Claire voisin e casada com jeanmichel coron, tendo o casal cinco filhos. What would eventually become the field of mirror symmetry began when physicists went looking for some extra dimensions. This monograph, based on lectures given at the nsfcbms conference on tropical geometry and mirror symmetry at kansas state university, aims to present a snapshot of ideas being developed by gross and siebert to understand mirror symmetry via tropical geometry. Alloftheabovementioned resultsrequire g tobeagroupofdiagonalsymmetries. The natural construction obtained is used to give the vanishing of a corresponding invariant metric. The term refers to a situation where two calabiyau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory. Parkes 6 which purports to give a count of the number of. Picardfuchs equation and canonical coordinates for the quintic mirror family. Therehasbeenmuch interest in understanding the mirror symmetry for when g is nonabelian, but until now therehas not been a clear way to determine the mirror model, since the dual group was only. Mirror symmetry ams bookstore american mathematical society. We use fermat motives to interpret the topological mirror symmetry phenonemon.
What are your views about the claire voisin s book on mirror symmetry. Lecture notes by claire voisin hodge theory and complex algebraic geometry i. Physics around mirror symmetry november 18 22, 20 workshop 4. Mathematician claire voisin awarded the cnrs 2016 gold. Pantev abstract we discuss the hodge theory of algebraic noncommutative spaces and analyze how this theory interacts with the calabiyau condition and with mirror symmetry. The first of two volumes offering a modern introduction to kaehlerian geometry and hodge structure. Relation with mixed hodge theory 222 bibliography 229 viii. Hodge theory, singularities and dmodules, claude sabbah. There is another interpretation of the kodairaspencer map h 1 x. We need the version for calabiyau manifolds and mirror hodge numbers. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. The first chapter is devoted to the geometry of calabiyau manifolds, and the second describes, as motivation, the ideas from quantum field. It uses symplectic geometry and is related to the first by two dimensional equivariant mirror symmetry. Mark gross, on theta functions and mirror symmetry.
I will comment on some recent ideas of claire voisin on constructing invariant measures so as to show that they vanish on some calabiyau varieties. Claire voisin jussieu johannes walcher mcgill kang zuo mainz there will be a concentrated graduate course the week of november 11 at the fields institute prior to workshop 4. Mirror symmetry has become an established branch of mathematics and mathematical physics, and research in the subject has resulted in brilliant developments. Claire voisin has just been awarded the prestigious shaw prize, also known as the nobel of the east, in mathematical sciences. Unlike previous symplectic geometry based approaches, it produces a bigraded homology theory. Mark green, claire voisin, jacob murre, algebraic cycles and hodge theory lecture notes in mathematics, 1594 1993.
Mathematician claire voisin awarded the cnrs 2016 gold medal. The book starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in. The first chapter is devoted to the geometry of calabiyau manifolds, and the second describes, as motivation, the ideas from quantum field theory. In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called calabiyau manifolds. The hodge conjecture is one of the seven clay mathematics institute millennium prize problems which. Mirror symmetry via 3tori for a class of calabiyau. And what is the present status of research in mirror symmetry, i mean what type of. Mirror symmetry says that there is another calabiyau manifold whose hodge diamond is the mirror image or 90 degree rotation of this one. Hodge theory and complex algebraic geometry i, ii by claire voisin pdf. Toric varieties form a class of algebraic varieties that plays an important role.
Selected titles in this series universite grenoble alpes. Variations of hodge structure on calabi yau threefolds. Voi99 claire voisin, mirror symmetry, smfams texts and monographs. Her most important achievement involves the kodaira theorem for surfaces. In 2002, voisin proved that the generalization of the hodge conjecture for compact kahler varieties is false. Gro11 mark gross, tropical geometry and mirror symmetry, cbms regional conference series in mathematics, 114. As far back as the late 1960s, physicists had tried to explain the existence of fundamental particles electrons, photons, quarks in terms of minuscule vibrating strings. This is the english translation of professor voisins book reflecting the discovery of the mirror symmetry phenomenon. This course will focus on various aspects of mirror symmetry. Order exclusively on the ams bookstoretitles are conveniently. Here we apply this process to a compact g2 manifold, constructed by joyce, and as a result we obtain a pair of borceavoisin calabiyau manifolds, which are known to be mirror duals of each other. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. Mirror symmetry definition of mirror symmetry by the free. Gromovwitten invariants in genus zero, as is predicted by mirror symmetry.
The book starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a more theoretical way than is usual in geometry. The schedule for the concentrated graduate course and for the workshop as well as titles and abstracts of talks will be posted on the fields program webpage. I am seeking a list of good references for syz conjecture, homological mirror symmetry, physics of the theory, modern developments and on its relation to other areas of mathematics and some original papers preferably in chronological order. Claire voisin wikipedia bahasa indonesia, ensiklopedia bebas. Pierre deligne, dennis sullivan, rene thom, all ihes professors, were the first to join the committee in 1980. Dec, 2016 claire voisin has just been awarded the prestigious shaw prize, also known as the nobel of the east, in mathematical sciences. Mirror symmetry give a short introduction to mirror symmetry, e. Morrison introduction there has been much recent excitement among mathematicians about a calculation made by a group of string theorists p. Mirror symmetry via 3tori for a class of calabiyau threefolds. Gromovwitten invariants amodel are related to the \closedstring bmodel.
