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The Decomposition of Global Conformal Invariants (AM-182)
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780691153476.jpg" alt="The Decomposition of Global Conformal Invariants (AM-182)"/><br/></td><td><dl><dt>Author:</dt><dd>Spyros Alexakis</dd><dt>ISBN:</dt><dd>9780691153476</dd><dt>Publisher:</dt><dd>Princeton University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology</dd><dt>DOI:</dt><dd>10.23943/princeton/9780691153476.001.0001</dd><dt>Published in print:</dt><dd>2012</dd><dt>Published Online:</dt><dd>2017-10-19</dd></dl></td></tr></table><p>This book addresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies. The question concerns conformally invariant functionals on the space of Riemannian metrics over a given manifold. These functionals act on a metric by first constructing a Riemannian scalar out of it, and then integrating this scalar over the manifold. Suppose this integral remains invariant under conformal re-scalings of the underlying metric. What information can one then deduce about the Riemannian scalar? This book asserts that the Riemannian scalar must be a linear combination of three obvious candidates, each of which clearly satisfies the required property: a local conformal invariant, a divergence of a Riemannian vector field, and the Chern–Gauss–Bonnet integrand. The book provides a proof of this conjecture. The result itself sheds light on the algebraic structure of conformal anomalies, which appear in many settings in theoretical physics. It also clarifies the geometric significance of the renormalized volume of asymptotically hyperbolic Einstein manifolds. The methods introduced here make an interesting connection between algebraic properties of local invariants—such as the classical Riemannian invariants and the more recently studied conformal invariants—and the study of global invariants, in this case conformally invariant integrals.</p>Spyros Alexakis2017-10-19Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780691160504.jpg" alt="Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)"/><br/></td><td><dl><dt>Author:</dt><dd>Claire Voisin</dd><dt>ISBN:</dt><dd>9780691160504</dd><dt>Publisher:</dt><dd>Princeton University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology</dd><dt>DOI:</dt><dd>10.23943/princeton/9780691160504.001.0001</dd><dt>Published in print:</dt><dd>2014</dd><dt>Published Online:</dt><dd>2017-10-19</dd></dl></td></tr></table><p>This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.</p>Claire Voisin2017-10-19Arithmetic Compactifications of PEL-Type Shimura Varieties
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780691156545.jpg" alt="Arithmetic Compactifications of PEL-Type Shimura Varieties"/><br/></td><td><dl><dt>Author:</dt><dd>Kai-Wen Lan</dd><dt>ISBN:</dt><dd>9780691156545</dd><dt>Publisher:</dt><dd>Princeton University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology</dd><dt>DOI:</dt><dd>10.23943/princeton/9780691156545.001.0001</dd><dt>Published in print:</dt><dd>2013</dd><dt>Published Online:</dt><dd>2017-10-19</dd></dl></td></tr></table><p>By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical foundation for several exciting recent developments. PEL-type Shimura varieties, which are natural generalizations of modular curves, are useful for studying the arithmetic properties of automorphic forms and automorphic representations, and they have played important roles in the development of the Langlands program. As with modular curves, it is desirable to have integral models of compactifications of PEL-type Shimura varieties that can be described in sufficient detail near the boundary, which this book explains in detail. Through the discussion, the book generalizes the theory of degenerations of polarized abelian varieties and the application of that theory to the construction of toroidal and minimal compactifications of Siegel moduli schemes over the integers (as developed by Mumford, Faltings, and Chai). The book is designed to be accessible to graduate students who have an understanding of schemes and abelian varieties.</p>Kai-Wen Lan2017-10-19Sasakian Geometry
