DOCSLIB.ORG

Calabi-Yau Manifolds at the Interface of Math and Physics by Shiu-Yuen Cheng*, Lizhen Ji†, Liping Wang‡, and Hao Xu§

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Calabi-Yau Manifolds at the Interface of Math and Physics by Shiu-Yuen Cheng*, Lizhen Ji†, Liping Wang‡, and Hao Xu§ There were once a young man and an older man. could be built with them? Maybe there was not Their mathematics paths crossed and their names be- enough structure. How about adding complex struc- came merged into the long-lasting notion of Calabi- tures? Would they admit canonical metrics? Or per- Yau manifolds. haps there was still not enough structure. How about How did this happen? Why did Calabi make his Kähler manifolds—structures between complex ge- famous conjecture? How did Yau prove it? Why are ometry and algebraic geometry? This was what Calabi Calabi-Yau manifolds so important? And how are asked 60 years ago, and what Yau answered 24 years they used? All these and other questions take time to later. He cracked a geometric problem by unraveling answer and real experts to explain. They can probably hard nonlinear differential equations. be best understood through the interactions between The Calabi-Yau metric has been an invaluable tool math and physics, and that’s where their true value in differential and algebraic geometry. Here is an in- lies. Although these questions have a long history, complete list of long-standing conjectures and prob- the future will undoubtedly bring more questions and lems solved by using the Calabi-Yau metric and Yau’s surprises regarding Calabi-Yau manifolds. technique of the a priori estimate. Indeed, one must go back to more than 150 years 1. The Severi conjecture, that every complex sur- ago, when Riemann introduced the concepts of Rie- face that is homotopic to the complex projective mann surfaces and Riemannian metrics. His map- 2 2 ping theorem motivated people to endow each Rie- plane CP is biholomorphic to CP . mann surface with a canonical Riemannian metric, 2. Bogomolov-Miyaoka-Yau Chern number inequal- and from there one of the greatest theorems in the ities and their associated uniformization theo- history of mathematics, the uniformization theorem rems in higher dimensions. n for Riemann surfaces and surfaces with Riemannian 3. Uniqueness of Kähler complex structure on CP . metrics, took shape. 4. Torelli theorem for K3 surfaces. What happened to higher dimensional Rieman- 5. Every K3 surface is Kähler. nian manifolds? What kind of geometric structures 6. The existence of Hermitian-Yang-Mills connec- tions in stable vector bundles (Donaldson- * Mathematical Sciences Center, Tsinghua University, Beijing, Uhlenbeck-Yau theorem). China 7. The existence of complete Kähler-Einstein met- Email: [email protected] † Department of Mathematics, University of Michigan, Ann rics on pseudoconvex domains. Arbor, MI, U.S.A. 8. Calabi-Yau manifolds are used in the context of Email: [email protected] string theory to make models of particle physics ‡ Higher Education Press, Beijing, China unified with quantum gravity. Email: [email protected] 9. Construction of non-flat compact simply- § Department of Mathematics, University of Pittsburg, PA, U.S.A. connected Kähler manifolds whose Ricci curva- Email: [email protected] ture is identically zero. 90 NOTICES OF THE ICCM VOLUME 2,NUMBER 2 10. Any compact Kähler manifold with non-negative SP(n) by reducing the construction of them to alge- first Chern class is covered holomorphically by braic conditions. the product of Cn and a compact simply con- 4. Algebraic geometry: nected Kähler manifold. (a) The second Chern class of such manifolds is nu- 11. The vanishing of c and c [ [ ]n−2 for a Kähler 1 2 w merically positive in the sense that the cup prod- manifold implies that the manifold admits a met- uct of it with the top number of products of any ric with zero curvature and is covered by the flat positive (1,1) class is positive unless it is covered torus. by the torus. 12. As conjectured by Yau, the existence of Kähler- (b) Calabi-Yau manifolds are stable and their tangent Einstein metrics on Fano manifolds has intimate bundles are also stable with respect to any polar- connections to the stability of manifolds. Impor- ization. tant progress has been made recently. (c) Counting the number of algebraic curves in Such beautiful results are appreciated not only by Calabi-Yau manifolds has become quite success- mathematicians but by physicists as well. For, accord- ful since the introduction of the concept of mir- ing to string theory, Calabi-Yau manifolds provide the ror symmetry, which has opened up many new hidden inner spaces of our universe. directions in mathematics. It is true that 60 is a magic number, but 65 is also (d) It gives a structure theorem for Compact Kähler a milestone in one’s life. Here is a poem by Profes- manifolds with non-negative first Chern class. sor Yau reflecting his experience with geometry and 5. Nonlinear partial differential equations: The physics. proof of Calabi conjecture gave the first