This edited volume presents a fascinating collection of lecture notes focusing on differential equations from two viewpoints: formal calculus (through the theory of Groebner bases) and geometry (via quiver theory). Groebner bases serve as effective models for computation in algebras of various types. Although the theory of Groebner bases was developed in the second half of the 20th century, many works on computational methods in algebra were published well before the introduction of the modern algebraic language. Since then, new algorithms have been developed and the theory itself has greatly expanded. In comparison, diagrammatic methods in representation theory are relatively new, with the quiver varieties only being introduced - with big impact - in the 1990s. Divided into two parts, the book first discusses the theory of Groebner bases in their commutative and noncommutative contexts, with a focus on algorithmic aspects and applications of Groebner bases to analysis on systems of partial differential equations, effective analysis on rings of differential operators, and homological algebra. It then introduces representations of quivers, quiver varieties and their applications to the moduli spaces of meromorphic connections on the complex projective line. While no particular reader background is assumed, the book is intended for graduate students in mathematics, engineering and related fields, as well as researchers and scholars.

This edited volume presents a fascinating collection of lecture notes focusing on differential equations from two viewpoints: formal calculus (through the theory of Groebner bases) and geometry (via quiver theory). Groebner bases serve as effective models for computation in algebras of various types. Although the theory of Groebner bases was developed in the second half of the 20th century, many works on computational methods in algebra were published well before the introduction of the modern algebraic language. Since then, new algorithms have been developed and the theory itself has greatly expanded. In comparison, diagrammatic methods in representation theory are relatively new, with the quiver varieties only being introduced - with big impact - in the 1990s. Divided into two parts, the book first discusses the theory of Groebner bases in their commutative and noncommutative contexts, with a focus on algorithmic aspects and applications of Groebner bases to analysis on systems of partial differential equations, effective analysis on rings of differential operators, and homological algebra. It then introduces representations of quivers, quiver varieties and their applications to the moduli spaces of meromorphic connections on the complex projective line. While no particular reader background is assumed, the book is intended for graduate students in mathematics, engineering and related fields, as well as researchers and scholars.

In the last two decades, many important developments and interactions in algebraic geometry and integrable systems have arisen from ideas in mirror symmetry. The conference "New developments in Algebraic Geometry, Integrable Systems and Mirror Symmetry" was held at RIMS, Kyoto University on January 7-11, 2008, to explore recent developments and interactions in various mathematical fields, such as algebraic geometry, integrable systems, Gromov-Witten theory and symplectic geometry, and in particular to explore the developments and interactions coming from ideas in mirror symmetry. This volume is the outcome of that conference, and consists of twelve contributed papers by invited speakers. Readers will find, within this volume, beneficial expositions on various aspects and interesting interactions in these mathematical fields.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

The present volume is the proceedings of the meeting 'Development of Moduli theory' which was held as the Seasonal Institute 2013 (The 6th MSJ-SI), the Mathematical Society of Japan and the 2013 Research Project of Research Institute of Mathematical Science, Kyoto University.This volume is dedicated to Professor Shigeru Mukai on the occasion of his sixtieth birthday.The volume consists of five survey articles and eight research articles given by the authors: I Dolgachev, E Looijenga, N Mestrano, C Simpson, H Nakajima, I Nakamura, V Alexeev, M Aprodu, G Farkas, A Ortega, I Ciocan-Fontanine, B Kim, B Hassett, A Kresch, Y Tschinkel, D Huybrechts, J H Keum, V Nikulin, K Yoshioka.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America