## BOISE EXTRAVAGANZA IN SET THEORY – Announcement 2, Call for papers

January 20, 2009

The 18-th annual meeting of BEST will be hosted at Boise State University during the weekend of March 27 (Friday) – March 29 (Sunday), 2009.

It is organized by Liljana Babinkostova, Andres Caicedo, Stefan Geschke, Richard Ketchersid and Marion Scheepers.

Contributed and invited talks will be held on Friday, Saturday and Sunday at the Department of Mathematics, Boise State University. The four invited speakers are:

Steve Jackson (University of North Texas)

Ljubisa Kocinac (University of Nis, Republic of Serbia)

Assaf Rinot (Tel Aviv University, Israel)

Grigor Sargsyan (University of California, Berkeley)

The conference webpage is available at URL

http://math.boisestate.edu/~best/best18

There are four important deadlines regarding the conference:

Lodging: The Hampton Inn & Suites is providing rooms at a reduced rate for BEST participants. To take advantage of the reduced rate, reservations must be made online by MARCH 12. Follow this link for the Hampton Inn’s online reservation site for BEST.

Financial support: Limited financial support is available to partially offset travel expenses of some participants. The criteria for granting support include whether the participant has alternative financial support for the conference, and whether the participant is presenting a talk at the conference. Preference is given to graduate students and early career researchers. The amount of support is contingent on the budget constraints. University accounting regulations require completing certain forms online BEFORE the conference, and submitting original receipts of expenses. Reimbursements will be sent after the conference. The deadline for applying for financial support is MARCH 3.

To apply for support, email the organizers at

best@diamond.boisestate.edu

Applications from graduate students must be supported by a separate email from their thesis advisor. Anyone interested in participating should contact the organizers as soon as possible by sending an email to

best@math.boisestate.edu

Abstracts: Atlas Conferences, Inc. is providing abstract services for BEST 18. The deadline for submitting an abstract for invited or contributed talk is MARCH 25. Links are available at the BEST 18 web site.

Call for papers: The organizers will be editors for a volume in the Contemporary Mathematics series. Research papers on topics related to Set Theory and its Applications will be considered for publication in this volume.

All papers will go through a thorough referee process. Former and current participants of the BEST conferences or their collaborators are especially encouraged to consider submitting a research paper. Anyone interested in submitting a paper should contact Marion Scheepers as soon as possible at

marion@math.boisestate.edu

with this information. Subsequently information regarding preparation of papers will be sent to contributing authors by Contemporary Mathematics. The deadline for submitting a paper is JULY 21.

The conference is supported by a grant from the National Science Foundation. Abstract services are provided by Atlas Conferences, Inc. Contemporary Mathematics is published by the American Mathematical Society. Reduced lodging rate is provided by The Hampton Inn & Suites. Support from these entities is gratefully acknowledged.

## Set theory seminar -Richard Ketchersid: Quasiiterations I. Iteration trees

January 19, 2009

In October 24-November 14 of 2008, Richard Ketchersid gave a nice series of talks on quasiiterations at the Set Theory Seminar. The theme is to correctly identify `nice’ branches through iteration trees, and to see how difficult it is for a model to compute these branches. Richard presented a prototypical result in this area (due to Woodin) and a nice application (due to Jackson and Ketchersid). This post will be far from self-contained, and only present some of the definitions.

[Edit Sep. 25, 2010: My original intention was to follow this post with two more notes, on Woodin’s result and on the Jackson-Ketchersid theorem, but I never found the time to polish the presentation to a satisfactory level, so instead I will let the interested reader find my drafts at Lucien’s library.]

I’ll assume known the notions of extender and Woodin cardinal, and associated notions like the length or strength of an extender. A good reference for this post is Donald Martin, John Steel, Iteration trees, Journal of the American Mathematical Society 7 (1) 1994, 1-73. As usual, all inaccuracies below are mine. Some of the notions below are slightly simpler than the official definitions. These notions are all due to Donald Martin, John Steel, and Hugh Woodin.

In this post I present the main notions (iteration trees and iterability) and close with a quick result about the height of tree orders. The order I follow is close to Richard’s but it differs from his presentation at a few places.

