[FOM] Asian Initiative for Infinity Graduate Summer School in Logic: 28 June--23 July 2010, National University of Singapore
Chi Tat Chong
chongct at gmail.com
Sat Mar 13 11:48:04 EST 2010
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.
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