Most objects in astrophysics are filled with highly conducting plasma and hence easily carry magnetic fields. The topological properties of these fields have important physical consequences. The atmospheres of the Sun, many types of stars, and accretion disks have magnetic fields rooted at the surface. The topological structure of the magnetic lines of force determines the possible equilibrium configurations of the field. Solar and stellar atmospheres are much hotter than expected given the surface temperature. A proposed model of heating involves tangled magnetic field lines, which release their energy in small flares. The degree of topological complexity of a magnetic field helps to determine how much energy it stores. Flares simplify the topology of the field and thereby release the stored energy. Topology is also important in understanding large–scale properties of the solar dynamo that generates the solar magnetic field. The magnetic helicity integral, which measures linking properties of the field, can be decomposed into contributions from different regions of the Sun and space. Transport of helicity from one region to another underlies many important processes in solar activity.
{"title":"Topological methods in astrophysics","authors":"M. Berger","doi":"10.1098/rsta.2001.0846","DOIUrl":"https://doi.org/10.1098/rsta.2001.0846","url":null,"abstract":"Most objects in astrophysics are filled with highly conducting plasma and hence easily carry magnetic fields. The topological properties of these fields have important physical consequences. The atmospheres of the Sun, many types of stars, and accretion disks have magnetic fields rooted at the surface. The topological structure of the magnetic lines of force determines the possible equilibrium configurations of the field. Solar and stellar atmospheres are much hotter than expected given the surface temperature. A proposed model of heating involves tangled magnetic field lines, which release their energy in small flares. The degree of topological complexity of a magnetic field helps to determine how much energy it stores. Flares simplify the topology of the field and thereby release the stored energy. Topology is also important in understanding large–scale properties of the solar dynamo that generates the solar magnetic field. The magnetic helicity integral, which measures linking properties of the field, can be decomposed into contributions from different regions of the Sun and space. Transport of helicity from one region to another underlies many important processes in solar activity.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1098/rsta.2001.0846","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72464616","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We review recent work in which perturbative, geometric and topological arguments are used to prove the existence of countable sets of orbits connecting equilibria in ordinary differential equations. We first consider perturbations of a three–dimensional integrable system possessing a line of degenerate saddle points connected by a two–dimensional manifold of homoclinic loops. We show that this manifold splits to create transverse homoclinic orbits, and then appeal to geometrical and symbolic dynamic arguments to show that homoclinic bifurcations occur in which ‘simple’ connecting orbits are replaced by a countable infinity of such orbits. We discover a rich variety of connections among equilibria and periodic orbits, as well as more exotic sets, including Smale horseshoes. The second problem is a four–dimensional Hamiltonian system. Using symmetries and classical estimates, we again find countable sets of connecting orbits. There is no small parameter in this case, and the methods are non–perturbative.
{"title":"Homoclinic orbits and chaos in three– and four–dimensional flows","authors":"P. Holmes, A. Doelman, G. Hek, G. Domokos","doi":"10.1098/rsta.2001.0845","DOIUrl":"https://doi.org/10.1098/rsta.2001.0845","url":null,"abstract":"We review recent work in which perturbative, geometric and topological arguments are used to prove the existence of countable sets of orbits connecting equilibria in ordinary differential equations. We first consider perturbations of a three–dimensional integrable system possessing a line of degenerate saddle points connected by a two–dimensional manifold of homoclinic loops. We show that this manifold splits to create transverse homoclinic orbits, and then appeal to geometrical and symbolic dynamic arguments to show that homoclinic bifurcations occur in which ‘simple’ connecting orbits are replaced by a countable infinity of such orbits. We discover a rich variety of connections among equilibria and periodic orbits, as well as more exotic sets, including Smale horseshoes. The second problem is a four–dimensional Hamiltonian system. Using symmetries and classical estimates, we again find countable sets of connecting orbits. There is no small parameter in this case, and the methods are non–perturbative.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82343474","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is known that solutions of the Euler equations of hydrodynamics correspond to geodesics on the group of volume–preserving diffeomorphisms of a compact manifold. We conjecture that, regardless of the dimension of the manifold, the associated Riemannian exponential map on the group is nonlinear Fredholm of index zero. Such a result has been established for the Riemannian exponential maps of natural Sobolev metrics on loop spaces and loop groups.
{"title":"A conjecture concerning the exponential map on Dμ(M)","authors":"G. Misiołek","doi":"10.1098/rsta.2001.0847","DOIUrl":"https://doi.org/10.1098/rsta.2001.0847","url":null,"abstract":"It is known that solutions of the Euler equations of hydrodynamics correspond to geodesics on the group of volume–preserving diffeomorphisms of a compact manifold. We conjecture that, regardless of the dimension of the manifold, the associated Riemannian exponential map on the group is nonlinear Fredholm of index zero. Such a result has been established for the Riemannian exponential maps of natural Sobolev metrics on loop spaces and loop groups.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80951643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A nonlinear model in three–dimensional space allowing for the solitons localized in the vicinity of a loop is presented. Two possible applications in real physics are discussed.
