All integral slopes can be Seifert fibered slopes for hyperbolic knots
stoz定理
stoz定理斯托兹定理（Stokes' theorem）是数学中的一个重要定理，它描述了曲面与曲面边界之间的关系。
该定理是格林定理的三维扩展，也是高维微积分的重要基础之一。
斯托兹定理在物理学、工程学以及计算机图形学等领域都有广泛的应用。
斯托兹定理的表述如下：对于一个光滑的曲面S，它的边界为曲线C。
假设在曲面S上有一个向量场F，那么斯托兹定理告诉我们，曲面S 上向量场F的通量等于曲线C上向量场F的环路积分。
换句话说，斯托兹定理将曲面上的积分问题转化为曲线上的积分问题。
为了更好地理解斯托兹定理，我们可以通过一个简单的例子来说明。
假设有一个平面曲面S，它被一个简单闭合曲线C所包围。
现在在曲面S上有一个向量场F，我们想要计算向量场F在曲面S上的通量。
根据斯托兹定理，我们可以将曲面S的通量转化为曲线C的环路积分。
具体计算方法如下：我们需要计算曲线C的方向与曲面S的法向量的点积。
如果曲线C 的方向与曲面S的法向量的点积大于零，那么曲线C的方向与曲面S的法向量是一致的；如果点积小于零，则方向相反。
然后，我们需要计算曲线C的环路积分。
最后，我们将曲线C的环路积分乘以曲线C的方向与曲面S的法向量的点积的绝对值，即可得到曲面S 上向量场F的通量。
斯托兹定理的应用非常广泛。
在物理学中，它可以用来描述电场和磁场的分布情况，以及电荷和电流的密度。
在工程学中，它可以用来计算液体和气体在曲面上的流动情况，以及材料的应力和应变分布。
在计算机图形学中，它可以用来计算曲面的法向量、曲面的光照效果以及曲面的纹理映射。
斯托兹定理的重要性在于它将曲面上的积分问题转化为曲线上的积分问题，简化了计算过程。
它不仅为我们理解和研究物理现象提供了数学工具，也为工程设计和计算机模拟提供了基础。
通过运用斯托兹定理，我们可以更加深入地理解曲面与曲线之间的关系，从而更好地解决实际问题。
斯托兹定理是数学中的一个重要定理，它描述了曲面与曲面边界之间的关系。
学数学的必看GTM经典著作下载三
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Felix是新鲁汶大学(法语鲁汶大学)的教授，第二作者是著名的华人逻辑学家王浩的学生。
206 Problems in Analytic Number Theory,M.Ram Murty 207 Algebraic Graph Theory,Godsil,Royle(代数图论)镜像下载(4062KB，英文版，DJVU格式，支持关键词检索，点击打开下载页面，支持迅雷、快车下载)Godsil是加拿大滑铁卢大学的教授，代数组合图论的权威。
曾任JAC的主编，现在是组合学期刊(JC)电子版的主编。
Royle是UWA的副教授。
208 Analysis for Applied Mathematics,Ward Cheney 209 AShort Course on Spectral Theory,William Arveson(谱理论简明教程)镜像下载(4366KB，英文版，DJVU格式，支持关键词检索，点击打开下载页面，支持迅雷、快车下载)本书给读者提供谱论-被称之为解决算子理论基本问题的基本工具，并主要计算了无限维空间特别是希尔伯特空间算子的谱。
数学专业英语词汇（T）_数学物理英语词汇
t distribution 学生t分布t number t数t statistic t统计量t test t检验t1topological space t1拓扑空间t2topological space t2拓扑空间t3topological space 分离空间t4topological space 正则拓扑空间t5 topological space 正规空间t6topological space 遗传正规空间table 表table of random numbers 随机数表table of sines 正弦表table of square roots 平方根表table of values 值表tabular 表的tabular value 表值tabulate 制表tabulation 造表tabulator 制表机tacnode 互切点tag 标签tame 驯顺嵌入tame distribution 缓增广义函数tamely imbedded 驯顺嵌入tangency 接触tangent 正切tangent bundle 切丛tangent cone 切线锥面tangent curve 正切曲线tangent function 正切tangent line 切线tangent of an angle 角的正切tangent plane 切平面tangent plane method 切面法tangent surface 切曲面tangent vector 切向量tangent vector field 切向量场tangent vector space 切向量空间tangential approximation 切线逼近tangential component 切线分量tangential curve 正切曲线tangential equation 切线方程tangential stress 切向应力tangents method 切线法tape 纸带tape inscription 纸带记录tariff 税tautology 重言taylor circle 泰勒圆taylor expansion 泰勒展开taylor formula 泰勒公式taylor series 泰勒级数technics 技术technique 技术telegraph equation 电报方程teleparallelism 绝对平行性temperature 温度tempered distribution 缓增广义函数tend 倾向tendency 瞧tension 张力tensor 张量tensor algebra 张量代数tensor analysis 张量分析tensor bundle 张量丛tensor calculus 张量演算法tensor density 张量密度tensor differential equation 张量微分方程tensor field 张量场tensor form 张量形式tensor form of the first kind 第一张量形式tensor function 张量函数tensor of torsion 挠率张量tensor product 张量乘积tensor product functor 张量乘积函子tensor representation 张量表示tensor space 张量空间tensor subspace 张量子空间tensor surface 张量曲面tensorial multiplication 张量乘法term 项term of higher degree 高次项term of higher order 高次项term of series 级数的项terminability 有限性terminable 