider's poset for adding a strong ω₂-chain of functions in ((ω₁)^ω₁, <_fin) and the Baumgarther-Shelah poset for adding a thin very tall superatomic Boolean algebra. Some open questions will be discussed at the end of the workshop.

Advanced references

James E. Baumgartner and Saharon Shelah, "Remarks on superatomic Boolean algebras", Annals of Pure and Applied Logic, 1987
Piotr Koszmider, "On strong chains of uncountable functions", Israel Journal of Mathematics, 2000
William J. Mitchell, "I[ω₂] can be the nonstationary ideal on Cof(ω₁)", Transactions of the American Mathematical Society 361 (2009) 561-601

Participant travel support

Funds provided by the NSF will be used to reimburse some participant transportation and lodging expenses. Priority will be given to students and postdocs, and to faculty who do not hold federal research grants. Please request such funds as far in advance of the meeting as possible by sending the following information to Ernest Schimmerling by email.

Your name, university affiliation, mailing address, phone number and email address
Your professional status and
- undergraduate students: please describe your background in set theory
- graduate students: please tell us your year and the name of your thesis advisor if you have one
- faculty: please tell us whether you hold a federal research grant
A brief statement about your interest in the workshop
An itemized estimate of your expected transportation expenses

Schedule

Workshop lectures run from 9:30 a.m. to 6 p.m. with morning refreshments before the workshop, starting at around 8:30 or 8:45 a.m., several coffee breaks during the workshop, and a lunch break from 12:30 to 2:30 p.m. There is no formal dinner but groups of participants will head out to restaurants after the talks.

Practicalities

The easiest place to park a car is the Halsted Street Parking Structure which is squeezed in between Halsted Street and Expressway 90/94 as indicated on the map of UIC. They charge per hour or approximately $13 for a whole day. It may be cheaper on weekends though.
The lectures will take place in Science and Engineering Offices (SEO), which is easily recognised as the 2nd highest building on campus and can be found at the center of the map of UIC, right west of the parking structure. There are three entrances, but they are all more or less next to each other. The lectures themselves will take place in Room 636 in SEO which is on the 6th floor of the building. With lectures beginning at 9:30 AM, the building should be open from 8:30 AM and will close again at 7:00 PM.

Refreshments will either be offered in 636 or 300 SEO. There will be some info on the door downstairs and on 6th floor to indicate where people should go. The breakfast (the standard fare including coffee, bagels, fruit, etc) should be ready 45 min - 1 hour before the first lecture at 9:30.

Dining: More info will be provided during the conference, but there are principally three areas for dining around campus; Taylor Street between the math department and Campus Housing, Halsted Street close to Crowne Plaza, which is the Greek area, and Halsted Street south of Campus. For those venturing further away, downtown is easily accessible by the Blue Line.

For participants staying at UIC Guest Housing:
- Arriving to O'hare Airport: Most flights arrive to O'hare International Airport, which is situated at the northwest end of the Blue L-line (Metro/Subway). To get from the airport to campus housing, by far the easiest method (other than taxi) is to take the Blue line from O'hare all the way to the Illinois Medical District stop as indicated on the Blue Line map.
- Arriving from Midway Airport: Midway Airport is at the end of the Orange L-line (Metro/Subway), which connects to the Pink Line in downtown (Stop: State/Lake). Continue on the Pink Line direction 54th/Cermak and get off at Polk, which is a short walk from Campus Housing.
- Arrival by other public transportation: The Greenhound Station is a short walk from the UIC HAlsted stop on the Blue L-line (Metro/ Subway). Take this in the direction of Forest Park and get off at Illinois Medical District, as indicated on the Blue Line map.
- Getting back and forth to the math department from Campus Housing: There are several buses and also the Blue L-line connecting the two. These are indicated on the Bus 7 map and Bus 157 map. Another option is to walk along Taylor Street directly from the Medical District all the way to the math department. Taylor Street is pretty busy with many restaurants and cafes, so it is a reasonably pleasant walk, but takes 30 min.
- Safety: The Medical District and the East Campus, where the math department is, are in relatively safe areas, but bordered by some bad neighbourhoods. The stretch between the two on Taylor Street is quite safe and will be busy on Saturday night, but I would recommend against going further south down to Roosevelt (see the map), especially at night. In general, at night it is best to walk in a group and travel on well-lit streets (Taylor) or alternatively take public transport.