提出了一个三维空间中允许孤子在环附近局部化的非线性模型。讨论了在实际物理中的两种可能的应用。
{"title":"Knotted solitons and their physical applications","authors":"L. Faddeev","doi":"10.1098/rsta.2001.0842","DOIUrl":"https://doi.org/10.1098/rsta.2001.0842","url":null,"abstract":"A nonlinear model in three–dimensional space allowing for the solitons localized in the vicinity of a loop is presented. Two possible applications in real physics are discussed.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74926165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
I shall describe the recent progress in the study of cohomology rings of spaces of knots in Rn, H*({knots in Rn}), with arbitrary n ⩾ 3. ‘Any dimensions’ in the title can be read as dimensions n of spaces Rn, as dimensions i of the cohomology groups Hi, and also as a parameter for different generalizations of the notion of a knot. An important subproblem is the study of knot invariants. In our context, they appear as zero–dimensional cohomology classes of the space of knots in R3. It turns out that our more general problem is never less beautiful. In particular, nice algebraic structures arising in the related homological calculations have equally (or maybe even more) compact description, of which the classical ‘zero–dimensional’ part can be obtained by easy factorization. There are many good expositions of the theory of related knot invariants. Therefore, I shall deal almost completely with results in higher (or arbitrary) dimensions.
{"title":"Homology of spaces of knots in any dimensions","authors":"V. Vassiliev","doi":"10.1098/rsta.2001.0838","DOIUrl":"https://doi.org/10.1098/rsta.2001.0838","url":null,"abstract":"I shall describe the recent progress in the study of cohomology rings of spaces of knots in Rn, H*({knots in Rn}), with arbitrary n ⩾ 3. ‘Any dimensions’ in the title can be read as dimensions n of spaces Rn, as dimensions i of the cohomology groups Hi, and also as a parameter for different generalizations of the notion of a knot. An important subproblem is the study of knot invariants. In our context, they appear as zero–dimensional cohomology classes of the space of knots in R3. It turns out that our more general problem is never less beautiful. In particular, nice algebraic structures arising in the related homological calculations have equally (or maybe even more) compact description, of which the classical ‘zero–dimensional’ part can be obtained by easy factorization. There are many good expositions of the theory of related knot invariants. Therefore, I shall deal almost completely with results in higher (or arbitrary) dimensions.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1098/rsta.2001.0838","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72434102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We survey some of the recent results on Legendrian knots and links in the standard contact 3–space and solid torus. These include the description of finite–order invariants and estimates of the self–linking number coming from the classical polynomial link invariants. We also describe the combinatorial invariant introduced by Chekanov and Pushkar, which allowed them to prove Arnold's conjecture on the necessity of four–cusp curves in generic eversions of a circular front in the plane.
{"title":"Plane curves, wavefronts and Legendrian knots","authors":"V. Goryunov","doi":"10.1098/rsta.2001.0837","DOIUrl":"https://doi.org/10.1098/rsta.2001.0837","url":null,"abstract":"We survey some of the recent results on Legendrian knots and links in the standard contact 3–space and solid torus. These include the description of finite–order invariants and estimates of the self–linking number coming from the classical polynomial link invariants. We also describe the combinatorial invariant introduced by Chekanov and Pushkar, which allowed them to prove Arnold's conjecture on the necessity of four–cusp curves in generic eversions of a circular front in the plane.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74804164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the global well–posedness and regularity of the (isotropic) Lagrangian averaged Navier–Stokes (LANS–α) equations on a three–dimensional bounded domain with a smooth boundary with no–slip boundary conditions for initial data in the set {u ∈ Hs ∩ H10| Au = 0 on ∂Ω, div u = 0}, s ∈ [3, 5), where A is the Stokes operator. As with the Navier–Stokes equations, one has parabolic–type regularity; that is, the solutions instantaneously become space–time smooth when the forcing is smooth (or zero). The equations are an ensemble average of the Navier–Stokes equations over initial data in an α–radius phase–space ball, and converge to the Navier–Stokes equations as α → 0. We also show that classical solutions of the LANS–α equations converge almost all in Hs for s ∈ 2.5, 3), to solutions of the inviscid equations (ν = 0), called the Lagrangian averaged Euler (LAE–α) equations, even on domains with boundary, for time–intervals governed by the time of existence of solutions of the LAE–α equations.