general method to solve the nonlinear Complex Ampere equa- Forty years of geometry, tion on a complex manifold. The equation solves a equation creates structure, very important inverse problem: given any volume el- analysis blends perfection and beauty. ement on any compact Kähler manifold, any Kähler metric can be deformed to a new Kähler metric whose Thirty years of physics, volume form is the given one up to a constant. The strings and branes drifting through methods of estimates have been fundamental to all hyperspace, constructions of Kähler-Einstein metrics, including breaking supersymmetry as they go. Kähler metrics with prescribed singularities. The sin- Quantum connects to gravity, gular metrics were used by Donaldson et al. to solve and nature is whole again. a famous conjecture of Yau on existence of Kähler- S.-T. Yau Einstein metric on Stable Fano manifolds. The Power of Calabi-Yau Manifolds Perspectives of Experts When Yau turned 65 this year, we asked experts From the Uniformization (Klein, Poincare, Koebe, all over the world, in both math and physics, to com- Brouwer) to Riemman-Hilbert Correspondence, ment on the applications and impacts of Calabi-Yau Picard-Fuchs Equations, and Kähler-Einstein manifolds and on the work of Yau. What followed is Metrics (Calabi-Yau) the mathematical garden of Yau and Calabi-Yau man- 1. Number theory: Calabi-Yau manifolds are nat- ifolds as seen through their eyes. We hope you will ural, higher-dimensional generalizations of elliptic enjoy a walk through this joyful, colorful, and mind- curves—the theory of which dominated number the- expanding garden. ory in the 20th century. Many believe that Calabi-Yau manifolds will be the main tool in number theory in In Mathematics… the 21st century. The related questions of modular forms and quasi-modular forms are now being devel- Calabi-Yau manifold is the treasure of human oped. civilization, although many of its mysteries are yet 2. Algebra: There is a natural category, which ap- to be explored. In view of its surrounding theo- peared in algebra, where the Calabi-Yau condition rems, tools, questions and conjectures, it has been plays an important role. It is used in classifying topo- an extremely active and important research field. logical field theories. Lo Yang 3. Differential geometry: The construction of the Academy of Mathematics and System Sciences, Calabi-Yau metric solves the old problem of classifi- Chinese Academy of Sciences cation of manifolds with holonomy group SU(n) and (Hua Loo-Keng Mathematics Prize 1997) DECEMBER 2014 NOTICES OF THE ICCM 91 There is no doubt that Calabi-Yau manifolds shown to be of great importance in physics, and have become justly famous. The mere fact that one can speculate that their arithmetic mysteries, they are now abbreviated to CY manifolds testifies still largely untouched by number-theorists, could to their importance. some day throw light on even some of the most In both geometry and physics the Kummer sur- intractable ancient problems of the subject. face and its relatives have been of central interest In addition to his own mathematical research, since the 19th century. They are the natural suc- Yau has played an immense role since the middle cessors to elliptic curves, and look how long these 1990’s in the rebuilding of mathematics in China. curves have fascinated mathematicians. Yau has His creation and organization of the ICCM’s has pioneered many important ideas in global geomet- brought together mathematicians of Chinese ori- ric analysis and has influenced much of the math- gin from all over the world, and built up a new es- ematics and physics of our time. prit de corps amongst them. He has known how Michael Atiyah to persuade the highest political voices in China University of Edinburgh that high-level mathematical research is of great (Fields Medal 1966; Abel Prize 2004; importance for the future of China, and deserves King Faisal International Prize 1987) good financial support. Above all, Yau has striven always to foster the very finest mathematical re- search, and to give gifted young Chinese mathe- Calabi-Yau manifolds form a very specific class maticians every chance to develop their talents. of mathematical objects, which play an increas- ingly important role in various branches of math Just one of many examples of Yau’s legacy in this and physics. They are fundamental in algebraic ge- direction, which I enormously appreciate, has been ometry, where they stand somehow in the middle his creation of an outstanding group of number- of the complexity hierarchy of algebraic varieties: theorists in the Chinese Academy of Sciences in not too simple, but still accessible. In differential Beijing. It seems no exaggeration to say that Yau’s geometry they provide the simplest examples of myriad contributions to all aspects of the mathe- the so-called Kähler-Einstein manifolds, a very par- matical world must place him amongst the greatest ticular and very rich structure.
Recommended publications
  • Kähler-Einstein Metrics and Algebraic Geometry

