28 June - 23 July 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 Gdel Completeness and Incompleteness Theorems and with the G del 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 Gdels 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.