Marc Levine - Böcker

This volume contains the proceedings of the Workshop on Motivic Homotopy Theory and Refined Enumerative Geometry, held from May 14-18, 2018, at the Universitat Duisburg-Essen, Essen, Germany.It constitutes an accessible yet swift introduction to a new and active area within algebraic geometry, which connects well with classical intersection theory. Combining both lecture notes aimed at the graduate student level and research articles pointing towards the manifold promising applications of this refined approach, it broadly covers refined enumerative algebraic geometry.

This book combines foundational constructions in the theory of motives and results relating motivic cohomology to more explicit constructions. Prerequisite for understanding the work is a basic background in algebraic geometry. The author constructs and describes a triangulated category of mixed motives over an arbitrary base scheme. Most of the classical constructions of cohomology are described in the motivic setting, including Chern classes from higher $K$-theory, push-forward for proper maps, Riemann-Roch, duality, as well as an associated motivic homology, Borel-Moore homology and cohomology with compact supports.

Departing from the political economy perspective taken by the vast majority of volumes devoted to Mesoamerican obsidian, Obsidian Reflections is an examination of obsidian's sociocultural dimensions—particularly in regard to Mesoamerican world view, religion, and belief systems. Exploring the materiality of this volcanic glass rather than only its functionality, this book considers the interplay among people, obsidian, and meaning and how these relationships shaped patterns of procurement, exchange, and use. An international group of scholars hailing from Belize, France, Japan, Mexico, and the United States provides a variety of case studies from Mexico, Guatemala, Belize, and Honduras. The authors draw on archaeological, iconographic, ethnographic, and ethnohistoric data to examine obsidian as a touchstone for cultural meaning, including references to sacrificial precepts, powerful deities, landscape, warfare, social relations, and fertility.Obsidian Reflections underscores the necessity of understanding obsidian from within its cultural context—the perspective of the indigenous people of Mesoamerica. It will be of great interest to Mesoamericanists as well as students and scholars of lithic studies and material culture.

Following Quillen's approach to complex cobordism, the authors introduce the notion of oriented cohomology theory on the category of smooth varieties over a fixed field. They prove the existence of a universal such theory (in characteristic 0) called Algebraic Cobordism. Surprisingly, this theory satisfies the analogues of Quillen's theorems: the cobordism of the base field is the Lazard ring and the cobordism of a smooth variety is generated over the Lazard ring by the elements of positive degrees. This implies in particular the generalized degree formula conjectured by Rost. The book also contains some examples of computations and applications.

This book is based on lectures given at a summer school on motivic homotopy theory at the Sophus Lie Centre in Nordfjordeid, Norway, in August 2002. Aimed at graduate students in algebraic topology and algebraic geometry, it contains background material from both of these fields, as well as the foundations of motivic homotopy theory. It will serve as a good introduction as well as a convenient reference for a broad group of mathematicians to this important and fascinating new subject. Vladimir Voevodsky is one of the founders of the theory and received the Fields medal for his work, and the other authors have all done important work in the subject.