有限的terminal decision 最后判决terminal edge 终结边terminal point 终点terminal unit 级端设备terminal vertex 悬挂点terminate 终止terminating chain 可终止的链terminating continued fraction 有尽连分数terminating decimal 有尽小数termination 终止terminology 专门名词termwise 逐项的termwise addition 逐项加法termwise differentiation 逐项微分termwise integration 逐项积分ternary 三元的ternary connective 三元联结ternary form 三元形式ternary notation 三进制记数法ternary number system 三进制数系ternary operation 三项运算ternary relation 三项关系ternary representation og numbers 三进制记数法tertiary obstruction 第三障碍tesseral harmonic 田形函数tesseral legendre function 田形函数test 检验test for additivity 加性检验test for uniform convergence 一致收敛检验test function 测试函数test of dispersion 色散检验test of goodness of fit 拟合优度检验test of hypothesis 假设检验test of independence 独立性检验test of linearity 线性检验test of normality 正规性检验test point 测试点test routine 检验程序test statistic 检验统计量tetracyclic coordinates 四圆坐标tetrad 四元组tetragon 四角形tetragonal 正方的tetrahedral 四面角tetrahedral angle 四面角tetrahedral co ordinates 四面坐标tetrahedral group 四面体群tetrahedral surface 四面曲面tetrahedroid 四面体tetrahedron 四面形tetrahedron equation 四面体方程theorem 定理theorem for damping 阻尼定理theorem of alternative 择一定理theorem of identity for power series 幂级数的一致定理theorem of implicit functions 隐函数定理theorem of mean value 平均值定理theorem of principal axes 轴定理theorem of residues 残数定理theorem of riemann roch type 黎曼洛赫型定理theorem on embedding 嵌入定理theorems for limits 极限定理theoretical curve 理论曲线theoretical model 理论模型theory of automata 自动机理论theory of cardinals 基数论theory of complex multiplication 复数乘法论theory of complexity of computations 计算的复杂性理论theory of correlation 相关论theory of differential equations 微分方程论theory of dimensions 维数论theory of elementary divisors 初等因子理论theory of elementary particles 基本粒子论theory of equations 方程论theory of errors 误差论theory of estimation 估计论theory of functions 函数论theory of games 对策论theory of hyperbolic functions 双曲函数论theory of judgment 判断论theory of numbers 数论theory of ordinals 序数论theory of perturbations 摄动理论theory of probability 概率论theory of proportions 比例论theory of relativity 相对论theory of reliability 可靠性理论theory of representations 表示论theory of sets 集论theory of sheaves 层理论theory of singularities 奇点理论theory of testing 检验论theory of time series 时间序列论theory of transversals 横断线论theory of types 类型论thermal 热的thermodynamic 热力学的thermodynamics 热力学theta function 函数theta series 级数thick 厚的thickness 厚度thin 薄的thin set 薄集third boundary condition 第三边界条件third boundary value problem 第三边界值问题third fundamental form 第三基本形式third isomorphism theorem 第三同构定理third proportional 比例第三项third root 立方根thom class 汤姆类thom complex 汤姆复形three body problem 三体问题three dimensional 三维的three dimensional space 三维空间three dimensional torus 三维环面three eighths rule 八分之三法three faced 三面的three figur 三位的three place 三位的three point problem 三点问题three series theorem 三级数定理three sheeted 三叶的three sided 三面的three sigma rule 三规则three termed 三项的three valued 三值的three valued logic 三值逻辑three valued logic calculus 三值逻辑学threshold logic 阈逻辑time interval 时程time lag 时滞time series analysis 时序分析timesharing 分时toeplitz matrix 托普利兹矩阵tolerance 容许tolerance distribution 容许分布tolerance