    Kähler-Einstein Metrics and Algebraic Geometry

    Current Developments in Mathematics, 2015 K¨ahler-Einstein metrics and algebraic geometry Simon Donaldson Abstract. This paper is a survey of some recent developments in the area described by the title, and follows the lines of the author’s lecture in the 2015 Harvard Current Developments in Mathematics meeting. The main focus of the paper is on the Yau conjecture relating the ex- istence of K¨ahler-Einstein metrics on Fano manifolds to K-stability. We discuss four different proofs of this, by different authors, which have ap- peared over the past few years. These involve an interesting variety of approaches and draw on techniques from different fields. Contents 1. Introduction 1 2. K-stability 3 3. Riemannian convergence theory and projective embeddings 6 4. Four proofs 11 References 23 1. Introduction General existence questions involving the Ricci curvature of compact K¨ahler manifolds go back at least to work of Calabi in the 1950’s [11], [12]. We begin by recalling some very basic notions in K¨ahler geometry. • All the K¨ahler metrics in a given cohomology class can be described in terms of some fixed reference metric ω0 and a potential function, that is (1) ω = ω0 + i∂∂φ. • A hermitian holomorphic line bundle over a complex manifold has a unique Chern connection compatible with both structures. A Her- −1 n mitian metric on the anticanonical line bundle KX =Λ TX is the same as a volume form on the manifold. When this volume form is derived from a K¨ahler metric the curvature of the Chern connection c 2016 International Press 1 2 S.
  • 978-961-293-071-4.Pdf PUBLIC LECTURES 53

    978-961-293-071-4.Pdf PUBLIC LECTURES 53

    CONTENTS 8th European Congress of Mathematics 20–26 June 2021 • Portorož, Slovenia PLENARY SPEAKERS 1 Presentation of Plenary, Invited, Public, Abel and Prize Speakers at the 8ECM Edited by INVITED SPEAKERS 11 Nino Bašic´ Ademir Hujdurovic´ Klavdija Kutnar THE EMS PRIZES 33 Tomaž Pisanski Vito Vitrih THE FELIX KLEIN PRIZE 43 Published by University of Primorska Press THE OTTO NEUGEBAUER PRIZE 45 Koper, Slovenia • www.hippocampus.si © 2021 University of Primorska ABEL LECTURE 49 Electronic Edition https://www.hippocampus.si/ISBN/978-961-293-071-4.pdf PUBLIC LECTURES 53 https://www.hippocampus.si/ISBN/978-961-293-072-1/index.html https://doi.org/10.26493/978-961-293-071-4 Kataložni zapis o publikaciji (CIP) pripravili v Narodni in univerzitetni knjižnici v Ljubljani COBISS.SI-ID = 65201411 ISBN 978-961-293-071-4 (pdf) ISBN 978-961-293-072-1 (html) PLENARY SPEAKERS 8th European Congress of Mathematics Plenary Speakers Peter Bühlmann Nirenberg, from the Courant Institute, New York University, 1994. Following his PhD, he has held the positions of Member of the Institute for Advanced ETH Zürich Study, Princeton, 1994–95; Habilitation à diriger des recherches, Université Pierre et Marie Curie-Paris VI, 1998; Harrington Faculty Fellow, The University of Texas at Austin, 2001–02; and Tenure Associate Professor, The University Biosketch of Texas at Austin, 2002–03. Since 2003, he has been an ICREA Research Peter Bühlmann is Professor of Mathematics and Professor at the Universitat Politècnica de Catalunya. He received the Kurt Statistics, and Director of Foundations of Data Science at ETH Zürich. He Friedrichs Prize, New York University, 1995, and is a Fellow of the American studied mathematics at ETH Zürich and received his doctoral degree in 1993 Mathematical Society, inaugural class of 2012.
  • 2019 AMS Prize Announcements

