RESEARCH STATEMENT 1. Introduction My Research Focuses
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FLOER HOMOLOGY and MIRROR SYMMETRY I Kenji FUKAYA 0
FLOER HOMOLOGY AND MIRROR SYMMETRY I Kenji FUKAYA Abstract. In this survey article, we explain how the Floer homology of Lagrangian submanifold [Fl1],[Oh1] is related to (homological) mirror symmetry [Ko1],[Ko2]. Our discussion is based mainly on [FKO3]. 0. Introduction. This is the first of the two articles, describing a project in progress to study mirror symmetry and D-brane using Floer homology of Lagrangian submanifold. The tentative goal, which we are far away to achiev, is to prove homological mirror symmetry conjecture by M. Kontsevich (see 3.) The final goal, which is yet very § very far away from us, is to find a new concept of spaces, which is expected in various branches of mathematics and in theoretical physics. Together with several joint authors, I wrote several papers on this project [Fu1], [Fu2], [Fu4], [Fu5], [Fu6], [Fu7], [FKO3], [FOh]. The purpose of this article and part II, is to provide an accesible way to see the present stage of our project. The interested readers may find the detail and rigorous proofs of some of the statements, in those papers. The main purose of Part I is to discribe an outline of our joint paper [FKO3] which is devoted to the obstruction theory to the well-definedness of Floer ho- mology of Lagrangian submanifold. Our emphasis in this article is its relation to mirror symmetry. So we skip most of its application to the geometry of Lagrangian submanifolds. In 1, we review Floer homology of Lagrangian submanifold in the form intro- § duced by Floer and Oh. They assumed various conditions on Lagrangian subman- ifold and symplectic manifold, to define Floer homology. -
Quantum Cohomology of Lagrangian and Orthogonal Grassmannians
QUANTUM COHOMOLOGY OF LAGRANGIAN AND ORTHOGONAL GRASSMANNIANS HARRY TAMVAKIS Abstract. This is the written version of my talk at the Mathematische Ar- beitstagung 2001, which reported on joint work with Andrew Kresch (papers [KT2] and [KT3]). The Classical Theory For the purposes of this talk we will work over the complex numbers, and in the holomorphic (or algebraic) category. There are three kinds of Grassmannians that we will consider, depending on their Lie type: • In type A,letG(k; N)=SLN =Pk be the Grassmannian which parametrizes N k-dimensional linear subspaces of C . • In type C,letLG = LG(n; 2n)=Sp2n=Pn be the Lagrangian Grassmannian. • In types B and D,letOG = OG(n; 2n +1)=SO2n+1=Pn denote the odd or- thogonal Grassmannian, which is isomorphic to the even orthogonal Grassmannian OG(n +1; 2n +2)=SO2n+2=Pn+1. We begin by defining the Lagrangian Grassmannian LG(n; 2n), and will discuss the Grassmannians for the other types later in the talk. Let V be the vector 2n V space C , equipped with a symplectic form. A subspace Σ of is isotropic if the restriction of the form to Σ vanishes. The maximal possible dimension of an isotropic subspace is n, and in this case Σ is called a Lagrangian subspace. The Lagrangian Grassmannian LG = LG(n; 2n) is the projective complex manifold which parametrizes Lagrangian subspaces Σ in V . There is a stratification of LG by Schubert varieties Xλ, one for each strict n ` ` λ partition λ =(λ1 >λ2 > ··· >λ` > 0) with λ1 6 . The number = ( )isthe n length of λ, and we let Dn denote the set of strict partitions λ with λ1 6 .To describe Xλ more precisely, fix an isotropic flag of subspaces Fi of V : 0 ⊂ F1 ⊂ F2 ⊂···⊂Fn ⊂ V with dim(Fi)=i for each i,andFn Lagrangian. -
Floer Homology of Lagrangian Submanifolds Are Ex- Pected to Enjoy Can Be Summarized As Follows
