What Determines an Algebraic Variety? by
A pioneering new nonlinear approach to a fundamental question in algebraic geometry One of the crowning achievements of nineteenth-century mathematics was the proof that the geometry of lines in space uniquely determines the Cartesian coordinates, up to a linear ambiguity. What Determines an Algebraic Variety? develops a nonlinear version of this theory, offering the first nonlinear generalization of the seminal work of Veblen and Young in a century. While the book uses cutting-edge techniques, the statements of its theorems would have been understandable a century ago; despite this, the results are totally unexpected. Putting geometry first in algebraic geometry, the book provides a new perspective on a classical theorem of fundamental importance to a wide range of fields in mathematics. Starting with basic observations, the book shows how to read off various properties of a variety from its geometry. The results get stronger as the dimension increases. The main result then says that a normal projective variety of dimension at least 4 over a field of characteristic 0 is completely determined by its Zariski topological space. There are many open questions in dimensions 2 and 3, and in positive characteristic.
Michael Abbott's Hydroinformatics: Poiesis of New Relationships with Water by
Hydroinformatics was established 30 years ago as a novel discipline in which computer modelling of water was combined with novel developments of Informational and Computational Technologies for purposes of introducing new and different approaches to water engineering, management, and decision support. The late Professor Michael B. Abbott is widely recognized as founding father of this field. The book will present the original ideas about the field of hydroinformatics, primarily presented in the works of Abbott, together with critical assessment of its current developments.The first chapter will introduce a general re-visit of the motivations and ideas for establishing the field of hydroinformatics, together with an assessment of current research and practice regarding the extent and characteristics that relate to the original ideas introduced by Abbott. Six following chapters will have more or less the same structure, each addressing a particular aspect of hydroinformatics, as follows: computational hydraulics and its role in establishing hydroinformatics, integration of artificial intelligence and computational hydraulics, hydroinformatics impact on hydrology, transformations of water professions and businesses by hydroinformatics, the role of hydroinformatics in realizing water-related social justice, and evolution and key characteristics of hydroinformatics education.Each chapter will be related to already published works of Abbott. All chapters are written by contributors who were past collaborators with Abbott and are still active in the field of hydroinformatics. The book will be a tribute to Abbott's contributions to hydroinformatics, and it will provide an assessment of the current status of the field perceived from within the context of Abbott's original ideas.
Discrete Quantum Walks on Graphs and Digraphs by
Discrete quantum walks are quantum analogues of classical random walks. They are an important tool in quantum computing and a number of algorithms can be viewed as discrete quantum walks, in particular Grover's search algorithm. These walks are constructed on an underlying graph, and so there is a relation between properties of walks and properties of the graph. This book studies the mathematical problems that arise from this connection, and the different classes of walks that arise. Written at a level suitable for graduate students in mathematics, the only prerequisites are linear algebra and basic graph theory; no prior knowledge of physics is required. The text serves as an introduction to this important and rapidly developing area for mathematicians and as a detailed reference for computer scientists and physicists working on quantum information theory.
The Calabi Problem for Fano Threefolds by
Algebraic varieties are shapes defined by polynomial equations. Smooth Fano threefolds are a fundamental subclass that can be thought of as higher-dimensional generalizations of ordinary spheres. They belong to 105 irreducible deformation families. This book determines whether the general element of each family admits a Kähler-Einstein metric (and for many families, for all elements), addressing a question going back to Calabi 70 years ago. The book's solution exploits the relation between these metrics and the algebraic notion of K-stability. Moreover, the book presents many different techniques to prove the existence of a Kähler-Einstein metric, containing many additional relevant results such as the classification of all Kähler-Einstein smooth Fano threefolds with infinite automorphism groups and computations of delta-invariants of all smooth del Pezzo surfaces. This book will be essential reading for researchers and graduate students working on algebraic geometry and complex geometry.
Rectifiability by
Rectifiable sets, measures, currents and varifolds are foundational concepts in geometric measure theory. The last four decades have seen the emergence of a wealth of connections between rectifiability and other areas of analysis and geometry, including deep links with the calculus of variations and complex and harmonic analysis. This short book provides an easily digestible overview of this wide and active field, including discussions of historical background, the basic theory in Euclidean and non-Euclidean settings, and the appearance of rectifiability in analysis and geometry. The author avoids complicated technical arguments and long proofs, instead giving the reader a flavour of each of the topics in turn while providing full references to the wider literature in an extensive bibliography. It is a perfect introduction to the area for researchers and graduate students, who will find much inspiration for their own research inside.