    2019 AMS Prize Announcements

    FROM THE AMS SECRETARY 2019 Leroy P. Steele Prizes The 2019 Leroy P. Steele Prizes were presented at the 125th Annual Meeting of the AMS in Baltimore, Maryland, in January 2019. The Steele Prizes were awarded to HARUZO HIDA for Seminal Contribution to Research, to PHILIppE FLAJOLET and ROBERT SEDGEWICK for Mathematical Exposition, and to JEFF CHEEGER for Lifetime Achievement. Haruzo Hida Philippe Flajolet Robert Sedgewick Jeff Cheeger Citation for Seminal Contribution to Research: Hamadera (presently, Sakai West-ward), Japan, he received Haruzo Hida an MA (1977) and Doctor of Science (1980) from Kyoto The 2019 Leroy P. Steele Prize for Seminal Contribution to University. He did not have a thesis advisor. He held po- Research is awarded to Haruzo Hida of the University of sitions at Hokkaido University (Japan) from 1977–1987 California, Los Angeles, for his highly original paper “Ga- up to an associate professorship. He visited the Institute for Advanced Study for two years (1979–1981), though he lois representations into GL2(Zp[[X ]]) attached to ordinary cusp forms,” published in 1986 in Inventiones Mathematicae. did not have a doctoral degree in the first year there, and In this paper, Hida made the fundamental discovery the Institut des Hautes Études Scientifiques and Université that ordinary cusp forms occur in p-adic analytic families. de Paris Sud from 1984–1986. Since 1987, he has held a J.-P. Serre had observed this for Eisenstein series, but there full professorship at UCLA (and was promoted to Distin- the situation is completely explicit. The methods and per- guished Professor in 1998).
  • Math, Physics, and Calabi–Yau Manifolds

    Math, Physics, and Calabi–Yau Manifolds

    Math, Physics, and Calabi–Yau Manifolds Shing-Tung Yau Harvard University October 2011 Introduction I’d like to talk about how mathematics and physics can come together to the benefit of both fields, particularly in the case of Calabi-Yau spaces and string theory. This happens to be the subject of the new book I coauthored, THE SHAPE OF INNER SPACE It also tells some of my own story and a bit of the history of geometry as well. 2 In that spirit, I’m going to back up and talk about my personal introduction to geometry and how I ended up spending much of my career working at the interface between math and physics. Along the way, I hope to give people a sense of how mathematicians think and approach the world. I also want people to realize that mathematics does not have to be a wholly abstract discipline, disconnected from everyday phenomena, but is instead crucial to our understanding of the physical world. 3 There are several major contributions of mathematicians to fundamental physics in 20th century: 1. Poincar´eand Minkowski contribution to special relativity. (The book of Pais on the biography of Einstein explained this clearly.) 2. Contributions of Grossmann and Hilbert to general relativity: Marcel Grossmann (1878-1936) was a classmate with Einstein from 1898 to 1900. he was professor of geometry at ETH, Switzerland at 1907. In 1912, Einstein came to ETH to be professor where they started to work together. Grossmann suggested tensor calculus, as was proposed by Elwin Bruno Christoffel in 1868 (Crelle journal) and developed by Gregorio Ricci-Curbastro and Tullio Levi-Civita (1901).
  • Round Table Talk: Conversation with Nathan Seiberg