FLOER HOMOLOGY OF LAGRANGIAN SUBMANIFOLDS KENJI FUKAYA Abstract. This paper is a survey of Floer theory, which the author has been studying jointly with Y.-G. Oh, H. Ohta, K. Ono. A general idea of the construction is outlined. We also discuss its relation to (homological) mirror symmetry. Especially we describe various conjectures on (homological) mirror symmetry and various partial results towards those conjectures. 1. Introduction This article is a survey of Lagrangian Floer theory, which the author has been studying jointly with Y.-G. Oh, H. Ohta, K. Ono. A major part of our study was published as [28]. Floer homology is invented by A. Floer in 1980’s. There are two areas where Floer homology appears. One is symplectic geometry and the other is topology of 3-4 dimensional manifolds (more specifically the gauge theory). In each of the two areas, there are several different Floer type theories. In symplectic geometry, there are Floer homology of periodic Hamiltonian system ([18]) and Floer homology of Lagrangian submanifolds. Moreover there are two different kinds of Floer theories of contact manifolds ([15, 78]). In gauge theory, there are three kinds of Floer homologies: one based on Yang-Mills theory ([17]), one based on Seiberg-Witten theory ([55, 51]), and Heegard Floer homology ([63]). Those three are closely related to each other. All the Floer type theories have common feature that they define some kinds of homology theory of /2 degree in dimensional space, based on Morse theory. There are many interesting∞ topics∞ to discuss on the general feature of Floer type theories. -
Arxiv:Math/0701246V1 [Math.AG] 9 Jan 2007 Suoooopi Curve Pseudoholomorphic Ihvrie (0 Vertices with a Curves Obtain Singular) to Points, Singular Hccurves
A NON-ALGEBRAIC PATCHWORK BenoˆıtBertrand Erwan Brugall´e Abstract Itenberg and Shustin’s pseudoholomorphic curve patchworking is in principle more flexible than Viro’s original algebraic one. It was natural to wonder if the former method allows one to construct non-algebraic objects. In this paper we construct the first examples of patchworked real pseudoholomorphic curves in Σn whose position with respect to the pencil of lines cannot be realised by any homologous real algebraic curve. 1 Introduction Viro’s patchworking has been since the seventies one of the most important and fruitful methods in topology of real algebraic varieties. It was applied in the proof of a lot of meaningful results in this field (e.g. [Vir84], [Vir89], [Shu99], [LdM],[Ite93], [Ite01], [Haa95],[Bru06], [Bih], [IV], [Ber06], [Mik05], and [Shu05]). Here we will only consider the case of curves in CP 2 and in rational geomet- rically ruled surfaces Σn. Viro Method allows to construct an algebraic curve A out of simpler curves Ai so that the topology of A can be deduced from the topology of the initial curves Ai. Namely one gets a curve with Newton polytope ∆ out of curves whose Newton polytopes are the 2-simplices of a subdivision σ of ∆, and one can see the curve A as a gluing of the curves Ai. Moreover, if all the curves Ai are real, so is the curve A. For a detailed account on the Viro Method we refer to [Vir84] and [Vir89] for example. One of the hypotheses of the Viro Method is that σ should be convex (i.e. -
Lecture 1: Basic Concepts, Problems, and Examples
LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES WEIMIN CHEN, UMASS, SPRING 07 In this lecture we give a general introduction to the basic concepts and some of the fundamental problems in symplectic geometry/topology, where along the way various examples are also given for the purpose of illustration. We will often give statements without proofs, which means that their proof is either beyond the scope of this course or will be treated more systematically in later lectures. 1. Symplectic Manifolds Throughout we will assume that M is a C1-smooth manifold without boundary (unless specific mention is made to the contrary). Very often, M will also be closed (i.e., compact). Definition 1.1. A symplectic structure on a smooth manifold M is a 2-form ! 2 Ω2(M), which is (1) nondegenerate, and (2) closed (i.e. d! = 0). (Recall that a 2-form ! 2 Ω2(M) is said to be nondegenerate if for every point p 2 M, !(u; v) = 0 for all u 2 TpM implies v 2 TpM equals 0.) The pair (M; !) is called a symplectic manifold. Before we discuss examples of symplectic manifolds, we shall first derive some im- mediate consequences of a symplectic structure. (1) The nondegeneracy condition on ! is equivalent to the condition that M has an even dimension 2n and the top wedge product !n ≡ ! ^ ! · · · ^ ! is nowhere vanishing on M, i.e., !n is a volume form. In particular, M must be orientable, and is canonically oriented by !n. The nondegeneracy condition is also equivalent to the condition that M is almost complex, i.e., there exists an endomorphism J of TM such that J 2 = −Id. -
