Publication list helmut hofer ias school of mathematics. Compactness results in symplectic field theory internet archive. The only known case in which the a priori estimates can be obtained is the case in which the almost complex structure is calibrated or tamed by a symplectic form. Let s sg be a compact riemann surface of genus g with a conformal. Symplectic field theory sketched in y eliashberg, a givental and h. Moreover, all curves near u are embedded, and t together with v in a foliation. The author presents a systematic introduction providing a. Compactness results in symplectic field theory core. An introduction to compactness results in symplectic field theory. Symplectic field theory informal graduate course at university college london 20152016 this is the archived homepage for a course that is now finished, but you may be interested in the lecture notes, which have now been assembled into a single file and uploaded to the arxiv, and will be appearing in book form in the ems lectures in mathematics series. For a threemanifold with parametrized boundary, the invariant comes in two different versions, one of which type d is a module over the algebra and the other of which type a is an ainfinity module. You are probably aware that the intersection theory of holomorphic curves plays an important role in 4dimensional symplectic topology, and you may also be aware that an extension of this theory for punctured holomorphic curves in the setting of symplectic field theory exists, and has interesting applications for contact 3manifolds e. Pdf compactness results in symplectic field theory semantic. In section 5 we study compactness and smoothness of the moduli space of closed curves, and explain a few application, including theorem a and theorem b.
This is one in a series of papers devoted to the foundations of. Holomorphic curves, planar open books and symplectic fillings a minicourse by chris wendl the overarching theme of this minicourse will be the properties of pseudoholomorphic curves and their use in proving global results about symplectic or contact manifolds based on more localized information. Finite energy cylinders of small area ergodic theory and. Introduction to compactness results in symplectic field theory the book grew out of lectures given by the author in 2005. Rutgers geometry, physics and mirror symmetry seminar.
A compactness result for the quilted atiyahfloer conjecture. As a general theory of symplectic invariants, sft has been outlined in introduction to symplectic field theory 2000, by y. In these lectures, m will denote a compact, orientable manifold of dimension 2n. In applications of gromovs theorem it is crucial to get an a priori bound on the area. Symplectic field theory and its applications international congress. It is a part of the symplectic field theory 6 of eliashberg, givental and hofer. Numerical simulation using eulerianeulerian method with kinetic theory of granular flow was adopted to analyze gassolid flow consisting in.
This is one in a series of papers devoted to the foundations of symplectic field theory sketched in y eliashberg, a givental and h hofer, introduction to symplectic field theory, geom. Introduction to compactness stochastic models in reliability. This is one in a series of papers devoted to the foundations of symplectic field theory sketched in y eliashberg, a givental and h hofer, introduction to. Compactness results in symplectic field theory with f. Effect of the crosssectional geometry on heat transfer and particle motion of circulating fluidized bed riser for co 2 capture was investigated. Kai cieliebak and yakov eliashberg, from stein to weinstein and back, american mathematical society colloquium publications, vol. Yangmills theory and geometry imperial college london. Introduction symplectic and contact topology is an active area of mathematics that combines ideas from dynamical systems, analysis, topology, several complex variables, and differential and algebraic geometry. Delignemumfordtype spaces with a view towards symplectic. Some knowledge about the properties of the delignemumford compacti.
A new proof for a result of ekeland and lasry concerning the number. Symplectic field theory 1 is a rather general framework for computing invariants of symplectic cobordisms. This is one in a series of papers devoted to the foundations of symplectic field theory sketched in y. Holomorphic foliations and lowdimensional symplectic field. Yakov eliashberg, crafoord prize laureate in mathematics 2016 at the royal swedish academy of sciences in stockholm sweden may 2016. Compactness results for the atiyahfloer conjecture for broken circle. Generalized steenrod homology theories are strong shape. The actual construction of sft is a hard analytical problem which will be overcome be means of the polyfold theory due to the present authors. Hyung hee cho, young mun lee, seungyeong choi, namkyu lee, dong il shim, yongki park abstract.
Based on the symplectic geometry method, the theory of hamilton system can be applied in the analysis of problem solved using the theory of elasticity and in the solution of elliptic partial differential equations. An introduction to compactness results in symplectic field theory casim abbas auth. The starting point of this theory are compactness results for holomorphic curves established in the last decade. Richard siefring, courant institute 2005, intersection theory of finite energy surfaces sam lisi, courant institute 2005, applications of symplectic geometry to hamiltonian mechanics joel fish, courant institute 2007, compactness results for pseudoholomorphic curves sunil pinnamaneni, courant institute december 2007, a com. Then u is immersed, and v is embedded with index 0. Holomorphic curves and real threedimensional dynamics, gafa 2000, special volume, part ii, pp674704 69. Symplectic field theory is a general framework for defining invariants of. Contact homology of good toric contact manifolds cambridge core. American mathematical society colloquium publications, 59. Symplectic and contact structures first arose in the study of classical mechanical systems, allowing one to describe the time evolution of both simple and complex systems such as. Twisted ktheory, khomology, and bivariant chernconnes type character of some infinite dimensional spaces mahanta, snigdhayan, kyoto journal of mathematics, 2014. In fact, all compactness results for holomorphic curves without boundary known to us, including the. Lectures on symplectic field theory chris wendl institut fur. Symplectic field theory sft attempts to approach the theory of holomorphic curves.