    Round Table Talk: Conversation with Nathan Seiberg

    Round Table Talk: Conversation with Nathan Seiberg Nathan Seiberg Professor, the School of Natural Sciences, The Institute for Advanced Study Hirosi Ooguri Kavli IPMU Principal Investigator Yuji Tachikawa Kavli IPMU Professor Ooguri: Over the past few decades, there have been remarkable developments in quantum eld theory and string theory, and you have made signicant contributions to them. There are many ideas and techniques that have been named Hirosi Ooguri Nathan Seiberg Yuji Tachikawa after you, such as the Seiberg duality in 4d N=1 theories, the two of you, the Director, the rest of about supersymmetry. You started Seiberg-Witten solutions to 4d N=2 the faculty and postdocs, and the to work on supersymmetry almost theories, the Seiberg-Witten map administrative staff have gone out immediately or maybe a year after of noncommutative gauge theories, of their way to help me and to make you went to the Institute, is that right? the Seiberg bound in the Liouville the visit successful and productive – Seiberg: Almost immediately. I theory, the Moore-Seiberg equations it is quite amazing. I don’t remember remember studying supersymmetry in conformal eld theory, the Afeck- being treated like this, so I’m very during the 1982/83 Christmas break. Dine-Seiberg superpotential, the thankful and embarrassed. Ooguri: So, you changed the direction Intriligator-Seiberg-Shih metastable Ooguri: Thank you for your kind of your research completely after supersymmetry breaking, and many words. arriving the Institute. I understand more. Each one of them has marked You received your Ph.D. at the that, at the Weizmann, you were important steps in our progress.
  • The Role of Partial Differential Equations in Differential Geometry

    The Role of Partial Differential Equations in Differential Geometry

    Proceedings of the International Congress of Mathematicians Helsinki, 1978 The Role of Partial Differential Equations in Differential Geometry Shing-Tung Yau In the study of geometric objects that arise naturally, the main tools are either groups or equations. In the first case, powerful algebraic methods are available and enable one to solve many deep problems. While algebraic methods are still important in the second case, analytic methods play a dominant role, especially when the defining equations are transcendental. Indeed, even in the situation where the geometric abject is homogeneous or algebraic, analytic methods often lead to important contributions. In this talk, we shall discuss a class of problems in dif­ ferential geometry and the analytic methods that are involved in solving such problems. One of the main purposes of differential geometry is to understand how a surface (or a generalization of it) is curved, either intrinsically or extrinsically. Naturally, the problems that are involved in studying such an object cannot be linear. Since curvature is defined by differentiating certain quantities, the equations that arise are nonlinear differential equations. In studying curved space, one of the most important tools is the space of tangent vectors to the curved space. In the language of partial differential equations, the main tool to study nonlinear equations is the use of the linearized operators. Hence, even when we are facing nonlinear objects, the theory of linear operators is unavoidable. Needless to say, we are then left with the difficult problem of how precisely a linear operator approximates a non­ linear operator. To illustrate the situation, we mention five important differential operators in differential geometry.
  • UC Santa Barbara UC Santa Barbara Electronic Theses and Dissertations