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780198564959.jpg" alt="Sasakian Geometry"/><br/></td><td><dl><dt>Author:</dt><dd>Charles Boyer, Krzysztof Galicki</dd><dt>ISBN:</dt><dd>9780198564959</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780198564959.001.0001</dd><dt>Published in print:</dt><dd>2007</dd><dt>Published Online:</dt><dd>2008-01-01</dd></dl></td></tr></table><p>Sasakian manifolds were first introduced in 1962. This book's main focus is on the intricate relationship between Sasakian and Kähler geometries, especially when the Kähler structure is that of an algebraic variety. The book is divided into three parts. The first five chapters carefully prepare the stage for the proper introduction of the subject. After a brief discussion of G-structures, the reader is introduced to the theory of Riemannian foliations. A concise review of complex and Kähler geometry precedes a fairly detailed treatment of compact complex Kähler orbifolds. A discussion of the existence and obstruction theory of Kähler-Einstein metrics (Monge-Ampère problem) on complex compact orbifolds follows. The second part gives a careful discussion of contact structures in the Riemannian setting. Compact quasi-regular Sasakian manifolds emerge here as algebraic objects: they are orbifold circle bundles over compact projective algebraic orbifolds. After a discussion of symmetries of Sasakian manifolds in Chapter 8, the book looks at Sasakian structures on links of isolated hypersurface singularities in Chapter 9. What follows is a study of compact Sasakian manifolds in dimensions three and five focusing on the important notion of positivity. The latter is crucial in understanding the existence of Sasaki-Einstein and 3-Sasakian metrics, which are studied in Chapters 11 and 13. Chapter 12 gives a fairly brief description of quaternionic geometry which is a prerequisite for Chapter 13. The study of Sasaki-Einstein geometry was the original motivation for the book. The final chapter on Killing spinors discusses the properties of Sasaki-Einstein manifolds, which allow them to play an important role as certain models in the supersymmetric field theories of theoretical physics.</p>Charles Boyer and Krzysztof Galicki2008-01-01Riemann Surfaces
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780198526391.jpg" alt="Riemann Surfaces"/><br/></td><td><dl><dt>Author:</dt><dd>Simon Donaldson</dd><dt>ISBN:</dt><dd>9780198526391</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology, Analysis</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780198526391.001.0001</dd><dt>Published in print:</dt><dd>2011</dd><dt>Published Online:</dt><dd>2013-12-17</dd></dl></td></tr></table><p>The theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an extremely useful part of mathematics, knowledge of which is needed by specialists in many other fields. It provides a model for a large number of more recent developments in areas including manifold topology, global analysis, algebraic geometry, Riemannian geometry, and diverse topics in mathematical physics. This text on Riemann surface theory proves the fundamental analytical results on the existence of meromorphic functions and the Uniformisation Theorem. The approach taken emphasises PDE methods, applicable more generally in global analysis. The connection with geometric topology, and in particular the role of the mapping class group, is also explained. To this end, some more sophisticated topics have been included, compared with traditional texts at this level. While the treatment is novel, the roots of the subject in traditional calculus and complex analysis are kept well in mind. Part I sets up the interplay between complex analysis and topology, with the latter treated informally. Part II works as a rapid first course in Riemann surface theory, including elliptic curves. The core of the book is contained in Part III, where the fundamental analytical results are proved.</p>Simon Donaldson2013-12-17Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181)
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780691153704.jpg" alt="Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181)"/><br/></td><td><dl><dt>Author:</dt><dd>Umberto ZannierDavidMasserDavid Masser</dd><dt>ISBN:</dt><dd>9780691153704</dd><dt>Publisher:</dt><dd>Princeton University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology</dd><dt>DOI:</dt><dd>10.23943/princeton/9780691153704.001.0001</dd><dt>Published in print:</dt><dd>2012</dd><dt>Published Online:</dt><dd>2017-10-19</dd></dl></td></tr></table><p>This book considers the so-called unlikely intersections, a topic that embraces well-known issues, such as Lang's and Manin–Mumford's, concerning torsion points in subvarieties of tori or abelian varieties. More generally, the book considers algebraic subgroups that meet a given subvariety in a set of