[FOM] Asian Initiative for Infinity Graduate Summer School in Logic: 28 June--23 July 2010, National University of Singapore

[Chi Tat Chong]

Asian Initiative for Infinity (AII)
AII Graduate Summer School in Logic
28 June---23 July 2010
National University of Singapore

The AII Graduate Summer School is organized by 
the Institute for Mathematical Sciences and the 
Department of Mathematics of the National 
University of Singapore, with funding from the 
John Templeton Foundation and the University. The 
Graduate Summer School bridges the gap between a 
general graduate education in mathematical logic 
and the specific preparation necessary to do 
research on problems of current interest in the 
subject. In general, students who attend the AII 
Summer School should have completed their first 
year, and in some cases, may already be working 
on a thesis. While a majority of the participants 
will be graduate students, some postdoctoral 
scholars and researchers may also be interested in attending.

Having completed at least one course in 
Mathematical Logic is required, and completion of 
an additional graduate course in either set 
theory or recursion theory is strongly 
recommended.  Students should be familiar with 
the Gödel Completeness and Incompleteness 
Theorems and with the Gödel and Cohen Independence Theorems in Set Theory.

The main activity of the AII Graduate Summer 
School will be a set of three intensive short 
courses offered by leaders in the field, designed 
to introduce students to exciting, current 
research topics. These lectures will not 
duplicate standard courses available elsewhere. 
Each course will consist of lectures with problem 
sessions.  On average, the participants of the 
AII Graduate Summer School meet twice each day 
for lectures and then again for a problem session.

Lectures will be conducted by Moti Gitik (Tel 
Aviv University), Menachem Magidor (Hebrew 
University of Jerusalem), and Denis Hirschfeldt 
(University of Chicago). In addition, Theodore A. 
Slaman and W. Hugh Woodin of the University of 
California, Berkeley, as well as two postdoctoral 
fellows supported by the John Templeton 
Foundation, will be in residence during the 
period of the AII Graduate Summer School.

Applications are invited from interested 
students. Each student selected for participation 
will be provided with a stipend of at least 
US$2000. Additional funding will be available to 
cover accommodation. Applications will be 
considered from 7 April 2010 and decisions made 
on a rolling basis, for as along as funds remain 
available. For further details, visit

<http://www2.ims.nus.edu.sg/Programs/010aiiss/index.php>http://www2.ims.nus.edu.sg/Programs/010aiiss/index.php

Course Titles and Descriptions
Moti Gitik, Tel Aviv University

Title: Prikry-type forcings and short extenders forcings

We plan to cover the following topics:  Basic 
Prikry forcing, tree Prikry forcing, supercompact 
Prikry forcing, negation of the Singular Cardinal 
Hypothesis via blowing up the power of a singular 
cardinal, Extender Based Prikry forcing, forcings 
with short extenders-gap 2, gap 3, arbitrary gap, 
dropping cofinalities, some further directions.

Denis Hirschfeldt, The University of Chicago

Title: Reverse Mathematics of Combinatorial Principles

Computability theory and reverse mathematics 
provide tools to analyze the relative strength of 
mathematical theorems. This analysis often 
reveals surprising relationships between results 
in different areas, such as the tight connection 
between nonstandard models of arithmetic, the 
compactness of Cantor space, and results as 
seemingly diverse as the existence of prime 
ideals of countable commutative rings, Brouwer's 
fixed point theorem, the separable Hahn-Banach 
Theorem, and Gödel’s completeness theorem, among 
many others. It also allows us to give 
mathematically precise versions of statements 
such as "Adding hypothesis A makes Theorem B 
strictly weaker", or "Technique X is essential to proving Theorem Y".

Combinatorial principles, such as versions of 
Ramsey's Theorem or results about partial and 
linear orders, are a particularly rich source of 
examples in computable mathematics and reverse 
mathematics.  This course will focus on 
fundamental techniques and themes in this area, 
with the goal of preparing students to tackle 
open problems, several of which will be discussed during the course.
Menachem Magidor, Hebrew University of Jerusalem
Title: The Theory of Possible Cofinalities (PCF) and some applications

The theory of Possible Cofinalities is the theory 
developed by Shelah which uncovers the deeper 
structure below cardinal arithmetic. The main 
concept is (for a set of regular cardinals A) the 
set of possible cofinalities of A (pcf(A)) which 
is the set of the regular cardinals that can be 
realized as the cofinality of some ultraproduct 
of A. It turned out that there are many deep 
results about this operation (as well as fascinating problems).
The Theory has many applications. This course 
will develop the basic concepts of the theory, 
will prove the main results like the bound on and 
(time permitting) will give some other 
applications like the existence of Jonson 
Algebras, the impossibility of certain cases of Chang's Conjecture and more.