estimation 容许估计tolerance factor 容许因子tolerance level 耐受水平tolerance limit 容许界限tolerance region 容许区域top digit 最高位数字topological 拓扑的topological abelian group 拓扑阿贝耳群topological algebra 拓扑代数topological cell 拓扑胞腔topological circle 拓扑圆topological completeness 拓扑完备性topological complex 拓扑复形topological convergence 拓扑收敛topological dimension 拓扑维topological direct sum 拓扑直和topological dynamics 拓扑动力学topological embedding 拓扑嵌入topological field 拓扑域topological group 拓扑群topological homeomorphism 拓扑同胚topological index 拓扑指数topological invariant 拓扑不变量topological limit 拓扑极限topological linear space 拓扑线性空间topological manifold 拓扑廖topological mapping 拓扑同胚topological pair 拓扑偶topological polyhedron 曲多面体topological product 拓扑积topological residue class ring 拓扑剩余类环topological ring 拓扑环topological simplex 拓扑单形topological skew field 拓扑非交换域topological space 拓扑空间topological sphere 拓扑球面topological structure 拓扑结构topological sum 拓扑和topological type 拓扑型topologically complete set 拓扑完备集topologically complete space 拓扑完备空间topologically equivalent space 拓扑等价空间topologically nilpotent element 拓扑幂零元topologically ringed space 拓扑环式空间topologically solvable group 拓扑可解群topologico differential invariant 拓扑微分不变式topologize 拓扑化topology 拓扑topology of bounded convergence 有界收敛拓扑topology of compact convergence 紧收敛拓扑topology of uniform convergence 一致收敛拓扑toroid 超环面toroidal coordinates 圆环坐标toroidal function 圆环函数torque 转矩torsion 挠率torsion coefficient 挠系数torsion form 挠率形式torsion free group 非挠群torsion group 挠群torsion module 挠模torsion of a curve 曲线的挠率torsion product 挠积torsion subgroup 挠子群torsion tensor 挠率张量torsion vector 挠向量torsionfree connection 非挠联络torsionfree module 无挠模torsionfree ring 无挠环torus 环面torus function 圆环函数torus group 环面群torusknot 环面纽结total 总和total correlation 全相关total curvature 全曲率total degree 全次数total differential 全微分total differential equation 全微分方程total error 全误差total graph 全图total image 全象total inspection 全检查total instability 全不稳定性total inverse image 全逆象total matrix algebra 全阵环total matrix ring 全阵环total order 全序total predicate 全谓词total probability 总概率total probability formula 总概率公式total regression 总回归total relation 通用关系total space 全空间total stability 全稳定性total step iteration 整步迭代法total step method 整步迭代法total stiefel whitney class 全斯蒂费尔惠特尼类total subset 全子集total sum 总和total variation 全变差totally bounded set 准紧集totally bounded space 准紧空间totally differentiable 完全可微分的totally differentiable function 完全可微函数totally disconnected 完全不连通的totally disconnected graph 完全不连通图totally disconnected groupoid 完全不连通广群totally disconnected set 完全不连通集totally disconnected space 完全不连通空间totally geodesic 全测地的totally nonnegative matrix 全非负矩阵totally ordered group 全有序群totally ordered set 线性有序集totally positive 全正的totally positive matrix 全正矩阵totally quasi ordered set 完全拟有序集totally real field 全实域totally reflexive relation 完全自反关系totally regular matrix method 完全正则矩阵法totally singular subspace 全奇异子空间totally symmetric loop 完全对称圈totally symmetric quasigroup 完全对称拟群touch 相切tournament 竞赛图trace 迹trace form 迹型trace function 迹函数trace of dyadic 并向量的迹trace of matrix 矩阵的迹trace of tensor 张量的迹tracing point 追迹点track 轨迹tractrix 曳物线trajectory 