unlikely dimension. The book is an expansion of the Hermann Weyl Lectures delivered by the author at the Institute for Advanced Study in Princeton in May 2010.The book consists of four chapters and seven brief appendixes. The first chapter considers multiplicative algebraic groups, presenting proofs of several developments, ranging from the origins to recent results, and discussing many applications and relations with other contexts. The second chapter considers an analogue in arithmetic and several applications of this. The third chapter introduces a new method for approaching some of these questions, and presents a detailed application of this to a relative case of the Manin–Mumford issue. The fourth chapter focuses on the André–Oort conjecture (outlining work by Pila).</p>Umberto Zannier2017-10-19Introduction to Symplectic Topology
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780198794899.jpg" alt="Introduction to Symplectic Topology"/><br/></td><td><dl><dt>Author:</dt><dd>Dusa McDuff, Dietmar Salamon</dd><dt>ISBN:</dt><dd>9780198794899</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Analysis, Geometry / Topology</dd><dt>DOI:</dt><dd>10.1093/oso/9780198794899.001.0001</dd><dt>Published in print:</dt><dd>2017</dd><dt>Published Online:</dt><dd>2017-06-22</dd></dl></td></tr></table><p>Over the past number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. The first edition of Introduction to Symplectic Topology was published in 1995. The book was the first comprehensive introduction to the subject and became a key text in the area. In 1998, a significantly revised second edition contained new sections and updates. This third edition includes both further updates and new material on this fast-developing area. All chapters have been revised to improve the exposition, new material has been added in many places, and various proofs have been tightened up. Copious new references to key papers have been added to the bibliography. In particular, the material on contact geometry has been significantly expanded, many more details on linear complex structures and on the symplectic blowup and blowdown have been added, the section on J-holomorphic curves in Chapter 4 has been thoroughly revised, there are new sections on GIT and on the topology of symplectomorphism groups, and the section on Floer homology has been revised and updated. Chapter 13 has been completely rewritten and has a new title (Questions of Existence and Uniqueness). It now contains an introduction to existence and uniqueness problems in symplectic topology, a section describing various examples, an overview of Taubes–Seiberg–Witten theory and its applications to symplectic topology, and a section on symplectic 4-manifolds. Chapter 14 on open problems has been added.</p>Dusa McDuff and Dietmar Salamon2017-06-22A Primer on Mapping Class Groups (PMS-49)
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780691147949.jpg" alt="A Primer on Mapping Class Groups (PMS-49)"/><br/></td><td><dl><dt>Author:</dt><dd>Benson Farb, Dan Margalit</dd><dt>ISBN:</dt><dd>9780691147949</dd><dt>Publisher:</dt><dd>Princeton University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology</dd><dt>DOI:</dt><dd>10.23943/princeton/9780691147949.001.0001</dd><dt>Published in print:</dt><dd>2011</dd><dt>Published Online:</dt><dd>2017-10-19</dd></dl></td></tr></table><p>The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.</p>Benson Farb and Dan Margalit2017-10-19Algebraic and Geometric Surgery
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780198509240.jpg" alt="Algebraic and Geometric Surgery"/><br/></td><td><dl><dt>Author:</dt><dd>Andrew Ranicki</dd><dt>ISBN:</dt><dd>9780198509240</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780198509240.001.0001</dd><dt>Published in print:</dt><dd>2002</dd><dt>Published Online:</dt><dd>2007-09-01</dd></dl></td></tr></table><p>This book is an introduction to surgery theory, the standard algebraic topology classification method for manifolds of dimension greater than 4. It is aimed at those who have already been on a basic topology course, and would now like to understand the topology of high-dimensional manifolds. This text contains entry-level accounts of the various prerequisites of both algebra and topology. Surgery theory expresses the manifold structure set in terms of the topological K-theory of vector bundles and the algebraic L-theory of quadratic forms. While concentrating on the basic mechanics of surgery, this book includes many worked examples, useful drawings for illustration of the algebra and references for further reading.</p>Andrew Ranicki2007-09-01Theories, Sites, Toposes