    UC Santa Barbara UC Santa Barbara Electronic Theses and Dissertations

    UC Santa Barbara UC Santa Barbara Electronic Theses and Dissertations Title Aspects of Emergent Geometry, Strings, and Branes in Gauge / Gravity Duality Permalink https://escholarship.org/uc/item/8qh706tk Author Dzienkowski, Eric Michael Publication Date 2015 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California University of California Santa Barbara Aspects of Emergent Geometry, Strings, and Branes in Gauge / Gravity Duality A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics by Eric Michael Dzienkowski Committee in charge: Professor David Berenstein, Chair Professor Joe Polchinski Professor David Stuart September 2015 The Dissertation of Eric Michael Dzienkowski is approved. Professor Joe Polchinski Professor David Stuart Professor David Berenstein, Committee Chair July 2015 Aspects of Emergent Geometry, Strings, and Branes in Gauge / Gravity Duality Copyright c 2015 by Eric Michael Dzienkowski iii To my family, who endured my absense for the better part of nine long years while I attempted to understand the universe. iv Acknowledgements There are many people and entities deserving thanks for helping me complete my dissertation. To my advisor, David Berenstein, for the guidance, advice, and support over the years. With any luck, I have absorbed some of your unique insight and intuition to solving problems, some which I hope to apply to my future as a physicist or otherwise. A special thanks to my collaborators Curtis Asplund and Robin Lashof-Regas. Curtis, it has been and will continue to be a pleasure working with you. Ad- ditional thanks for various comments and discussions along the way to Yuhma Asano, Thomas Banks, Frederik Denef, Jim Hartle, Sean Hartnoll, Matthew Hastings, Gary Horowitz, Christian Maes, Juan Maldacena, John Mangual, Don Marolf, Greg Moore, Niels Obers, Joe Polchisnki, Jorge Santos, Edward Shuryak, Christoph Sieg, Eva Silverstein, Mark Srednicki, and Matthias Staudacher.
  • 1. GIT and Μ-GIT

    1. GIT and Μ-GIT

    1. GIT and µ-GIT Valentina Georgoulas Joel W. Robbin Dietmar A. Salamon ETH Z¨urich UW Madison ETH Z¨urich Dietmar did the heavy lifting; Valentina and I made him explain it to us. • A key idea comes from Xiuxiong Chen and Song Sun; they were doing an • analogous infinite dimensional problem. I learned a lot from Sean. • The 1994 edition of Mumford’s book lists 926 items in the bibliography; I • have read fewer than 900 of them. Dietmar will talk in the Geometric Analysis Seminar next Monday. • Follow the talk on your cell phone. • Calc II example of GIT: conics, eccentricity, major axis. • Many important problems in geometry can be reduced to a partial differential equation of the form µ(x)=0, where x ranges over a complexified group orbit in an infinite dimensional sym- plectic manifold X and µ : X g is an associated moment map. Here we study the finite dimensional version.→ Because we want to gain intuition for the infinite dimensional problems, our treatment avoids the structure theory of compact groups. We also generalize from projective manifolds (GIT) to K¨ahler manifolds (µ-GIT). In GIT you start with (X, J, G) and try to find Y with R(Y ) R(X)G. • ≃ In µ-GIT you start with (X,ω,G) and try to solve µ(x) = 0. • GIT = µ-GIT + rationality. • The idea is to find analogs of the GIT definitions for K¨ahler manifolds, show that the µ-GIT definitions and the GIT definitions agree for projective manifolds, and prove the analogs of the GIT theorems in the K¨ahler case.
  • Council Congratulates Exxon Education Foundation

    Council Congratulates Exxon Education Foundation

    from.qxp 4/27/98 3:17 PM Page 1315 From the AMS ics. The Exxon Education Foundation funds programs in mathematics education, elementary and secondary school improvement, undergraduate general education, and un- dergraduate developmental education. —Timothy Goggins, AMS Development Officer AMS Task Force Receives Two Grants The AMS recently received two new grants in support of its Task Force on Excellence in Mathematical Scholarship. The Task Force is carrying out a program of focus groups, site visits, and information gathering aimed at developing (left to right) Edward Ahnert, president of the Exxon ways for mathematical sciences departments in doctoral Education Foundation, AMS President Cathleen institutions to work more effectively. With an initial grant Morawetz, and Robert Witte, senior program officer for of $50,000 from the Exxon Education Foundation, the Task Exxon. Force began its work by organizing a number of focus groups. The AMS has now received a second grant of Council Congratulates Exxon $50,000 from the Exxon Education Foundation, as well as a grant of $165,000 from the National Science Foundation. Education Foundation For further information about the work of the Task Force, see “Building Excellence in Doctoral Mathematics De- At the Summer Mathfest in Burlington in August, the AMS partments”, Notices, November/December 1995, pages Council passed a resolution congratulating the Exxon Ed- 1170–1171. ucation Foundation on its fortieth anniversary. AMS Pres- ident Cathleen Morawetz presented the resolution during —Timothy Goggins, AMS Development Officer the awards banquet to Edward Ahnert, president of the Exxon Education Foundation, and to Robert Witte, senior program officer with Exxon.
  • About the Calabi-Yau Theorem and Its Applications

