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

Miami University in Oxford OH

270 Engineering Building, which is building no. 43 in cell F-3 on this campus map

Andreas Blass : "Ultrafilters and cardinal characteristics of the continuum"

Workshop description

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.

References

• 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 http://www.math.lsa.umich.edu/~ablass/hbk.pdf.
• Finally, for connections between cardinal characteristics and ultrafilters, the beginning of the story is told in my STACY paper:
  • "Applications of superperfect forcing and its relatives" (Set Theory and its Applications, Springer Lecture Notes in Math 1401 (1989) 18-40)
  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).

Lodging

• 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.

Parking

1. Please refer to this marked up map
2. Flat Lot across from the Business School (Cook Field) --- Short walking distance to the Engineering Building
3. Flat Lot directly behind the Engineering Building --- Even shorter distance than Cook Field
4. Parking Garage near and directly the Engineering Building --- there is a charge to park here 50 cents an hour
5. DO NOT PARK in the Miami Inn parking lot unless you are a registered guest there

Places to eat

Dinner on High Street

• Steinkeller
• Kona Bistro
• Wild Bistro

Cafes

• Patterson's Cafe: 103 West Spring Street (513) 523-0770
• Morning Sun Cafe and Bakery: 109 West High Street (513) 524-6786
• Kofenya

Post workshop materials

• List of participants
• Lecture notes from this workshop by Andreas Blass and Nicholas Rupprecht (PDF revised November 2015)