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780198758914.jpg" alt="Theories, Sites, ToposesRelating and studying mathematical theories through topos-theoretic 'bridges'"/><br/></td><td><dl><dt>Author:</dt><dd>Olivia Caramello</dd><dt>ISBN:</dt><dd>9780198758914</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology, Pure Mathematics</dd><dt>DOI:</dt><dd>10.1093/oso/9780198758914.001.0001</dd><dt>Published in print:</dt><dd>2017</dd><dt>Published Online:</dt><dd>2018-03-22</dd></dl></td></tr></table><p>This book is devoted to a general study of geometric theories from a topos-theoretic perspective. After recalling the necessary topos-theoretic preliminaries, it presents the main methodology it uses to extract ‘concrete’ information on theories from properties of their classifying toposes—the ‘bridge’ technique. As a first implementation of this methodology, a duality is established between the subtoposes of the classifying topos of a geometric theory and the geometric theory extensions (also called ‘quotients’) of the theory. Many concepts of elementary topos theory which apply to the lattice of subtoposes of a given topos are then transferred via this duality into the context of geometric theories. A second very general implementation of the ‘bridge’ technique is the investigation of the class of theories of presheaf type (i.e. classified by a presheaf topos). After establishing a number of preliminary results on flat functors in relation to classifying toposes, the book carries out a systematic investigation of this class resulting in a number of general results and a characterization theorem allowing one to test whether a given theory is of presheaf type by considering its models in arbitrary Grothendieck toposes. Expansions of geometric theories and faithful interpretations of theories of presheaf type are also investigated. As geometric theories can always be written (in many ways) as quotients of presheaf type theories, the study of quotients of a given theory of presheaf type is undertaken. Lastly, the book presents a number of applications in different fields of mathematics of the theory it develops.</p>Olivia Caramello2018-03-22Spaces of PL Manifolds and Categories of Simple Maps (AM-186)
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780691157757.jpg" alt="Spaces of PL Manifolds and Categories of Simple Maps (AM-186)"/><br/></td><td><dl><dt>Author:</dt><dd>Friedhelm Waldhausen, Bjørn Jahren, John Rognes</dd><dt>ISBN:</dt><dd>9780691157757</dd><dt>Publisher:</dt><dd>Princeton University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology</dd><dt>DOI:</dt><dd>10.23943/princeton/9780691157757.001.0001</dd><dt>Published in print:</dt><dd>2013</dd><dt>Published Online:</dt><dd>2017-10-19</dd></dl></td></tr></table><p>Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing the author's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a “desingularization,” improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.</p>Friedhelm Waldhausen, Bjørn Jahren, and John Rognes2017-10-19Geometry and Physics: Volume II
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780198802020.jpg" alt="Geometry and Physics: Volume IIA Festschrift in honour of Nigel Hitchin"/><br/></td><td><dl><dt>Author:</dt><dd>AndrewDancerAndrew DancerProfessor of Mathematics, Mathematical Institute, Oxford UniversityJørgenEllegaard AndersenJørgen Ellegaard AndersenProfessor of Mathematics, Aarhus UniversityOscarGarcía-PradaOscar García-PradaCSIC Research Professor, Consejo Superior de Investigaciones Cientificas</dd><dt>ISBN:</dt><dd>9780198802020</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology</dd><dt>DOI:</dt><dd>10.1093/oso/9780198802020.001.0001</dd><dt>Published in print:</dt><dd>2018</dd><dt>Published Online:</dt><dd>2018-12-20</dd></dl></td></tr></table><p>These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and Savilian Professor of Geometry at Oxford. The proceedings contain twenty-nine articles, including three by Fields medallists (Donaldson, Mori and Yau). The articles cover a wide range of topics in geometry and mathematical physics, including the following: Riemannian geometry, geometric analysis, special holonomy, integrable systems, dynamical systems, generalized complex structures, symplectic and Poisson geometry, low-dimensional topology, algebraic geometry, moduli spaces, Higgs bundles, geometric Langlands programme, mirror symmetry and string theory. These volumes will be of interest to researchers and graduate students both in geometry and mathematical physics.</p>Andrew Dancer, Jørgen Ellegaard Andersen, and Oscar García-Prada2018-12-20Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)
