This volume is a collection of written versions of the talks given at the Workshop on Computational Prospects of Infinity, held at the Institute for Mathematical Sciences from 18 June to 15 August 2005. It consists of contributions from many of the leading experts in recursion theory (computability theory) and set theory. Topics covered include the structure theory of various notions of degrees of unsolvability, algorithmic randomness, reverse mathematics, forcing, large cardinals and inner model theory, and many others.

This volume is a collection of written versions of the talks given at the Workshop on Computational Prospects of Infinity, held at the Institute for Mathematical Sciences from 18 June to 15 August 2005. It consists of contributions from many of the leading experts in recursion theory (computability theory) and set theory. Topics covered include the structure theory of various notions of degrees of unsolvability, algorithmic randomness, reverse mathematics, forcing, large cardinals and inner model theory, and many others.

This volume is a collection of written versions of the talks given at the Workshop on Computational Prospects of Infinity, held at the Institute for Mathematical Sciences from 18 June to 15 August 2005. It consists of contributions from many of the leading experts in recursion theory (computability theory) and set theory. Topics covered include the structure theory of various notions of degrees of unsolvability, algorithmic randomness, reverse mathematics, forcing, large cardinals and inner model theory, and many others.

The Institute for Mathematical Sciences at the National University of Singapore hosted a Spring School on Fluid Dynamics and Geophysics of Environmental Hazards from 19 April to 2 May 2009. This volume contains the content of the nine short lecture courses given at this School, with a focus mainly on tropical cyclones, tsunamis, monsoon flooding and atmospheric pollution, all within the context of climate variability and change.The book provides an introduction to these topics from both mathematical and geophysical points of view, and will be invaluable for graduate students in applied mathematics, geophysics and engineering with an interest in this broad field of study, as well as for seasoned researchers in adjacent fields.

The Institute for Mathematical Sciences at the National University of Singapore hosted a two-month research program on "Mathematical Theory and Numerical Methods for Computational Materials Simulation and Design" from 1 July to 31 August 2009. As an important part of the program, tutorials and special lectures were given by leading experts in the fields for participating graduate students and junior researchers. This invaluable volume collects four expanded lecture notes with self-contained tutorials. They cover a number of aspects on multiscale modeling, analysis and simulations for problems arising from materials science including some critical components in computational prediction of materials properties such as the multiscale properties of complex materials, properties of defects, interfaces and material microstructures under different conditions, critical issues in developing efficient numerical methods and analytic frameworks for complex and multiscale materials models. This volume serves to inspire graduate students and researchers who choose to embark into original research work in these fields.

This book aims to describe, for readers uneducated in science, the development of humanity's desire to know and understand the world around us through the various stages of its development to the present, when science is almost universally recognized - at least in the Western world - as the most reliable way of knowing. The book describes the history of the large-scale exploration of the surface of the earth by sea, beginning with the Vikings and the Chinese, and of the unknown interiors of the American and African continents by foot and horseback. After the invention of the telescope, visual exploration of the surfaces of the Moon and Mars were made possible, and finally a visit to the Moon. The book then turns to our legacy from the ancient Greeks of wanting to understand rather than just know, and why the scientific way of understanding is valued. For concreteness, it relates the lives and accomplishments of six great scientists, four from the nineteenth century and two from the twentieth. Finally, the book explains how chemistry came to be seen as the most basic of the sciences, and then how physics became the most fundamental.

This volume is based on lectures given during the program “Complex Quantum Systems” held at the National University of Singapore's Institute for Mathematical Sciences from 17 February to 27 March 2010. It guides the reader through two introductory expositions on large Coulomb systems to five of the most important developments in the field: derivation of mean field equations, derivation of effective Hamiltonians, alternative high precision methods in quantum chemistry, modern many-body methods originating from quantum information, and — the most complex — semirelativistic quantum electrodynamics.These introductions are written by leaders in their fields; amongst them are Volker Bach, Rafael Benguria, Thomas Chen, and Jan Philip Solovej. Together, they fill a gap between current textbooks and the vast modern literature on complex quantum systems.

This monograph presents recursion theory from a generalized point of view centered on the computational aspects of definability. A major theme is the study of the structures of degrees arising from two key notions of reducibility, the Turing degrees and the hyperdegrees, using techniques and ideas from recursion theory, hyperarithmetic theory, and descriptive set theory. The emphasis is on the interplay between recursion theory and set theory, anchored on the notion of definability. The monograph covers a number of fundamental results in hyperarithmetic theory as well as some recent results on the structure theory of Turing and hyperdegrees. It also features a chapter on the applications of these investigations to higher randomness.

Reverse mathematics studies the complexity of proving mathematical theorems and solving mathematical problems. Typical questions include: Can we prove this result without first proving that one? Can a computer solve this problem? A highly active part of mathematical logic and computability theory, the subject offers beautiful results as well as significant foundational insights. This text provides a modern treatment of reverse mathematics that combines computability theoretic reductions and proofs in formal arithmetic to measure the complexity of theorems and problems from all areas of mathematics. It includes detailed introductions to techniques from computable mathematics, Weihrauch style analysis, and other parts of computability that have become integral to research in the field. Topics and features: Provides a complete introduction to reverse mathematics, including necessary background from computability theory, second order arithmetic, forcing, induction, and model construction Offers a comprehensive treatment of the reverse mathematics of combinatorics, including Ramsey's theorem, Hindman's theorem, and many other results Provides central results and methods from the past two decades, appearing in book form for the first time and including preservation techniques and applications of probabilistic arguments Includes a large number of exercises of varying levels of difficulty, supplementing each chapter The text will be accessible to students with a standard first year course in mathematical logic. It will also be a useful reference for researchers in reverse mathematics, computability theory, proof theory, and related areas. Damir D. Dzhafarov is an Associate Professor of Mathematics at the University of Connecticut, CT, USA. Carl Mummert is a Professor of Computer and Information Technology at Marshall University, WV, USA.

Surveys on recent developments in the theory of algorithmic randomness and its interactions with other areas of mathematics.