In recent years, there has been lots of exciting progress in many branches of low dimensional topology, including heegard floer and khovanov homology, small 4manifolds, tqft, knot concordance and. Pdf problems in lowdimensional topology semantic scholar. There are other areas of topology including geometric topology, algebraic topology, differential topology, low dimensional topology, high dimensional topology, and various others. Problems in low dimensional contact topology john b. All of this has now grown into an enormous field, in which ideas from lowdimensional topology and symplectic topology intertwine, along. Akbuluts corks and hcobordisms of smooth, simply connected 4manifolds. Feynman diagrams and lowdimensional topology springerlink. In april, 1977 when my rst problem list 38,kirby,1978 was nished, a good topologist could reasonably hope to understand the main topics in all of low dimensional topology. Section 3 concerns handlebody structures on 4manifolds. This can be regarded as a part of geometric topology. Problems in lowdimensional topology 380 pages the above file is distributed in postscript format because of the large amount of graphics involved. The skein module based on the homflypt skein relation. Siebenmann he invented the kirbysiebenmann invariant for classifying the piecewise linear structures on a topological manifold. The course will allow students to see the exciting geometric ideas of topology and cultivate the.
Representative topics are the structure theory of 3manifolds and 4manifolds, knot theory, and braid groups. Problems in low dimensional contact topology request pdf. Recent success with the fourdimensional poincare conjecture has revived interest in lowdimensional topology, especially the threedimensional poincare conjecture and other aspects of the problems of classifying threedimensional manifolds. Pdf problems in low dimensional contact topology josh. Nov 06, 2015 structures in low dimensional topology and low dimensional geometry often combined with ideas from quantum field theory can explain and inspire concepts in algebra and in representation. Some open problem in low dimensional topology are maintained at thelow dimen sional topology 3 page. Intelligence of low dimensional topology 2006 series on. Beginning with the introduction of hyperbolic geometry into knots and 3manifolds by w. In 1993, paulin showed that the topology of the boundary of a gromov hyperbolic space, together with its quasimobius structure, determines the space up to quasiisometry. Boundaries of hyperbolic spaces have played a key role in low dimensional topology and geometric group theory.
This problem is still wide open, even for any evendimensional link. Ohtsuki1 this is a list of open problems on lowdimensional topology with expositions of their history, background, signi cance, or importance. The author has been involved in only some of these developments, but it seems more illuminating not to confine the discussion to. At regular intervals, the journal publishes a section entitled, open problems in topology, edited by j. This is immediately followed by a workshop that builds on the topics covered in the prior week and. This course is an introduction to some topics in geometric topology, including knots and links, surfaces and other manifolds, fixed points, and the fundamental group. The iaspark city mathematics institute has its own home page as ias.
At the core of lowdimensional topology has been the classification of knots and links in the 3sphere and the classification of 3 and 4dimensional manifolds see wikipedia for the definitions of basic topological terms. John harer uses computational topology to study a wide range of problems of an applied flavor. Thislist was made by editing manuscripts written by contributors of open problems to the problem session of the conference intelligence of. When i joined the mathematical community in the late 1960s the development of higherdimensional topology was in full swing. Problems in low dimensional topology 99 limit problem. Introduction in april, 1977 when my rst problem list 38,kirby,1978 was nished, a good topologist could reasonably hope to understand the main topics in all of low dimensional topology.
Biology, analysis, geometry, energies, links bagel19. We outline some questions in three different areas which seem to the author interesting. This is a list of open problems on lowdimensional topology with expositions of. As pointed out in an earlier comment, low dimensional topology is really really vast and you can spend more than a lifetime reading literature in either dimension 3 or 4.
The goal of the conference was to promote the exchange of methods and ideas across disciplines and generations, from graduate students to senior researchers, and to explore fundamental research problems in the broad fields of knot theory and low dimensional topology. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Rosebrock, problems in lowdimensional topology, encyclopaedia of mathematics, springer. Problems on lowdimensional topology, 2011 edited by t. Care was taken to prepare a source document for later translation in html, pdf, and other. Our goal for today will be to contextualize the initial advances in lowdimensional topology due to gauge theory by giving a picture of the state of affairs before its introduction.