//princeton.universitypressscholarship.com/view/10.23943/princeton/9780691144771.001.0001/upso-9780691144771
<table><tr><td width="200px"><img width="150px" src="/view/covers/9780691144771.jpg" alt="Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)"/><br/></td><td><dl><dt>Author:</dt><dd>Paula TretkoffHans-ChristophIm HofHans-Christoph Im Hof</dd><dt>ISBN:</dt><dd>9780691144771</dd><dt>Publisher:</dt><dd>Princeton University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology</dd><dt>DOI:</dt><dd>10.23943/princeton/9780691144771.001.0001</dd><dt>Published in print:</dt><dd>2016</dd><dt>Published Online:</dt><dd>2017-10-19</dd></dl></td></tr></table><p>This book introduces the theory of complex surfaces through a comprehensive look at finite covers of the projective plane branched along line arrangements. It emphasizes those finite coverings that are free quotients of the complex 2-ball. The book also includes a background on the classical Gauss hypergeometric function of one variable, and a chapter on the Appell two-variable F1 hypergeometric function. The book began as a set of lecture notes, taken by the author, of a course given by Friedrich Hirzebruch at ETH Zürich in 1996. The lecture notes were then considerably expanded over a number of years. In this book, the author has expanded those notes even further, still stressing examples offered by finite covers of line arrangements. The book is largely self-contained and foundational material is introduced and explained as needed, but not treated in full detail. References to omitted material are provided for interested readers. Aimed at graduate students and researchers, this is an accessible account of a highly informative area of complex geometry.</p>Paula Tretkoff2017-10-19Flips for 3-folds and 4-folds
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780198570615.jpg" alt="Flips for 3-folds and 4-folds"/><br/></td><td><dl><dt>Author:</dt><dd>AlessioCortiAlessio CortiDepartment of Mathematics, Imperial College, Londonhttp://www2.imperial.ac.uk/~acorti/</dd><dt>ISBN:</dt><dd>9780198570615</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780198570615.001.0001</dd><dt>Published in print:</dt><dd>2007</dd><dt>Published Online:</dt><dd>2007-09-01</dd></dl></td></tr></table><p>The minimal model program in algebraic geometry is a conjectural sequence of algebraic surgery operations that simplifies any algebraic variety to a point where it can be decomposed into pieces with negative, zero, and positive curvature, in a similar vein as the geometrization program in topology decomposes a three-manifold into pieces with a standard geometry. The last few years have seen dramatic advances in the minimal model program for higher dimensional algebraic varieties, with the proof of the existence of minimal models under appropriate conditions, and the prospect within a few years of having a complete generalization of the minimal model program and the classification of varieties in all dimensions, comparable to the known results for surfaces and 3-folds. This edited collection of chapters, authored by leading experts, provides a complete and self-contained construction of 3-fold and 4-fold flips, and n-dimensional flips assuming minimal models in dimension n-1. A large part of the text is an elaboration of the work of Shokurov, and a complete and pedagogical proof of the existence of 3-fold flips is presented. The book contains a self-contained treatment of many topics that could only be found, with difficulty, in the specialized literature. The text includes a ten-page glossary.</p>Alessio Corti2007-09-01Office Hours with a Geometric Group Theorist