Gromov--Witten Invariants and Quantum Cohomology
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 116, No. 4, November 2003, pp. 459–476. Printed in India Gromov–Witten invariants and quantum cohomology AMIYA MUKHERJEE Stat-Math Division, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, India E-mail: [email protected] Dedicated to Professor K B Sinha on the occasion of his 60th birthday Abstract. This article is an elaboration of a talk given at an international confer- ence on Operator Theory, Quantum Probability, and Noncommutative Geometry held during December 20–23, 2004, at the Indian Statistical Institute, Kolkata. The lecture was meant for a general audience, and also prospective research students, the idea of the quantum cohomology based on the Gromov–Witten invariants. Of course there are many important aspects that are not discussed here. Keywords. J-holomorphic curves; moduli spaces; Gromov–Witten classes; quantum cohomology. 1. Introduction Quantum cohomology is a new mathematical discipline influenced by the string theory as a joint venture of physicists and mathematicians. The notion was first proposed by Vafa [V], and developed by Witten [W] and others [B]. The theory consists of some new approaches to the problem of constructing invariants of compact symplectic manifolds and algebraic varieties. The approaches are related to the ideas of a (1 + 1)-dimensional topological quantum field theory, which indicate that the general principle of constructing invariants should be as follows: The invariants of a manifold M should be obtained by integrating cohomology classes over certain moduli space M associated to M. In our case the manifold is a symplectic manifold (M,ω), and the moduli space M is the space of 1 certain J-holomorphic spheres σ: CP M in a given homology class A H2(M;Z). -
Duality and Integrability in Topological String Theory
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by SISSA Digital Library International School of Advanced Studies Mathematical Physics Sector Academic Year 2008–09 PhD Thesis in Mathematical Physics Duality and integrability in topological string theory Candidate Andrea Brini Supervisors External referees Prof. Boris Dubrovin Prof. Marcos Marino˜ Dr. Alessandro Tanzini Dr. Tom Coates 2 Acknowledgements Over the past few years I have benefited from the interaction with many people. I would like to thank first of all my supervisors: Alessandro Tanzini, for introducing me to the subject of this thesis, for his support, and for the fun we had during countless discussions, and Boris Dubrovin, from whose insightful explanations I have learned a lot. It is moreover a privilege to have Marcos Mari˜no and Tom Coates as referees of this thesis, and I would like to thank them for their words and ac- tive support in various occasions. Special thanks are also due to Sara Pasquetti, for her help, suggestions and inexhaustible stress-energy transfer, to Giulio Bonelli for many stimulating and inspiring discussions, and to my collaborators Renzo Cava- lieri, Luca Griguolo and Domenico Seminara. I would finally like to acknowledge the many people I had discussions/correspondence with: Murad Alim, Francesco Benini, Marco Bertola, Vincent Bouchard ;-), Ugo Bruzzo, Mattia Cafasso, Hua-Lian Chang, Michele Cirafici, Tom Claeys, Andrea Collinucci, Emanuel Diaconescu, Bertrand Ey- nard, Jarah Evslin, Fabio Ferrari Ruffino, Barbara -
Relative Floer and Quantum Cohomology and the Symplectic Topology of Lagrangian Submanifolds
Relative Floer and quantum cohomology and the symplectic topology of Lagrangian submanifolds Yong-Geun Oh Department of Mathematics University of Wisconsin Madison, WI 53706 Dedicated to the memory of Andreas Floer Abstract: Floer (co)homology of the symplectic manifold which was originally introduced by Floer in relation to the Arnol’d conjecture has recently attracted much attention from both mathematicians and mathematical physicists either from the algebraic point of view or in relation to the quantum cohomology and the mirror symmetry. Very recent progress in its relative version, the Floer (co)homology of Lagrangian submanifolds, has revealed quite different