Lowdimensional topology, problems in encyclopedia of. Introduction to lowdimensional topology catalog description. Structures in lowdimensional topology and lowdimensional geometry often combined with ideas from quantum field theory can explain and. Rob kirbys home page university of california, berkeley. Topology of manifolds arise from, or have to do with, attempts to lift algebraic operations performed on the chain complex of a universal covering complex to geometric operations on the complex here and below, complex means a complex, i. The modern field of topology draws from a diverse collection of core areas of mathematics. Thurston in the late 1970s, geometric tools have become vital to the subject. Lowdimensional topology has long been a fertile area for the interaction of many different disciplines of mathematics, including differential geometry, hyperbolic geometry, combinatorics, representation theory, global analysis, classical mechanics, and theoretical physics. But at that time bill thurston was already starting to greatly in. Problems in lowdimensional topology mathematical sciences.
Low dimensional topology american mathematical society. The motivation behind topology is that some geometric problems do not depend on the exact shape of an. This list was made by editing manuscripts written by contributors of. The first is a weeklong summer school for graduate students and early career researchers that emphasizes both the interplay of low dimensional topology, geometry and knot theory, and their applications to other disciplines. Tirasan khandhawit, dale koenig, nick owad, robert tang, anastasiia tsvietkova oist this workshop is supported by oist s c h e d u l e m o n d a y, m a rc h 1 2 transportation from. We will thus spend this post establishing some basic terms and ideas for the uninitiated. These problems have a driving force, and have generated a great body of research, as well as insight. Thank you for your interest in our low dimensional topology workshop. In this paper the focus is on a shorter list of \tool questions, whose solution could. Without question, low dimensional topology is among the most popular areas of mathematics these days. Request pdf problems in low dimensional contact topology during the 2001 georgia international topology conference, two problem ses sions were held concerning contact geometry. This thesis investigates some lowdimensional topological problems from the point of view of a theoretical computer scientist. Pdf structures in lowdimensional topology and lowdimensional geometry. Problems in lowdimensional topology 99 limit problem.
Mini symposium computational problems in lowdimensional topology march 1214, 2018 oist conference center organizers. David kraines has worked on higher algebraic operations in homology and cohomology. The fundamental group of the image of the data can be compressed into arbitrarilyfarout terms in the lower central series of the fundamentalgroup of m. Problems on lowdimensional topology, 2019 edited by t. What are some interesting low dimensional topological results that dont. What distinguishes low dimensional topology from just topology in general. Pdf lowdimensional topology, lowdimensional field theory. Feynman diagrams and lowdimensional topology maxim kontsevich october 6, 2006 we shall describe a program here relating feynman diagrams, topology of manifolds, homotopical algebra, noncommutative geometry and several kinds of topological physics. Finally section 4 poses a triangulation problem for certain low dimensional stratified spaces. Proceedings of georgia topology conference, part 2. In fact theres quite a bit of structure in what remains, which is the principal subject of study in topology. There are other areas of topology including geometric topology, algebraic topology, differential topology, lowdimensional topology, highdimensional topology, and various others. Outstanding problems in lowdimensional topology and group.
This is a status report on the 1100 problems listed in the book of. In 2006 the theme was low dimensional topology the cornell topology festival, held each may. Low dimensional topology has long been a fertile area for the interaction of many different disciplines of mathematics, including differential geometry, hyperbolic geometry, combinatorics, representation theory, global analysis, classical mechanics, and theoretical physics. Ohtsuki1 this is a list of open problems on lowdimensional topology with expositions of their history, background, signi. Ohtsuki l this is a list of open problems on lowdimensional topology with expositions of their history, background, signi. Jul 24, 2017 our goal for today will be to contextualize the initial advances in low dimensional topology due to gauge theory by giving a picture of the state of affairs before its introduction. The goal of the conference was to promote the exchange of methods and ideas across disciplines and generations, from graduate students to senior researchers, and to explore fundamental research problems in the broad fields of knot theory and lowdimensional topology. In mathematics, low dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Low dimensional topology mathematical association of america. He also proved the fundamental result on the kirby calculus, a method for. Finally section 4 poses a triangulation problem for certain lowdimensional stratified spaces. May 17, 2011 at the core of low dimensional topology has been the classification of knots and links in the 3sphere and the classification of 3 and 4 dimensional manifolds see wikipedia for the definitions of basic topological terms. The lehigh geometrytopology conference is held each summer at lehigh univ.