Holomorphic foliations and lowdimensional symplectic. Symplectic field theory sketched in y eliashberg, a givental and h hofer, introduction to symplectic field theory, geom. Zehnder, compactness results in symplectic field theory. With the advent of electromagnetic theory, most notably with the work of fara. We prove compactness results for moduli spaces of holomorphic curves arising in symplectic field theory. This ebook involves elements, diversified in shape yet comparable in spirit. Contact homology of good toric contact manifolds volume 148 issue 1 miguel abreu, leonardo macarini. String, dilaton, and divisor equation in symplectic field theory. Floer field theory is a construction principle for e. Pdf compactness results in symplectic field theory. Electromagnetic field theory is often the least popular course in the electrical engineering curriculum. W are good index 2 curves and converge to a multiple cover u v. Field theory 3 about these notes the purpose of these notes is to give a treatment of the theory of elds.
Chris wendl is known among symplectic topologists for contributions to the study of symplectic fillability of contact manifolds, and for transversality results in the theory of pseudoholomorphic curves. This is one in a series of papers devoted to the foundations of symplectic field theory sketched in 4. The compactness results for holomorphic curves proved in this paper cover a variety of applications, from the original gromov compactness theorem for holomorphic curves 8, to floer homology theory 6, 7, and to symplectic field theory 4. Compactness results in symplectic field theory 801 by the bound on another quantity, called energy. The symplectic structure on moduli spaces of flat connections was a vital feature of wittens work on 3manifold and knot invariants 49. Heavy reli ance on vector and integral calculus can obscure physical. Compactness results in symplectic field theory arxiv. Existence and multiplicity result for a class of second order elliptic equations. The theorems generalize gromovs compactness theorem in gr as well as compactness theorems in floer homology theory, f1, f2, and in contact geometry, h, hwz8. An introduction to compactness results in symplectic field. Symplectic field theory is a new important subject which is currently being developed. Field theoretic notions appeared, even though still implicitly, in the modern theory of solvability of polynomial equations, as introduced by abel and galois in the early nineteenth century. The starting point of this theory are compactness results for holomorphic curves established in 2004. Indeed, this result implies that the 0dimensional moduli spaces consist of a.
For gromovs compactness theorem in riemannian geometry, see that article in the mathematical field of symplectic topology, gromovs compactness theorem states that a sequence of pseudoholomorphic curves in an almost complex manifold with a uniform energy bound must have a subsequence which limits to a pseudoholomorphic curve which may have nodes or a finite tree of bubbles. Besides providing a unified view on established pseudoholomorphic curve theories such as symplectic floer homology, contact homology, and gromovwitten theory, it leads to numerous new. Symplectic field theory, lecture course at ucl, 201516. From dynamics to contact and symplectic topology and back. He is currently professor of differential geometry and global analysis at the humboldt university in berlin, and is also the author of two. The starting point of this theory are compactness results for holomorphic. Holomorphic curves, planar open books and symplectic fillings. Aug 19, 2003 symplectic field theory sketched in y eliashberg, a givental and h hofer, introduction to symplectic field theory, geom. Holomorphic curves in low dimensions from symplectic. The equality d 0 trivially implies r d 0, and hence the corresponding j is symmetric. We give examples of explicit mmp runnings and descriptions of floer nontrivial tori in the case of toric manifolds, polygon spaces, and moduli spaces of flat bundles on punctured twospheres moduli of. Compactness results in symplectic field theory nasaads.
String, dilaton, and divisor equation in symplectic field. Nevertheless, for the convenience of the reader we have summarized these results in appendix 5. Pseudoholomorphic curves and dynamics in three dimensions with k. Contact homology of good toric contact manifolds volume 148 issue 1 miguel abreu, leonardo macarini skip to main content we use cookies to distinguish you from other users and to provide you with a better experience on our websites. We prove compactness results for moduli spaces of holomorphic. The theorems generalize gromovs compactness theorem in m. Is there an symplectic field theory compactness theorem applicable in the context of floer cohomology of a symplectomorphism. American mathematical society, providence, ri, 2012. This naturally leads to new algebraic structures which seems to have. Electromagnetic field theory a problemsolving approach. Representations of the fundamental group and floer theory.
Givental, gafa 2000, special volume, part ii, pp560673 68. Dusa mcduff and katrin wehrheim, smooth kuranishi atlases with isotropy, geom. In section 6 we study curves with boundary and their moduli spaces, and prove theorem c. Symplectic field theorist because math is hard, so we. Orienting moduli spaces of flow trees for symplectic field theory. Compactness results in symplectic field theory 803 example 2. Tropical graphs arise by studying the behavior of pseudoholomorphic disks under a multiple cut operation on a symplectic manifold that produces a collection of cut spaces each containing relative normal crossing divisors, following works of. The theory associates to an oriented, parametrized twomanifold a differential graded algebra. Hofer, introduction to symplectic field theory, part ii. Besides providing a unified view on established pseudoholomorphic curve theories such as symplectic floer homology, contact homology, and gromovwitten theory. We prove that small blowups or reverse flips in the sense of the minimal model program of rational symplectic manifolds with trivial centers create floernontrivial lagrangian tori. Finite energy cylinders of small area volume 22 issue 5 h.
407 1230 1183 58 308 292 294 514 460 743 1468 732 1070 152 219 905 820 793 1491 1023 1162 1356 228 1337 816 1072 506 1523 171 746 1218 751 1149 1067 851 286 1485 417 702 1 32