Appalachian set theory

Saturday, November 22, 2014

9:30 a.m. - 6 p.m. with coffee and lunch breaks

University of Illinois at Chicago

Lectures in SEO 636

Refreshments in SEO 300

Breakfast and coffee starting at 8:30

Slawomir Solecki : "Ultrafilter space methods in Infinite Ramsey Theory"

Description

We will start with stating two concrete Ramsey results of the type we are interested in: Gowers' tetris theorem and Furstenberg and Katznelson's generalization of the Hales-Jewett theorem. We will describe a general framework for such results in terms of partial semigroups. We will then introduce new algebraic structures appropriate for formalizing such theorems: this will involve in a crucial way ultrafilter spaces over partial semigroups. We will phrase and prove a general Ramsey theorem on the existence of appropriately defined basic sequences for such structures. We will proceed to describing concrete examples of such algebraic structures, which will involve monoid actions by endomorphisms on compact left-topological semigroups and operations of tensor product and substructure. As applications, we will derive the Gowers and Furstenberg-Katznelson theorems. Time permitting, we will finish with some further applications.

Suggested reading

• Basics:
  • The ultrafilter entry on Wikipedia.
  • Pages 409-413 in W. Rudin, Homogeneity problems in the theory of Čech compactifications, Duke Math. J. 23 (1956), 409-419.
• Intermediate:
  • Sections 2 and 3 in V. Bergelson, A. Blass, N. Hindman, Partition theorems for spaces of variable  words, Proc. London Math. Soc. 68 (1994), 449-476.
• Advanced:
  • H. Furstenberg, Y. Katznelson, Idempotents in compact semigroups and Ramsey theory,   Israel J. Math. 68 (1989), 257-270.
  • Chapter 2 in S. Todorčević, Introduction to Ramsey Spaces, Annals of Mathematics Studies, 174. Princeton University Press, 2010.
• Supplementary:
  • Chapters 1-5 in  N. Hindman, D. Strauss, Algebra in the Stone-Čech Compactification, de Gruyter  Expositions in Mathematics 27, Walter de Gruyter & Co., 2012.

Local information

The UIC mathematics department is located in the Science and Engineering Offices Building (SEO) on the UIC campus. Here is a map.

Participant travel support

Funds provided by the National Science Foundation will be used to reimburse some participant transportation and lodging expenses. Priority will be given to students and 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 the email address appalachiansettheory@gmail.com

• Your name, university affiliation, mailing address, phone number and email address
• Your nationality and visa status
• Your professional status and some additional information:
  • 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

Participant lodging

Lodging is available at UIC visitor housing, which can be booked online at the UIC guest housing web page.