轨道transcendence 超越性transcendence basis 超越基transcendence degree 超越次数transcendency 超越性transcendental element 超越元素transcendental equation 超越方程transcendental function 超越函数transcendental integral function 超越整函数transcendental number 超越数transcendental singularity 超越奇点transcendental surface 超越曲面transfer 转移transfer function 转移函数transfinite 超限的transfinite diameter 超限直径transfinite induction 超限归纳法transfinite number 超限序数transfinite ordinal 超限序数transform 变换transformation 变换transformation equation 变换方程transformation factor 变换因子transformation formulas of the coordinates 坐标的变换公式transformation function 变换函数transformation group 变换群transformation of air mass 气团变性transformation of coordinates 坐标的变换transformation of parameter 参数变换transformation of state 状态变换transformation of the variable 变量的更换transformation rules 变换规则transformation theory 变换论transformation to principal axes 轴变换transgression 超渡transient response 瞬态响应transient stability 瞬态稳定性transient state 瞬态transient time 过渡时间transition function 转移函数transition graph 转换图transition matrix 转移矩阵transition probability 转移函数transitive closure 传递闭包transitive graph 传递图transitive group of motions 可迁运动群transitive law 可迁律transitive permutation group 可迁置换群transitive relation 传递关系transitive set 可递集transitivity 可递性transitivity laws 可迁律translatable design 可旋转试验设计translate 转移translation 平移translation curve 平移曲线translation group 平移群translation invariant 平移不变的translation invariant metric 平移不变度量translation number 殆周期translation of axes 坐标轴的平移translation operator 平移算子translation surface 平移曲面translation symmetry 平移对称translation theorem 平移定理transmission channel 传输通道transmission ratio 传输比transport problem 运输问题transportation algorithm 运输算法transportation matrix 运输矩阵transportation network 运输网络transportation problem 运输问题transpose 转置transposed inverse matrix 转置逆矩阵transposed kernel 转置核transposed map 转置映射transposed matrix 转置阵transposition 对换transversal 横截矩阵胚transversal curve 横截曲线transversal field 模截场transversal lines 截线transversality 横截性transversality condition 横截条件transverse axis 横截轴transverse surface 横截曲面trapezium 不规则四边形trapezoid 不规则四边形trapezoid formula 梯形公式trapezoid method 梯形公式traveling salesman problem 转播塞尔斯曼问题tree 树trefoil 三叶形trefoil knot 三叶形纽结trend 瞧trend line 瞧直线triad 三元组trial 试验triangle 三角形triangle axiom 三角形公理triangle condition 三角形公理triangle inequality 三角形公理triangulable 可三角剖分的triangular decomposition 三角分解triangular form 三角型triangular matrix 三角形矩阵triangular number 三角数triangular prism 三棱柱triangular pyramid 四面形triangular surface 三角曲面triangulate 分成三角形triangulation 三角剖分triaxial 三轴的triaxial ellipsoid 三维椭面trichotomy 三分法trident of newton 牛顿三叉线tridiagonal matrix 三对角线矩阵tridimensional 三维的trigammafunction 三函数trigonometric 三角的trigonometric approximation polynomial 三角近似多项式trigonometric equation 三角方程trigonometric function 三角函数trigonometric moment problem 三角矩问题trigonometric polynomial 三角多项式trigonometric series 三角级数trigonometrical interpolation 三角内插法trigonometry 三角学trihedral 三面形的trihedral angle 三面角trihedron 三面体trilateral 三边的trilinear 三线的trilinear coordinates 三线坐标trilinear form 三线性形式trinomial 三项式;三项式的trinomial equation 三项方程triplanar point 三切面重点 ?triple 三元组triple curve 三重曲线triple integral 三重积分triple point 三重点triple product 纯量三重积triple product of vectors 向量三重积triple root 三重根triple series 三重级数triple tangent 三重切线triply orthogonal system 三重正交系triply tangent 三重切线的trirectangular spherical triangle 三直角球面三角形trisecant 三度割线trisect 把...