mathematical perspective: The Floer (co)homology theory of Lagrangian submanifolds is a powerful tool in the study of symplectic topology of Lagrangian submanifolds, just as the classical (co)homology theory in topology has been so in the study of differential topology of submanifolds on differentiable manifolds. In this survey, we will review the Floer theory of Lagrangian submanifolds and explain the recent progress made by Chekanov and by the present author in this relative Floer theory, which have found several applications of the Floer theory to the symplectic topology of Lagrangian submanifolds which demonstrates the above perspective: They include a Floer theoretic proof of Gromov’s non-exactness theorem, an optimal lower bound for the symplectic disjunction energy, construction of new symplectic invariants, proofs of the non-degeneracy of Hofer’s distance on the space of Lagrangian submanifolds in the cotangent bundles and new results on the Maslov class obstruction to the Lagrangian embedding in Cn. Each of these applications accompanies new development in the Floer theory itself: localizations, semi-infinite cycles and minimax theory in the Floer theory and a spectral sequence as quantum cor- rections and others. -
1. Eigenvalues of Hermitian Matrices and Schubert Calculus
AROUND THE HORN CONJECTURE L MANIVEL Abstract We discuss the problem of determining the p ossible sp ectra of a sum of Hermitian matrices each with known sp ectrum We explain how the Horn conjecture which gives a complete answer to this question is related with algebraic geometry symplectic geometry and representation theory The rst lecture is an introduction to Schubert calculus from which one direction of Horns conjecture can b e deduced The reverse direction follows from an application of geometric invariant theory this is treated in the second lecture Finally we explain in the third lecture how a version of Horns problem for sp ecial unitary matrices is related to the quantum cohomology of Grassmannians Notes for Gael VI I I CIRM March Eigenvalues of hermitian matrices and Schubert calculus The problem Let A B C b e complex Hermitian n by n matrices Denote the set of eigenvalues or sp ectrum of A by A A A and similarly by B and C the sp ectra n of B and C The main theme of these notes is the following question Suppose that A B C What can be their spectra A B C There are obvious relations like trace C trace A trace B or C A B But a complete answer to this longstanding question was given only recently and combines works and ideas from representation theory symplectic and algebraic geometry Weyls inequalities There are various characterizations of the eigenvalues of Hermitian ma trices many of which are variants of the minimax principle Let j b e the standard Hermitian n pro duct on C If s n r -
Computations with Pseudoholomorphic Curves
COMPUTATIONS WITH PSEUDOHOLOMORPHIC CURVES J. EVANS 1. Introduction Symplectic topology is renowned for its invariants which are either incomputable or only conjecturally well-defined. For algebraic geometers this is something of an anaethema. The aim of this talk is to give a sketch of why we can prove anything in symplectic topology, using friendly words like \Dolbeaut cohomology" and \positivity of intersections". By the end of the talk we should be able to prove (amongst other things) that there is a unique line through two points in the (complex projective) plane. Everything I will say is a) due to Gromov [Gro85] and b) explained more thoroughly and clearly in the big book by McDuff and Salamon [MS04]. 2. Some linear algebra Recall that a symplectic vector space is a 2n-dimensional vector space with a non-degenerate alternating form !. The group Spn(R) of symplectic matrices (which preserve this form) is a non-compact Lie group containing the unitary group U(n) as a maximal compact subgroup. In fact, there is a homotopy equivalence 2n ∼ n Spn(R) ! U(n). Identifying the vector space with R = C , this unitary group is the subgroup of symplectic matrices which additionally commute with the complex structure (multiplication by i). The point is that together ! and i define a metric by g(v; w) = !