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| Author | : Yukinobu Toda |
| Publisher | : Springer Nature |
| Total Pages | : 318 |
| Release | : |
| Genre | : |
| ISBN | : 3031617053 |
| Author | : Yukinobu Toda |
| Publisher | : Springer Nature |
| Total Pages | : 110 |
| Release | : 2021-12-15 |
| Genre | : Science |
| ISBN | : 9811678383 |
This book is an exposition of recent progress on the Donaldson–Thomas (DT) theory. The DT invariant was introduced by R. Thomas in 1998 as a virtual counting of stable coherent sheaves on Calabi–Yau 3-folds. Later, it turned out that the DT invariants have many interesting properties and appear in several contexts such as the Gromov–Witten/Donaldson–Thomas conjecture on curve-counting theories, wall-crossing in derived categories with respect to Bridgeland stability conditions, BPS state counting in string theory, and others. Recently, a deeper structure of the moduli spaces of coherent sheaves on Calabi–Yau 3-folds was found through derived algebraic geometry. These moduli spaces admit shifted symplectic structures and the associated d-critical structures, which lead to refined versions of DT invariants such as cohomological DT invariants. The idea of cohomological DT invariants led to a mathematical definition of the Gopakumar–Vafa invariant, which was first proposed by Gopakumar–Vafa in 1998, but its precise mathematical definition has not been available until recently. This book surveys the recent progress on DT invariants and related topics, with a focus on applications to curve-counting theories.
| Author | : Yukinobu Toda |
| Publisher | : |
| Total Pages | : 0 |
| Release | : 2021 |
| Genre | : |
| ISBN | : 9789811678394 |
This book is an exposition of recent progress on the Donaldson-Thomas (DT) theory. The DT invariant was introduced by R. Thomas in 1998 as a virtual counting of stable coherent sheaves on Calabi-Yau 3-folds. Later, it turned out that the DT invariants have many interesting properties and appear in several contexts such as the Gromov-Witten/Donaldson-Thomas conjecture on curve-counting theories, wall-crossing in derived categories with respect to Bridgeland stability conditions, BPS state counting in string theory, and others. Recently, a deeper structure of the moduli spaces of coherent sheaves on Calabi-Yau 3-folds was found through derived algebraic geometry. These moduli spaces admit shifted symplectic structures and the associated d-critical structures, which lead to refined versions of DT invariants such as cohomological DT invariants. The idea of cohomological DT invariants led to a mathematical definition of the Gopakumar-Vafa invariant, which was first proposed by Gopakumar-Vafa in 1998, but its precise mathematical definition has not been available until recently. This book surveys the recent progress on DT invariants and related topics, with a focus on applications to curve-counting theories.
| Author | : Dominic D. Joyce |
| Publisher | : |
| Total Pages | : 199 |
| Release | : 2011 |
| Genre | : Calabi-Yau manifolds |
| ISBN | : 9780821887523 |
Donaldson-Thomas invariants DTα(τ) are integers which `count' τ-stable coherent sheaves with Chern character α α on a Calabi-Yau 3-fold X, where τ denotes Gieseker stability for some ample line bundle on X. They are unchanged under deformations of X. The conventional definition works only for classes α containing no strictly τ-semistable sheaves. Behrend showed that DTα(τ) can be written as a weighted Euler characteristic χ(Mstα(τ),νMstα(τ)) of the stable moduli scheme Mstα(τ) by a constructible function νMstα(τ) we call the `Behrend function'. This book studies generalized Donaldson-Thomas invariants DT ̄α(τ). They are rational numbers which `count' both τ-stable and τ-semistable coherent sheaves with Chern character α on X; strictly τ-semistable sheaves must be counted with complicated rational weights. The DT ̄α(τ) are defined for all classes α α, and are equal to DTα(τ) when it is defined. They are unchanged under deformations of X, and transform by a wall-crossing formula under change of stability condition τ. To prove all this we study the local structure of the moduli stack M of coherent sheaves on X. We show that an atlas for M may be written locally as Crit(f) for f:U→C holomorphic and U smooth, and use this to deduce identities on the Behrend function νM. We compute our invariants DT ̄α(τ) in examples, and make a conjecture about their integrality properties. We also extend the theory to abelian categories Q with relations I I coming from a superpotential W on Q, and connect our ideas with Szendrői's noncommutative Donaldson-Thomas invariants, and work by Reineke and others on invariants counting quiver representations. Our book is closely related to Kontsevich and Soibelman's independent paper Stability structures, motivic Donaldson-Thomas invariants and cluster transformations.