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780691158662.jpg" alt="Office Hours with a Geometric Group Theorist"/><br/></td><td><dl><dt>Author:</dt><dd>DanMargalitDan MargalitGeorgia Institute of TechnologyMattClayMatt ClayUniversity of Arkansas</dd><dt>ISBN:</dt><dd>9780691158662</dd><dt>Publisher:</dt><dd>Princeton University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology</dd><dt>DOI:</dt><dd>10.23943/princeton/9780691158662.001.0001</dd><dt>Published in print:</dt><dd>2017</dd><dt>Published Online:</dt><dd>2018-05-24</dd></dl></td></tr></table><p>Geometric group theory is the study of the interplay between groups and the spaces they act on, and has its roots in the works of Henri Poincaré, Felix Klein, J.H.C. Whitehead, and Max Dehn. This book brings together leading experts who provide one-on-one instruction on key topics in this exciting and relatively new field of mathematics. It's like having office hours with your most trusted math professors. An essential primer for undergraduates making the leap to graduate work, the book begins with free groups—actions of free groups on trees, algorithmic questions about free groups, the ping-pong lemma, and automorphisms of free groups. It goes on to cover several large-scale geometric invariants of groups, including quasi-isometry groups, Dehn functions, Gromov hyperbolicity, and asymptotic dimension. It also delves into important examples of groups, such as Coxeter groups, Thompson's groups, right-angled Artin groups, lamplighter groups, mapping class groups, and braid groups. The tone is conversational throughout, and the instruction is driven by examples. It features numerous exercises and in-depth projects designed to engage readers and provide jumping-off points for research projects.</p>Dan Margalit and Matt Clay2018-05-24Differential Geometry
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780199605880.jpg" alt="Differential GeometryBundles, Connections, Metrics and Curvature"/><br/></td><td><dl><dt>Author:</dt><dd>Clifford Henry Taubes</dd><dt>ISBN:</dt><dd>9780199605880</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology, Mathematical Physics</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780199605880.001.0001</dd><dt>Published in print:</dt><dd>2011</dd><dt>Published Online:</dt><dd>2013-12-17</dd></dl></td></tr></table><p>Bundles, connections, metrics, and curvature are the ‘lingua franca’ of modern differential geometry and theoretical physics. Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, Lie groups, and Grassmanians are all presented here. Other material covered includes the basic theorems about geodesics and Jacobi fields, the classification theorem for flat connections, the definition of characteristic classes, and also an introduction to complex and Kähler geometry. The book uses many of the classical examples from, and applications of, the subjects it covers, in particular those where closed form expressions are available, to bring abstract ideas to life.</p>Clifford Henry Taubes2013-12-17New Perspectives in Stochastic Geometry
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780199232574.jpg" alt="New Perspectives in Stochastic Geometry"/><br/></td><td><dl><dt>Author:</dt><dd>Wilfrid S.KendallWilfrid S. KendallDepartment of Statistics, University of Warwick, UKIlyaMolchanovIlya MolchanovDepartment of Mathematical Statistics and Actuarial Science, University of Bern, Switzerland</dd><dt>ISBN:</dt><dd>9780199232574</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780199232574.001.0001</dd><dt>Published in print:</dt><dd>2009</dd><dt>Published Online:</dt><dd>2010-02-01</dd></dl></td></tr></table><p>Stochastic geometry is a subject with roots stretching back at least 300 years, but one which has only been formed as an academic area in the last 50 years. It covers the study of random patterns, their probability theory, and the challenging problems raised by their statistical analysis. It has grown rapidly in response to challenges in all kinds of applied science, from image analysis through to materials science. Recently, still more stimulus has arisen from exciting new links with rapidly developing areas of mathematics, from fractals through percolation theory to randomized allocation schemes. Coupled with many ongoing developments arising from all sorts of applications, the area is changing and developing rapidly. This book is intended to lay foundations for future research directions by collecting together seventeen chapters contributed by leading researchers in the field, both theoreticians and people involved in applications, surveying these new developments both in theory and in applications. It will introduce and lay foundations for appreciating the fresh perspectives, new ideas, and interdisciplinary connections now arising from stochastic geometry and from other areas of mathematics now connecting to this area.</p>Wilfrid S. Kendall and Ilya Molchanov2010-02-01Hodge Theory (MN-49)