The georgia topology conference, held each summer at the university of georgia, athens, ga. In knot theory, and larger, in the lowdimensional topology, there have been several studies on random links manifolds recently. Ng during the 2001 georgia international topology conference, two problem sessions were held concerning contact geometry. In recent years, there has been lots of exciting progress in many branches of lowdimensional topology, including heegard floer and khovanov homology, small 4manifolds, tqft, knot concordance and. Some open problem in low dimensional topology are maintained at thelow dimen sional topology3 page. Geometric topology is very much motivated by lowdimensional phenomena and the very notion of lowdimensional phenomena being special is due to the existence of a big tool called the whitney trick, which allows one to readily convert certain problems in manifold theory into sometimes quite complicated. In 2006 the theme was lowdimensional topology the cornell topology festival, held each may. This in turn led to novikovs discovery of the undecidability of the word problem for nitely presented groups 24 and the undecidability of the. Floer homology, gauge theory, and lowdimensional topology. Four dimensional topology is in an unsettled state. The lehigh geometry topology conference is held each summer at lehigh univ.
Is it simply topology concerned about low dimensional spaces if so, then how low is low. Dehn formulated the word and the isomorphism problems for groups in an attempt to solve this question the isomorphism problem was stated by tietze 32 in 1908 as well. This list was made by editing manuscripts written by contributors of open problems to the problem session of the. This is the first time ive heard the term low dimensional topology. This paper was developed from a lecture given at the international conference on surgery and controlled topology, held at josai university in september 1996. I would like to expand on the dedication of this paper to c. If it could be pushed into the intersection the general conjecture would follow. Differential geometric methods in lowdimensional topology. Low dimensional topology hardcover september 1, 1999 by karoly boroczky editor see all formats and editions hide other formats and editions. Problems on lowdimensional topology, 2015 rims, kyoto. The approach is inspired by the results of arnold and rokhlin and involves techniques from reidemeister torsion and intersection in homology, braid and. Pacific journal of mathematics topology topology and its applications topology. I hope you will consider attending other workshops here in the near future.
Problems in low dimensional contact topology georgia institute. Problems on lowdimensional topology, 2014 edited by t. Mini symposium computational problems in lowdimensional topology. Problems on lowdimensional topology, 2017 edited by tomotada ohtsuki1 and tetsuya ito2 this is a list of open problems on lowdimensional topology with expositions of their history, background, signi cance, or importance. Problems in lowdimensional topology edited by rob kirby berkeley 22 dec 95. During the 2001 georgia international topology conference, two problem. Some problems in differential geometry and topology. What happens if one allows geometric objects to be stretched or squeezed but not broken. This is altogether reasonable on several counts, including the fact that it resonates with the world of our ordinary experience at least to some extent. Problems on lowdimensional topology, 2019 rims, kyoto.
The lectures centered on several topics, including heegaard floer theory, knot theory, symplectic and contact topology, and seibergwitten theory. In part, the hope was to foster dialogue across closely related disciplines, some of which were developing in relative isolation until fairly recently. Problemson lowdimensionaltopology,2015 edited by t. Robion cromwell kirby born february 25, 1938 is a professor of mathematics at the university of california, berkeley who specializes in lowdimensional topology. Lenhard ng and adam levine study low dimensional topology, the topology of three and four dimensional spaces, via knot theory and symplectic geometry. Donaldson july 9, 2008 1 introduction this is a survey of various applications of analytical and geometric techniques to problems in manifold topology. As is common, the problems that have seemed to be most di. Differential geometric methods in low dimensional topology s. All solutions of problems are put in the end of the book. Geometric topology is more motivated by objects it wants to prove theorems about. Differential geometric methods in lowdimensional topology s. The first is a weeklong summer school for graduate students and early career researchers that emphasizes both the interplay of lowdimensional topology, geometry and knot theory, and their applications to other disciplines. Problems in low dimensional topology 380 pages the above file is distributed in postscript format because of the large amount of graphics involved.
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