| Author | : Ricardo Castano-Bernard |
| Publisher | : Springer |
| Total Pages | : 445 |
| Release | : 2014-10-07 |
| Genre | : Mathematics |
| ISBN | : 3319065149 |
The relationship between Tropical Geometry and Mirror Symmetry goes back to the work of Kontsevich and Y. Soibelman (2000), who applied methods of non-archimedean geometry (in particular, tropical curves) to Homological Mirror Symmetry. In combination with the subsequent work of Mikhalkin on the “tropical” approach to Gromov-Witten theory and the work of Gross and Siebert, Tropical Geometry has now become a powerful tool. Homological Mirror Symmetry is the area of mathematics concentrated around several categorical equivalences connecting symplectic and holomorphic (or algebraic) geometry. The central ideas first appeared in the work of Maxim Kontsevich (1993). Roughly speaking, the subject can be approached in two ways: either one uses Lagrangian torus fibrations of Calabi-Yau manifolds (the so-called Strominger-Yau-Zaslow picture, further developed by Kontsevich and Soibelman) or one uses Lefschetz fibrations of symplectic manifolds (suggested by Kontsevich and further developed by Seidel). Tropical Geometry studies piecewise-linear objects which appear as “degenerations” of the corresponding algebro-geometric objects.
| Author | : Clay Mathematics Institute. Summer School |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 396 |
| Release | : 2004 |
| Genre | : Mathematics |
| ISBN | : 9780821837153 |
Contains selection of expository and research article by lecturers at the school. Highlights current interests of researchers working at the interface between string theory and algebraic supergravity, supersymmetry, D-branes, the McKay correspondence andFourer-Mukai transform.
| Author | : Ana Cannas da Silva |
| Publisher | : Springer |
| Total Pages | : 240 |
| Release | : 2004-10-27 |
| Genre | : Mathematics |
| ISBN | : 354045330X |
The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.
| Author | : David A. Cox |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 498 |
| Release | : 1999 |
| Genre | : Mathematics |
| ISBN | : 082182127X |
Mirror symmetry began when theoretical physicists made some astonishing predictions about rational curves on quintic hypersurfaces in four-dimensional projective space. Understanding the mathematics behind these predictions has been a substantial challenge. This book is the first completely comprehensive monograph on mirror symmetry, covering the original observations by the physicists through the most recent progress made to date. Subjects discussed include toric varieties, Hodge theory, Kahler geometry, moduli of stable maps, Calabi-Yau manifolds, quantum cohomology, Gromov-Witten invariants, and the mirror theorem. This title features: numerous examples worked out in detail; an appendix on mathematical physics; an exposition of the algebraic theory of Gromov-Witten invariants and quantum cohomology; and, a proof of the mirror theorem for the quintic threefold.
| Author | : László Lovász |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 495 |
| Release | : 2012 |
| Genre | : Mathematics |
| ISBN | : 0821890859 |
Recently, it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks. To develop a mathematical theory of very large networks is an important challenge. This book describes one recent approach to this theory, the limit theory of graphs, which has emerged over the last decade. The theory has rich connections with other approaches to the study of large networks, such as ``property testing'' in computer science and regularity partition in graph theory. It has several applications in extremal graph theory, including the exact formulations and partial answers to very general questions, such as which problems in extremal graph theory are decidable. It also has less obvious connections with other parts of mathematics (classical and non-classical, like probability theory, measure theory, tensor algebras, and semidefinite optimization). This book explains many of these connections, first at an informal level to emphasize the need to apply more advanced mathematical methods, and then gives an exact development of the theory of the algebraic theory of graph homomorphisms and of the analytic theory of graph limits. This is an amazing book: readable, deep, and lively. It sets out this emerging area, makes connections between old classical graph theory and graph limits, and charts the course of the future. --Persi Diaconis, Stanford University This book is a comprehensive study of the active topic of graph limits and an updated account of its present status. It is a beautiful volume written by an outstanding mathematician who is also a great expositor. --Noga Alon, Tel Aviv University, Israel Modern combinatorics is by no means an isolated subject in mathematics, but has many rich and interesting connections to almost every area of mathematics and computer science. The research presented in Lovasz's book exemplifies this phenomenon. This book presents a wonderful opportunity for a student in combinatorics to explore other fields of mathematics, or conversely for experts in other areas of mathematics to become acquainted with some aspects of graph theory. --Terence Tao, University of California, Los Angeles, CA Laszlo Lovasz has written an admirable treatise on the exciting new theory of graph limits and graph homomorphisms, an area of great importance in the study of large networks. It is an authoritative, masterful text that reflects Lovasz's position as the main architect of this rapidly developing theory. The book is a must for combinatorialists, network theorists, and theoretical computer scientists alike. --Bela Bollobas, Cambridge University, UK
| Author | : |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 698 |
| Release | : 2009 |
| Genre | : Mathematics |
| ISBN | : 0821838482 |
Research in string theory has generated a rich interaction with algebraic geometry, with exciting work that includes the Strominger-Yau-Zaslow conjecture. This monograph builds on lectures at the 2002 Clay School on Geometry and String Theory that sought to bridge the gap between the languages of string theory and algebraic geometry.