(v; iw). The unitary group can then be thought of as the complex (or symplectic) isometries of this metric. This is known as the 3-in-2 property of the unitary group: Spn(R) \ O(2n) = Spn(R) \ GLn(C) = GLn(C) \ O(2n) = U(n) If one considers the set J of all possible matrices J such that J 2 = −1 and such that !(v; Jw) defines a (J-invariant) metric, it turns out to be a homogeneous space J = Spn(R)=U(n) This is quite easy to see: if one picks a vector v, then v and Jv span a symplectic subspace, and passing to its symplectic orthogonal complement one obtains a lower- dimensional space and (by induction) symplectic coordinates on the whole space for which J and ! look like the standard complex and symplectic structures. -
Research Statement for Daniel Pomerleano 1. Background And
Research statement for Daniel Pomerleano 1. Background and motivating questions My research centers around algebraic and geometric structures arising in mirror symmetry. Mirror symmetry originates in string theory and predicts astonishing and intricate relationships between seemingly disparate notions such as coherent sheaves, which originate in algebraic geometry and Lagrangian submanifolds, which are native to symplectic geometry. More precisely, let X be a smooth projective (or affine) variety. The variety X inherits a natural symplectic structure by n n restriction of the usual symplectic form on P (or C ). Kontsevich's Homological Mirror Symmetry conjecture applied to X proposes that there exists a mirror to X,(X;ˇ w), with Xˇ a smooth variety 1 and w : Xˇ ! A a regular function, such that symplectic invariants on X are exchanged with algebraic invariants on (X;ˇ w). The richest version of this conjecture asserts that we have an equivalence of categories: F uk(X) =∼ MF (X;ˇ w) Here F uk(X) denotes a suitable Fukaya category and MF (X;ˇ w) denotes a category of matrix factorizations of w. This conjecture serves as a powerful guiding light for taking structures which are transparent on one side of the mirror and revealing new, unexpected structures on the mirror side. This has lead to the development of tools in both fields that would otherwise have been difficult to imagine (for notable examples see [Bri, GroHacKeeKon, SeiTho]). My two main research projects to date aim to exploit mirror symmetry to develop new structures in both algebraic and symplectic geometry. The first concerns the symplectic cohomology ring, SH∗(X), of affine varieties X. -
Applications and Combinatorics in Algebraic Geometry Frank Sottile Summary
Applications and Combinatorics in Algebraic Geometry Frank Sottile Summary Algebraic Geometry is a deep and well-established field within pure mathematics that is increasingly finding applications outside of mathematics. These applications in turn are the source of new questions and challenges for the subject. Many applications flow from and contribute to the more combinatorial and computational parts of algebraic geometry, and this often involves real-number or positivity questions. The scientific development of this area devoted to applications of algebraic geometry is facilitated by the sociological development of administrative structures and meetings, and by the development of human resources through the training and education of younger researchers. One goal of this project is to deepen the dialog between algebraic geometry and its appli- cations. This will be accomplished by supporting the research of Sottile in applications of algebraic geometry and in its application-friendly areas of combinatorial and computational algebraic geometry. It will be accomplished in a completely different way by supporting Sottile’s activities as an officer within SIAM and as an organizer of scientific meetings. Yet a third way to accomplish this goal will be through Sottile’s training and mentoring of graduate students, postdocs, and junior collaborators. The intellectual merits of this project include the development of applications of al- gebraic geometry and of combinatorial and combinatorial aspects of algebraic geometry. Specifically, Sottile will work to develop the theory and properties of orbitopes from the perspective of convex algebraic geometry, continue to investigate linear precision in geo- metric modeling, and apply the quantum Schubert calculus to linear systems theory.