    About the Calabi-Yau Theorem and Its Applications

    About the Calabi-Yau theorem and its applications { Krageomp program { Julien Keller L.A.T.P., Marseille University { Universit´ede Provence October 21, 2009 1 Any comments, remarks and suggestions are welcome. These notes are summing up the series of lectures given by the author during the Krageomp program (2009). Please feel free to contact the author at [email protected] 2 1 About complex and K¨ahlermanifolds In this lecture, starting with the background of Riemannian geometry, we introduce the notion of complex manifolds: • Definition using charts and holomorphic transition functions • Almost complex structure, complex structure and Newlander-Nirenberg theorem • Examples: the projective space , S2, hypersurfaces, weighted projective spaces • Uniformization theorem for Riemann surfaces, Chow's theorem (com- n plex compact analytic submanifolds of CP are actually algebraic vari- eties). We introduce the language of tensors on complex manifolds, and some natural invariants of the complex structure: • k-forms, exterior diffentiation operator, d = @ +@¯ decomposition, (p; q) forms • De Rham cohomology, Dolbeault cohomology, Hodge numbers, Euler characteristic invariant Using all the previous material, we can explain what is a K¨ahlermanifold. In particular we will formulate the Calabi conjecture on that space. • K¨ahlerforms, K¨ahlerclasses, K¨ahlercone n • The Fubini-Study metric on CP is a K¨ahlermetric Finally, we sketched the notion of holomorphic line bundle, the correspon- dence with divisor and Chern classes. This very classical material can be read in different books: - F. Zheng, Complex Differential Geometry, Studies in Adv. Maths, AMS (2000) 3 - W. Ballmann, Lectures on K¨ahlermanifolds, Lectures in Math and Physics, EMS (2006).
  • Public Recognition and Media Coverage of Mathematical Achievements

    Public Recognition and Media Coverage of Mathematical Achievements

    Journal of Humanistic Mathematics Volume 9 | Issue 2 July 2019 Public Recognition and Media Coverage of Mathematical Achievements Juan Matías Sepulcre University of Alicante Follow this and additional works at: https://scholarship.claremont.edu/jhm Part of the Arts and Humanities Commons, and the Mathematics Commons Recommended Citation Sepulcre, J. "Public Recognition and Media Coverage of Mathematical Achievements," Journal of Humanistic Mathematics, Volume 9 Issue 2 (July 2019), pages 93-129. DOI: 10.5642/ jhummath.201902.08 . Available at: https://scholarship.claremont.edu/jhm/vol9/iss2/8 ©2019 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. Public Recognition and Media Coverage of Mathematical Achievements Juan Matías Sepulcre Department of Mathematics, University of Alicante, Alicante, SPAIN [email protected] Synopsis This report aims to convince readers that there are clear indications that society is increasingly taking a greater interest in science and particularly in mathemat- ics, and thus society in general has come to recognise, through different awards, privileges, and distinctions, the work of many mathematicians.
  • Jeff Cheeger

    Jeff Cheeger

    Progress in Mathematics Volume 297 Series Editors Hyman Bass Joseph Oesterlé Yuri Tschinkel Alan Weinstein Xianzhe Dai • Xiaochun Rong Editors Metric and Differential Geometry The Jeff Cheeger Anniversary Volume Editors Xianzhe Dai Xiaochun Rong Department of Mathematics Department of Mathematics University of California Rutgers University Santa Barbara, New Jersey Piscataway, New Jersey USA USA ISBN 978-3-0348-0256-7 ISBN 978-3-0348-0257-4 (eBook) DOI 10.1007/978-3-0348-0257-4 Springer Basel Heidelberg New York Dordrecht London Library of Congress Control Number: 2012939848 © Springer Basel 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc.