The two symposia, the Hayashibara Forum and the MSJ/IHES joint workshop, were held at the Institute des Hautes Etudes Scientifiques (IHES) in November, 2006. The Hayashibara Forum focused on singularity theory, which has been one of the research areas that has over the years been well represented at IHES. The MSJ/IHES Joint Workshop, focused on the broad area of noncommutativity, with an emphasis on noncommutative geometry as one of the fundamental themes of 21st century mathematics. This volume contains papers presented at the symposia in the form of invited lectures and contributing talks by young researchers. We believe that the scope of this volume well reflects a new development for singularity theory, and a new direction in mathematics through noncommutativity. This volume are aimed to inspire not only the specialists in these fields but also a wider audience of mathematicians.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America