Quantum mirror symmetry for borceavoisin threefolds. Mirror symmetry definition of mirror symmetry by the. Claire voisin born 4 march 1962 is a french mathematician known for her work in algebraic. Finally, there is an introduction to the hodge conjecture, and the construction of intermediate jacobians. Quanta magazine has an excellent article by kevin hartnett on the state of efforts by mathematicians to understand mirror symmetry. Yukawa couplings and numbers of rational curves on the quintic. She had received the 2016 cnrs gold medal, frances highest scientific distinction, for her research in complex algebraic geometry. Claire voisin born 4 march 1962 is a french mathematician known for her work in algebraic geometry.
Math 6180complex manifoldsspring 2018 syllabus hodge. The rise of mathematical interest dates back to the early 1990s, when candelas, ossa. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. In both cases, mirror symmetry, and techniques developed by physicists and mathematicians to understand it play a crucial role. The book starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a. Sheel ganatra, tim perutz, and nick sheridan, mirror symmetry. Claire voisin, hodge theory and the topology of complex kahler and complex projective manifolds survey, pdf claire voisin, hodge theory and complex algebraic geometry i,ii, cambridge stud. Mirror pairs of calabiyau threefolds from mirror pairs of. This volume consists of articles by speakers at the conference on geometry and topology held at harvard university in 2014. Introduction to hodge theory download pdfepub ebook.
She is noted for her work in algebraic geometry particularly as it pertains to variations of hodge structures and mirror symmetry, and has written several books on hodge theory. What are your views about the claire voisins book on mirror symmetry. We develop an abstract theory of noncommutative hodge structures, investigate ex. American mathematical society, providence ri, 2011. Overview 1 generalities on nonhomological mirror symmetry. References bar01 serguei barannikov, quantum periods, i. Simon donaldson, on kahlereinstein metrics and algebraic structures on limit spaces.
In fact one considers mirror symmetry for degenerating families for calabiyau 3folds in large volume limit which may be expressed precisely via the gromovhausdorff metric. She is noted for her work in algebraic geometry particularly as it pertains to variations of hodge structures and mirror symmetry, and has written several books on hodge. This sizable book contains essentially some polished lecture notes of a seminar series in mirror symmetry that was given in the spring of 2000. Room 127, department of mathematics graduate school of science bldg no. There is another interpretation of the kodairaspencer map h. Mirror symmetry and related topics 2017 previous workshops 2014, 2015, 2016 dates. Gps15 sheel ganatra, tim perutz, and nick sheridan, mirror symmetry. Mirror symmetry for exceptional unimodular singularities 28 appendix a. They were followed by jean bourgain in 1987, mikhail gromov in 1982, maxim kontsevich in 1996, laurent lafforgue in 1999 and claire voisin in 2007. In this case, the mirror of a fano manifold is not a compact manifold, but rather a landauginzburg model, a noncompact manifold equipped with a regular function called superpotential. These are obtained from a k3 surface s with holomorphic involution. For k3 surfaces, topological mirror symmetry seems to be trivial at first sight. Math 6180complex manifoldsspring 2018 syllabus hodge theory.
Mirror symmetry string theory, a relation between two calabiyau manifolds in string theory homological mirror symmetry, a mathematical conjecture about calabiyau manifolds made by maxim kontsevich. The latest notices has an interesting and extensive article about claire voisin. This is the english translation of professor voisin s book reflecting the discovery of the mirror symmetry phenomenon. The english translation of professor voisin s book reflecting the discover. Reflection symmetry, a geometrical symmetry with respect to reflection. Calabi yau manifolds and, mirror symmetry by jose bertin and chris peters. Claire voisin, miroirs et involutions sur les surfaces k3, asterisque 1993. Mirror symmetry is another area of algebraic geometry that drew her interest, and where, among other things, she constructed explicit mirror families. Batyrevs construction which exhibits the mirror symmetry phenomenon between.
The term refers to a situation where two calabiyau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory mirror symmetry was originally discovered by physicists. Shortly after mirror symmetry was formulated as a duality between calabiyau manifolds, it was suggested that fano manifolds also may exhibit mirror symmetry. Variations of hodge structure, calabiyaus and mirror symmetry, in introduction to hodge theory, bertin, demailly, illusie, peters. The geometric concepts needed to formulate various mathematical versions of mirror symmetry will be introduced along the way, in variable levels of detail and rigor. Hodge theory and complex algebraic geometry i by claire voisin. Gro11 mark gross, tropical geometry and mirror symmetry, cbms regional conference. The mirror symmetry conjecture roughly claims that every calabiyau 3fold has a mirror. Mirror symmetry claire voisin, universite pierre et marie curie ams, 1999, 120 pp. Part 5 is devoted to advanced topics in mirror symmetry, including the role of dbranes in the context of mirror symmetry, and some of their applications in physics and mathematics. What is meant by the proof of mirror symmetry 463 20.