我们证明了(各向同性)拉格朗日平均Navier-Stokes (LANS -α)方程在具有光滑边界和无滑移边界条件的三维有界域上的全局适性和正则性，对于集合{u∈Hs∩H10| Au = 0 on∂Ω， div u = 0}， s∈[3,5]中的初始数据，其中a是Stokes算子。与Navier-Stokes方程一样，它具有抛物型正则性;也就是说，当强迫是平滑的(或零)时，解瞬间变成时空平滑的。该方程是α -半径相空间球中Navier-Stokes方程对初始数据的系综平均，并在α→0时收敛于Navier-Stokes方程。我们还证明了LANS -α方程的经典解在s∈2.5,3时几乎全部收敛于ν = 0的无粘方程(称为拉格朗日平均欧拉(LAE -α)方程)的解，甚至在有边界的区域上，对于由LAE -α方程解的存在时间控制的时间间隔。
{"title":"Global well–posedness for the Lagrangian averaged Navier–Stokes (LANS–α) equations on bounded domains","authors":"J. Marsden, S. Shkoller","doi":"10.1098/rsta.2001.0852","DOIUrl":"https://doi.org/10.1098/rsta.2001.0852","url":null,"abstract":"We prove the global well–posedness and regularity of the (isotropic) Lagrangian averaged Navier–Stokes (LANS–α) equations on a three–dimensional bounded domain with a smooth boundary with no–slip boundary conditions for initial data in the set {u ∈ Hs ∩ H10| Au = 0 on ∂Ω, div u = 0}, s ∈ [3, 5), where A is the Stokes operator. As with the Navier–Stokes equations, one has parabolic–type regularity; that is, the solutions instantaneously become space–time smooth when the forcing is smooth (or zero). The equations are an ensemble average of the Navier–Stokes equations over initial data in an α–radius phase–space ball, and converge to the Navier–Stokes equations as α → 0. We also show that classical solutions of the LANS–α equations converge almost all in Hs for s ∈ 2.5, 3), to solutions of the inviscid equations (ν = 0), called the Lagrangian averaged Euler (LAE–α) equations, even on domains with boundary, for time–intervals governed by the time of existence of solutions of the LAE–α equations.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84158159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Berry & Robbins, in their discussion of the spin–statistics theorem in quantum mechanics, were led to ask the following question. Can one construct a continuous map from the configuration space of n distinct particles in 3–space to the flag manifold of the unitary group U(n)? I shall discuss this problem and various generalizations of it. In particular, there is a version in which U(n) is replaced by an arbitrary compact Lie group. It turns out that this can be treated using Nahm's equations, which are an integrable system of ordinary differential equations arising from the self–dual Yang-Mills equations. Our topological problem is therefore connected with physics in two quite different ways, once at its origin and once at its solution.
{"title":"Configurations of points","authors":"M. Atiyah","doi":"10.1098/rsta.2001.0840","DOIUrl":"https://doi.org/10.1098/rsta.2001.0840","url":null,"abstract":"Berry & Robbins, in their discussion of the spin–statistics theorem in quantum mechanics, were led to ask the following question. Can one construct a continuous map from the configuration space of n distinct particles in 3–space to the flag manifold of the unitary group U(n)? I shall discuss this problem and various generalizations of it. In particular, there is a version in which U(n) is replaced by an arbitrary compact Lie group. It turns out that this can be treated using Nahm's equations, which are an integrable system of ordinary differential equations arising from the self–dual Yang-Mills equations. Our topological problem is therefore connected with physics in two quite different ways, once at its origin and once at its solution.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1098/rsta.2001.0840","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72424491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In string theory space–time comes equipped with an additional geometric structure called a B–field or ‘gerbe’. I describe this structure, mention its relationship with noncommutative geometry, and explain how to use the B–field to define a twisted version of the K–theory of space–time. String–theoretical space–time can contain topologically non–trivial dynamical structures called D–branes. These are simply accounted for in the framework of conformal field theory. In a highly simplified limiting casetopological field theory with a finite gauge group—the D–branes naturally represent elements of the twisted K–theory of space–time: the K–theory class is the ‘charge’ of the D–brane.
{"title":"Topological structures in string theory","authors":"G. Segal","doi":"10.1098/rsta.2001.0841","DOIUrl":"https://doi.org/10.1098/rsta.2001.0841","url":null,"abstract":"In string theory space–time comes equipped with an additional geometric structure called a B–field or ‘gerbe’. I describe this structure, mention its relationship with noncommutative geometry, and explain how to use the B–field to define a twisted version of the K–theory of space–time. String–theoretical space–time can contain topologically non–trivial dynamical structures called D–branes. These are simply accounted for in the framework of conformal field theory. In a highly simplified limiting casetopological field theory with a finite gauge group—the D–branes naturally represent elements of the twisted K–theory of space–time: the K–theory class is the ‘charge’ of the D–brane.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90246879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this almost non–technical note, mostly aimed at mathematicians, we review the construction of instantons on the non–commutative R4 and explain how their existence is tied up with the existence of M–theory.
{"title":"Instantons and the 11th dimension","authors":"Nikita Nekrasov","doi":"10.1098/rsta.2001.0843","DOIUrl":"https://doi.org/10.1098/rsta.2001.0843","url":null,"abstract":"In this almost non–technical note, mostly aimed at mathematicians, we review the construction of instantons on the non–commutative R4 and explain how their existence is tied up with the existence of M–theory.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81904377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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