Appalachian set theory -- Andreas Blass at Miami U on January 30, 2010
Appalachian set theory
Saturday, January 30, 2010
Registration and morning refreshments 9 - 9:30 a.m.
Lectures 9:30 a.m. - 12:30 p.m. and 2:30 - 6 p.m. with breaks
270 Engineering Building, which is
building no. 43 in cell F-3 on this campus map
"Ultrafilters and cardinal characteristics of the continuum"
This workshop will be in three approximately equal-sized parts: one
part ultrafilters, one part cardinal characteristics of the continuum,
and one part connections between ultrafilters and cardinal
characteristics. My (present and still subject to change) plans for
the three parts are as follows.
In the first part, I would like to decribe several ways of thinking
about ultrafilters, coming from combinatorics, topology, and model
theory. Then I want to describe some of the connections between
ultrafilters and partition theorems. Third, I'll briefly describe
some of the structure of the world of ultrafilters --- the
Rudin-Keisler ordering, limits, and tensor products. Finally, I'll
describe some ultrafilter-related axioms, independent of ZFC, that
have been useful in various contexts.
In the second part, I'll describe a few specific cardinal
characteristics and connections between them. I'll explain some of
the ways that one thinks about these, including Vojtas's
``generalized Galois-Tukey connections'' and Zapletal's ``tame''
characteristics. I'll also introduce the groupwise density number, a
non-tame characteristic with rich connections to ultrafilters.
In the last part, I'll explore some of those connections, as well as
other results that relate cardinal characteristics of the continuum to
cardinals associated with ultrafilters, for example the minimal number
of generators of an ultrafilter or the cofinality of an ultrapower.
Some of those connections also involve the partition properties and
other structural notions discussed in the first part.
- The standard reference for ultrafilters is the book of
Comfort and Negrepontis, "The Theory of Ultrafilters,"
which contains far more than is relevant to my workshop.
In fact, the union of just chapters 7, 9, 11, and 16 properly includes
all the relevant background material.
More directly connected with
material that I plan to discuss, but still containing other material in
addition, are a couple of papers of mine:
- "A model-theoretic view of some special ultrafilters" (Logic Colloquium '77, North-Holland (1978) pp. 79-90)
- "Ultrafilters: where topological dynamics = algebra = combinatorics" (Topology Proceedings 18 (1993) 33-56)
For cardinal characteristics of the continuum, the standard published references are:
- E. van Douwen, "The integers and topology" (Handbook of Topology, North-Holland (1984) 111-167)
- J. Vaughan, "Small uncountable cardinals and topology" (Open Problems in Topology, North-Holland (1990) 195-218)
There's also my chapter for the Handbook of Set Theory, available from my web site
Finally, for connections between cardinal characteristics and ultrafilters, the beginning of the story is told in my STACY paper:
More recent material has, as far as I know, not been collected into survey papers, and I don't think it would be useful to give a long list of original research
articles here (especially as the particular topics I'll discuss may yet change a little).
- "Applications of superperfect forcing and its relatives" (Set Theory and its Applications, Springer Lecture Notes in Math 1401 (1989) 18-40)
- Rooms at the Marcum Conference Center & Miami Inn are being held until January 15.
Their phone number is 513-529-2104.
Rooms w/ 1 Double bed at Marcum = $85/room/night + tax.
Rooms w/ 1 Queen bed at the Inn = $115/room/night + tax.
Rooms w/ 2 Queen Beds at the Inn = $115/room/night + tax.
Rooms w/ King Suite at the Inn = $135/room/night + tax.
Rooms with two beds are being held at the
Best Western at 6 E. Sycamore Street until January 15.
These cost $99/room/night + tax.
Their phone number is 513-523-0000.
- Other Oxford hotels are listed here.
Please contact Paul Larson if you need a ride between
Cincinnati (airport or bus terminal) and your hotel.
The parking office is aware this event is going on.
No one attending the workshop should need a parking permit.
If you receive a ticket, please give it to Paul Larson or
Richard Ketchersid as soon as possible to get it voided.
- Please refer to this marked up map
- Flat Lot across from the Business School (Cook Field) --- Short
walking distance to the Engineering Building
- Flat Lot directly behind the Engineering Building --- Even
shorter distance than Cook Field
- Parking Garage near and directly the Engineering
Building --- there is a charge to park here 50 cents an hour
- DO NOT PARK in the Miami Inn parking lot unless you are a registered
Places to eat
Dinner on High Street
- Patterson's Cafe: 103 West Spring Street (513) 523-0770
- Morning Sun Cafe and Bakery: 109 West High Street (513) 524-6786
Post workshop materials