Open problems in symplectic topology pdf

Indeed, the developments in 4manifold theory since the 1980s have moved in parallel with developments in the general. One of the basic problems of topology is to determine when two given geometric objects are homeomorphic. Lectures on symplectic geometry ana cannas da silva. Research networking programme contact and symplectic. The fight to fix symplectic geometry quanta magazine. Cambridge core geometry and topology symplectic topology and floer homology by yonggeun oh. However, the problem of the complete description of biinvariant nondegenerate finsler metrics on the group ham is still open. Microlocal theory of sheaves and symplectic topology. A significantly revised second edition was published in 1998 introducing new sections and updates on the fastdeveloping area. We give a short introduction to some open problems in symplectic topol ogy, including existence of symplectic. For an upperlevel undergraduate or beginning graduate student, lectures on symplectic geometry remains, in my opinion, an ideal starting point into an exciting, active and growing area of mathematics. However there is a very appealing combinatorial trans lation of this in the case x is, for example, a topological sum of countably many compact spaces.

The dictionary of arithmetic topology, appendix 14, in hakenness and b 1, preprint, durham university. Lecture 3, which describes the simplest example of equivariant mirror symmetry, may be the most noteworthy one, since this subject has received considerable interest recently at a level far deeper than what we are. Some of these developments were brand new when the book was first published in 1974 and are briefly discussed in the appendices. Some open questions about symplectic 4manifolds, singular plane curves, and braid group factorizations denis auroux abstract. Introduction to symplectic topology oxford graduate texts. A new chapter 14 on open problems in the field has been added. The jones polynomial of which khovanov homology is a. The basic objects of study in this book are the geometry of lagrangian submanifolds and the dynamics of hamiltonian di. Lectures on symplectic geometry ana cannas da silva springer. In both theories there are strong connections with algebraic geometry, and the long history of enumerative problems there, and with mathematical physics. This was partly because symplectic geometry began to run out of problems that could be solved independently of the foundational issues, but also because, in 2012, a pair of researchers dusa mcduff, a prominent symplectic geometer at barnard college and author of a pair of canonical textbooks in the field, and katrin wehrheim, a. In symplectic topology the analogous invariants are the gromovwitten invariants, defined by counting holomorphic curves, which are solutions of a variant of the cauchyriemann equations. Prizes are often awarded for the solution to a longstanding problem, and lists of unsolved problems such as the list of millennium prize problems receive considerable attention. Some problems may belong to more than one discipline of mathematics and be studied using techniques from different areas.

Introduction to symplectic topology hardcover dusa mcduff. Introduction to symplectic topology hardcover dusa. Symplectic topology and floer homology by yonggeun oh. Every set in ois open in the usual topology, but not vice versa. Published in two volumes, this is the first book to provide a thorough and systematic explanation of symplectic topology, and the analytical details and techniques used in applying the machinery arising from floer theory as a whole. We will explain how the microlocal theory of sheaves may be applied to solve some problems of symplectic topology and could provide a substitute to the construction of the fukaya category. Quantitative symplectic geometry kai cieliebak, helmut hofer, janko latschev, and felix schlenk dedicated to anatole katok on the occasion of his sixtieth birthday a symplectic manifold. Therefore, understanding symplectic structures with respect to xed submanifolds and developing. The topology of symplectic 4manifolds is related to that of singular plane curves via the concept of branched covers. Introduction to symplectic topology oxford scholarship. Symplectic topology and algebraic geometry interact in many fruitful ways.

Z so that the moduli space of all jspheres in the class anecessarily has \excess dimension, i. If youre an algebraic geometer, i hope to give you a avour of some simple techniques in symplectic topology. We now give a better description of the set of lagrangians proposition 1. A subset uof a metric space xis closed if the complement xnuis open. Wednesday morning focused on tfts, wednesday afternoon was free, and wednesday evening we discussed open problems related to tfts. We will explain how the microlocal theory of sheaves may be applied to solve some problems of symplectic topology and could provide a substitute to. Suppose that s1 acts symplectically on the closed symplectic manifold m. Symplectic and euclidean geometry are related via complex numbers. I have made a note of some problems in the area of nonabelian algebraic topology and homological algebra in 1990, and in chapter 16 of the book in the same area and advertised here, with free pdf, there is a note of 32 problems and questions in this area which had occurred to me. A symplectic structure on the plane is just an area form. For example, for a broad class of symplectic manifolds almost all. Over the last number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results.

I ii some fundamental results on symplectic embeddings i iii open questions. The book was the first comprehensive introduction to the subject and became a key text in the area. University of georgia lowdimensional and symplectic topology. Open problems in algebraic topology and homotopy theory. Throughout the discussion we mention many open problems. Symplectic geometry, lagrangian submanifolds, minimal submani. Pdf the symplectic camel and poincare superrecurrence. System upgrade on feb 12th during this period, ecommerce and registration of new users may not be available for up to 12 hours. Sep 29, 2014 the origins of symplectic topology lie in classical dynamics, and the search for periodic orbits of hamiltonian systems. Introductory topics of pointset and algebraic topology are covered in a series of. List of unsolved problems in mathematics wikipedia.

