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Mathematisches Institut Georg-August-Universität Göttingen
This volume contains lecture notes from the seminars "Number Theory", "Algebraic Geometry" and "Twisted Cohomology Theories" which took place at the Mathematics Institute of the University of Göttingen during the Winter Term 2004/2005. Most contributions report on recent work by the authors.
Mathematisches Institut Georg-august-universität Göttingen, Seminars Summer Term 2004
This volume contains lecture notes from the seminars [alpha]Number Theory", [alpha]Algebraic Geometry" and [alpha]Geometric methods in representation theory" which took place at the Mathematics Institute of the University of Göttingen during the Summer Term 2004. Most contributions report on recent work by the authors.
Mathematisches Institut Georg-august-universität Göttingen, Seminars Summer 2003/2004
This volume contains lecture notes from the seminars [alpha]Number Theory", [alpha]Algebraic Geometry" and [alpha]Geometric methods in representation theory" which took place at the Mathematics Institute of the University of Göttingen during the Winter Term 2003-2004. Most contributions report on recent work by the authors.
Rational Points on Algebraic Varieties
This book is devoted to the study of rational and integral points on higher-dimensional algebraic varieties. It contains carefully selected research papers addressing the arithmetic geometry of varieties which are not of general type, with an emphasis on how rational points are distributed with respect to the classical, Zariski and adelic topologies. The present volume gives a glimpse of the state of the art of this rapidly expanding domain in arithmetic geometry. The techniques involve explicit geometric constructions, ideas from the minimal model program in algebraic geometry as well as analytic number theory and harmonic analysis on adelic groups.
Arithmetic of Higher-Dimensional Algebraic Varieties
This text offers a collection of survey and research papers by leading specialists in the field documenting the current understanding of higher dimensional varieties. Recently, it has become clear that ideas from many branches of mathematics can be successfully employed in the study of rational and integral points. This book will be very valuable for researchers from these various fields who have an interest in arithmetic applications, specialists in arithmetic geometry itself, and graduate students wishing to pursue research in this area.
Cohomological and Geometric Approaches to Rationality Problems
Rationality problems link algebra to geometry, and the difficulties involved depend on the transcendence degree of $K$ over $k$, or geometrically, on the dimension of the variety. A major success in 19th century algebraic geometry was a complete solution of the rationality problem in dimensions one and two over algebraically closed ground fields of characteristic zero. Such advances has led to many interdisciplinary applications to algebraic geometry. This comprehensive book consists of surveys of research papers by leading specialists in the field and gives indications for future research in rationality problems. Topics discussed include the rationality of quotient spaces, cohomological invariants of quasi-simple Lie type groups, rationality of the moduli space of curves, and rational points on algebraic varieties. This volume is intended for researchers, mathematicians, and graduate students interested in algebraic geometry, and specifically in rationality problems. Contributors: F. Bogomolov; T. Petrov; Y. Tschinkel; Ch. Böhning; G. Catanese; I. Cheltsov; J. Park; N. Hoffmann; S. J. Hu; M. C. Kang; L. Katzarkov; Y. Prokhorov; A. Pukhlikov
Martin Kneser Collected Works
This volume presents the collection of mathematical articles by Martin Kneser, reprinted in the original language – mostly German –, including one yet unpublished. Moreover, also included is an article by Raman Parimala, discussing Kneser’s work concerning algebraic groups and the Hasse principle, which has been written especially for this volume, as well as an article by Rudolf Scharlau about Kneser’s work on quadratic forms, published elsewhere before. Another commentary article, written by Günter M. Ziegler especially for this volume, describes the astounding influence on the field of combinatorics of what was published as “Aufgabe 360” and its subsequent solution in 1955 resp. 1957 in the “Jahresbericht der Deutschen Mathematiker-Vereinigung”. However, as the titles of the articles show, Kneser’s mathematical interests were much broader, which is beautifully discussed in an obituary by Ulrich Stuhler, included as well in this volume.
Geometry Over Nonclosed Fields
Based on the Simons Symposia held in 2015, the proceedings in this volume focus on rational curves on higher-dimensional algebraic varieties and applications of the theory of curves to arithmetic problems. There has been significant progress in this field with major new results, which have given new impetus to the study of rational curves and spaces of rational curves on K3 surfaces and their higher-dimensional generalizations. One main recent insight the book covers is the idea that the geometry of rational curves is tightly coupled to properties of derived categories of sheaves on K3 surfaces. The implementation of this idea led to proofs of long-standing conjectures concerning birational properties of holomorphic symplectic varieties, which in turn should yield new theorems in arithmetic. This proceedings volume covers these new insights in detail.
Algebra, Arithmetic, and Geometry
EMAlgebra, Arithmetic, and Geometry: In Honor of Yu. I. ManinEM consists of invited expository and research articles on new developments arising from Manin’s outstanding contributions to mathematics.
Algebra, Geometry, and Physics in the 21st Century
This volume is a tribute to Maxim Kontsevich, one of the most original and influential mathematicians of our time. Maxim’s vision has inspired major developments in many areas of mathematics, ranging all the way from probability theory to motives over finite fields, and has brought forth a paradigm shift at the interface of modern geometry and mathematical physics. Many of his papers have opened completely new directions of research and led to the solutions of many classical problems. This book collects papers by leading experts currently engaged in research on topics close to Maxim’s heart. Contributors: S. Donaldson A. Goncharov D. Kaledin M. Kapranov A. Kapustin L. Katzarkov A. Noll P. Pandit S. Pimenov J. Ren P. Seidel C. Simpson Y. Soibelman R. Thorngren