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780691161341.jpg" alt="Hodge Theory (MN-49)"/><br/></td><td><dl><dt>Author:</dt><dd>Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, Lê Dung Tráng</dd><dt>ISBN:</dt><dd>9780691161341</dd><dt>Publisher:</dt><dd>Princeton University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology</dd><dt>DOI:</dt><dd>10.23943/princeton/9780691161341.001.0001</dd><dt>Published in print:</dt><dd>2014</dd><dt>Published Online:</dt><dd>2017-10-19</dd></dl></td></tr></table><p>This book provides a comprehensive and up-to-date introduction to Hodge theory—one of the central and most vibrant areas of contemporary mathematics—from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch–Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students and researchers. The exposition is as accessible as possible and does not require a deep background. At the same time, the book presents some topics at the forefront of current research. The book is divided between introductory and advanced lectures. The introductory lectures address Kähler manifolds, variations of Hodge structure, mixed Hodge structures, the Hodge theory of maps, period domains and period mappings, algebraic cycles (up to and including the Bloch–Beilinson conjecture) and Chow groups, sheaf cohomology, and a new treatment of Grothendieck's algebraic de Rham theorem. The advanced lectures address a Hodge-theoretic perspective on Shimura varieties, the spread philosophy in the study of algebraic cycles, absolute Hodge classes (including a new, self-contained proof of Deligne's theorem on absolute Hodge cycles), and variation of mixed Hodge structures.</p>Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dung Tráng2017-10-19Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780691161686.jpg" alt="Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)"/><br/></td><td><dl><dt>Author:</dt><dd>Ehud Hrushovski, François Loeser</dd><dt>ISBN:</dt><dd>9780691161686</dd><dt>Publisher:</dt><dd>Princeton University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology</dd><dt>DOI:</dt><dd>10.23943/princeton/9780691161686.001.0001</dd><dt>Published in print:</dt><dd>2016</dd><dt>Published Online:</dt><dd>2017-10-19</dd></dl></td></tr></table><p>Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed and model-theoretic prerequisites are reviewed in the first sections.</p>Ehud Hrushovski and François Loeser2017-10-19Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra (AM-194)
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780691170541.jpg" alt="Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra (AM-194)"/><br/></td><td><dl><dt>Author:</dt><dd>Isroil A. Ikromov, Detlef Müller</dd><dt>ISBN:</dt><dd>9780691170541</dd><dt>Publisher:</dt><dd>Princeton University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology</dd><dt>DOI:</dt><dd>10.23943/princeton/9780691170541.001.0001</dd><dt>Published in print:</dt><dd>2016</dd><dt>Published Online:</dt><dd>2017-10-19</dd></dl></td></tr></table><p>This is the first book to present a complete characterization of Stein–Tomas-type Fourier restriction estimates for large classes of smooth hypersurfaces in three dimensions, including all real-analytic hypersurfaces. The range of Lebesgue spaces for which these estimates are valid is described in terms of Newton polyhedra associated to the given surface. The book begins with Elias M. Stein's concept of Fourier restriction and some relations between the decay of the Fourier transform of the surface measure and Stein–Tomas-type restriction estimates. Varchenko's ideas relating Fourier decay to associated Newton polyhedra are briefly explained, particularly the concept of adapted coordinates and the notion of height. It turns out that these classical tools essentially suffice already to treat the case where there exist linear adapted coordinates, and thus the book concentrates on the remaining case. Here the notion of r-height is introduced, which proves to be the right new concept. The book then describes decomposition techniques and related stopping time algorithms that allow to partition the given surface into various pieces, which can eventually be handled by means of oscillatory integral estimates. Different interpolation techniques are presented and used, from complex to more recent real methods by Bak and Seeger. Fourier restriction plays an important role in several fields, in particular in real and harmonic analysis, number theory, and PDEs. This book will interest graduate students and researchers working in such fields.</p>Isroil A. Ikromov and Detlef Müller2017-10-19