## 580 -Syllabus

January 12, 2009

Mathematics 580: Topics in Set Theory: Combinatorial Set Theory.

Section 1.
Instructor:
Andres Caicedo.
Time: MWF 3:40-4:30 pm.
Place: Education building, Room 330.

We will cover diverse topics in combinatorial set theory, depending on time and the interests of the audience, including partition calculus (a generalization of Ramsey theory), cardinal arithmetic, and infinite trees. Time permitting, we can also cover large cardinals, determinacy and infinite games, or cardinal invariants (the study of sizes of sets of reals), among others. I’m open to suggestions for topics.

Recommended background: Knowledge of cardinals and ordinals. A basic course on set theory (like 502: Logic and Set Theory) would be ideal but is not required.

Textbook: There is no official textbook. The following suggested references may be useful, but are not required:

• Set theory. By T. Jech. Springer (2006), ISBN-10: 3540440852  ISBN-13: 978-3540440857
• Set theory. An introduction to independence proofs. By K. Kunen. North Holland (1983), ISBN-10: 0444868399  ISBN-13: 978-0444868398
• Set theory for the working mathematician. By K. Ciesielski. Cambridge U. Press (1997), ISBN-10: 0521594650  ISBN-13: 978-0521594653
• Set theory. By A. Hajnal and P. Hamburger. Cambridge U. Press (1999), ISBN-10: 052159667X  ISBN-13: 978-0521596671
• Discovering modern set theory. By W. Just and M. Weese. Vol I. AMS (1995), ISBN-10: 0821802666 ISBN-13: 978-0821802663. Vol II. AMS (1997), ISBN-10: 0821805282 ISBN-13: 978-0821805282
• Problems and theorems in classical set theory. By P. Komjath and V. Totik. Springer (2006), ISBN-10: 038730293X  ISBN-13: 978-0387302935
• Notes on set theory. By Y. Moschovakis. Springer (2005), ISBN-10: 038728723X  ISBN-13: 978-0387287232

## 305 -Syllabus

January 12, 2009

Mathematics 305: Abstract Algebra I.

Section 1.
Instructor: Andres Caicedo.
Time: MWF 10:40-11:30 am.
Place: Education building, Room 221.

Text: Redfield, Robert H. Abstract Algebra. A concrete introduction. Addison Wesley, 2001. ISBN: 0-201-43721-X. If needed, I will provide references for additional topics not covered by the textbook.

Contents:  The usual syllabus for this course lists

Introduction to abstract algebraic systems – their motivation, definitions, and basic properties. Primary emphasis is on group theory (permutation and cyclic groups, subgroups, homomorphism, quotient groups), followed by a brief survey of rings, integral domains, and fields.

## Set theory seminar -Stefan Geschke: Cofinalities of algebraic structures

January 6, 2009

This is a short overview of a talk given by Stefan Geschke on November 21, 2008. Stefan’s topic, Cofinalities of algebraic structures and coinitialities of topological spaces, very quickly connects set theory with other areas, and leads to well-known open problems. In what follows, compact always includes Hausdorff. Most of the arguments I show below are really only quick sketches rather than complete proofs. Any mistakes or inaccuracies are of course my doing rather than Stefan’s, and I would be grateful for comments, corrections, etc.

Definition. Let $A$ be a (first order) structure in a countable language. Write ${\rm cf}(A)$ for the smallest $\delta$ such that $A=\bigcup_{\alpha<\delta}A_\alpha$ for a strictly increasing union of proper substructures.

Since the structures $A_\alpha$ need to be proper, ${\rm cf}(A)$ is not defined if $A$ is finite. It may also fail to exist if $A$ is countable, but it is defined if $A$ is uncountable. Moreover, if ${\rm cf}(A)$ exists, then

1. ${\rm cf}(A)\le|A|$, and
2. ${\rm cf}(A)$ is a regular cardinal.

Example 1. Groups can have arbitrarily large cofinality. This is not entirely trivial, as the sets $A_\alpha$ may have size $|A|$.

Question 1. Is every regular cardinal realized this way?