Open problems in geometric topology 5 if the manifold is hyperbolic and the surgery coe. All of this has now grown into an enormous field, in which ideas from lowdimensional topology and symplectic topology intertwine, along with much else. Find an example of a symplectic manifold xand a2h 2x. Problems about the uniform structures of topological groups 361 chapter 39. Gromovs symplectic nonsqueezing theorem, a fundamental property from symplectic topology, is applied to the study of uncertainty analysis in hamiltonian dynamical systems with a particular.

M is a smooth manifold m endowed with a nondegenerate and closed 2form by darbouxs theorem such a manifold. Notes on smooth topology and symplectic embedding problems these notes were written for participants in the symplectic geometry group in the 2018 berkeley geometry reu. The orientability problem in open gromovwitten theory, geom. However, due to transit disruptions in some geographies, deliveries may be delayed.

Hamiltonian dynamical systems tend to have infinitely many periodic orbits. Symplectic topology today joint mathematics meetings. The hamiltonian seifert conjecture is the existence problem for regular compact energy levels without periodic orbits. Symplectic geometry has its origins in the hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic. Lowdimensional and symplectic topology, volume 82 american. It is also worth mentioning that arnold was largely responsible for the reawakening of interest to symplectic geometry at the end of 1960s and pioneered the study of symplectic topology. We then have the notion of symplectic for a smooth nonlinear map simply by looking at the derivative. Lectures on categorical dynamics and symplectic topology. Georgia international topology conference, may 1829, 2009, university of georgia, athens, georgia michael usher, editor. D salamon over the last number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important. Chingli chaievery ordinary symplectic isogeny class in positive characteristic is.

The relative hprinciple and extension problems mathoverflow. Chingli chaievery ordinary symplectic isogeny class in positive characteristic is dense in the moduli. An introduction to symplectic topology through sheaf theory princeton, fall 2010new york,spring2011 c. Co nite topology we declare that a subset u of r is open i either u. Open problems in topology edited by jan van mill free university amsterdam, the netherlands george m. It comes with a surprising mixture of both rigid and flexible behavior. It contains open problems and questions covering the a number of topics including. Alex gorodniks page contains an open problems survey. Resolved problems from this section may be found in solved problems. Notes on introductory pointset topology allen hatcher chapter 1.

Open problem t159 models, but in dimension 4 or more the curvature tensor has another component. X which is not in the closure of any nowhere dense subset of x. I ceremade, universit6de parisdauphine, place du m. Parameterized morse theory in lowdimensional and symplectic. Although the nonsqueezing theorem may seem just like a curiosity, it turns out to be acornerstone of the modern theory. Open problems in dynamical systems and related fields. It now contains an introduction to existence and uniqueness problems in symplectic topology, a section describing various examples, an overview of taubesseibergwitten theory and its applications to symplectic topology, and a section on symplectic 4manifolds. U v, where u c v is open, is called symplectic iff x e spv for every x 9 u. Edmund hall oxford university oxford, united kingdom. Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds. On some problems in symplectic topology springerlink. Nikos frantzikinakiss survey of open problems on nonconventional ergodic averages. Symplectic topology is at the crossroads of several mathematical disciplines such as lowdimensional topology, algebraic geometry, representation theory, hamiltonian dynamics, integrable systems, mirror symmetry, and string theory.

The first edition of introduction to symplectic topology was published in 1995. An introduction to symplectic topology through sheaf theory. There is a huge variety of open problems in sym plectic geometry, and we will only mention a few. Symplectic geometry is also called symplectic topology although the latter is really a subfield concerned with important global questions in symplectic geometry. Research networking programme contact and symplectic topology. The following is a relative hprinciple theorem in y. Thirty open problems in the theory of homogeneous continua 347 part 4. Some problems in differential geometry and topology. It is now understood to arise naturally in algebraic geometry, in lowdimensional topology, in representation theory and in string theory. For example,for a broad class of symplectic manifolds almost all levels of a proper smooth hamiltonian carry periodic orbits. Mathematics 490 introduction to topology winter 2007 what is this. As a beginner for hprinciples, i want to know why the relative hprinciple cannot imply a positive solution to the problems for extending symplectic structures.

These problems may well seem narrow, andor outofline of current trends, but i thought. The set of all linear symplectic maps will be denoted by spv. This is a collection of topology notes compiled by math 490 topology students at the university of michigan in the winter 2007 semester. By a neighbourhood of a point, we mean an open set containing that point. The direct sum of n symplectic planes has a symplectic structure. This is probably the problem that i would most like to see answered. The origins of symplectic topology lie in classical dynamics, and the search for periodic orbits of hamiltonian systems. While a lot of progress has been made on some of the big problems in the field. Thursday focused on symplectic and contact topology, ending with. On some special classes of continuous maps 369 chapter 40.