三等分trisection 三等分trisection of an angle 角的三等分trisectrix 三等分角线trivalent map 三价地图trivector 三向量trivial 平凡的trivial character 单位特贞trivial cohomology functor 平凡上同弹子trivial extension 平凡扩张trivial fibre bundle 平凡纤维丛trivial graph 平凡图trivial homogeneous ideal 平凡齐次理想trivial knot 平凡纽结trivial solution 平凡解trivial subset 平凡子集trivial topology 密着拓扑trivial valuation 平凡赋值triviality 平凡性trivialization 平凡化trochoid 摆线trochoidal 余摆线的trochoidal curve 摆线true error 真误差true formula 真公式true proposition 真命题true sign 直符号true value 真值truncated cone 截锥truncated cylinder 截柱truncated distribution 截尾分布truncated pyramid 截棱锥truncated sample 截样本truncated sequence 截序列truncation 舍位truncation error 舍位误差truncation point 舍位点truth 真值truth function 真值函项truth matrix 真值表truth set 真值集合truth symbol 真符号truth table 真值表truthvalue 真值tube 管tubular knot 管状纽结tubular neighborhood 管状邻域tubular surface 管状曲面turbulence 湍流turbulent 湍聊turing computability 图灵机可计算性turing computable 图灵机可计算的turing machine 图录机turn 转向turning point 转向点twice 再次twice differentiable function 二次可微函数twin primes 素数对twisted curve 空间曲线twisted torus 挠环面two address 二地址的two address code 二地址代码two address instruction 二地址指令two body problem 二体问题two decision problem 二判定问题two digit 二位的two dimensional 二维的two dimensional laplace transformation 二重拉普拉斯变换two dimensional normal distribution 二元正态分布two dimensional quadric 二维二次曲面two dimensional vector space 二维向量空间two fold transitive group 双重可迁群two person game 两人对策two person zero sum game 二人零和对策two phase sampling 二相抽样法two place 二位的two point distribution 二点分布two point form 两点式two sample method 二样本法two sample problem 二样本问题two sample test 双样本检验two sheet 双叶的two sided condition 双边条件two sided decomposition 双边分解two sided divisor 双边因子two sided ideal 双边理想two sided inverse 双边逆元two sided module 双边模two sided neighborhood 双侧邻域two sided surface 双侧曲面two sided test 双侧检定two stage sampling 两阶段抽样法two termed expression 二项式two valued logic 二值逻辑two valued measure 二值测度two variable matrix 双变量矩阵two way array 二向分类two way classification 二向分类twopoint boundary value problem 两点边值问题type 型type problem 类型问题typenumber 型数typical mean 典型平均。
低维拓扑简介
考研论坛»数学»低维拓扑knight51发表于2005-7-28 08:34低维拓扑<P>下面说说低维拓扑的内容：低维拓扑是微分拓扑的一部分，主要研究3，4维流形与纽结理论。
又叫几何拓扑。
主要以代数拓扑与微分拓扑为工具。
它与微分几何和动力系统关系密切。
国外搞这个方向的也几乎都搞微分几何和动力系统。
我国这个方向北大最牛，美国是伯克利和普林斯顿最牛。
比起代数几何来，它比较好入门。
初学者只需要代数拓扑，微分拓扑，黎曼几何的知识就行了。
美国这方面比较牛，几乎每个搞基础数学研究的都会低维拓扑。
</P><DIV class=postcolor>纠正一下上面的错误，美国也不是每个搞基础数的都精通低维拓扑，而是懂一些低维拓扑的知识。
如果入门后还想更加深入了解它，那还需要读一些双曲几何和拓扑动力系统的书。
</DIV><!-- THE POST --><!-- THE POST --><DIV class=postcolor>下面介绍一下这方面的牛人：Bill Thurston studied at New College, Sarasota, Florida. He received his B.S. from there in 1967 and moved to the University of California at Berkeley to undertake research under Morris Hirsch's and Stephen Smale's supervision. He was awarded his doctorate in 1972 for a thesis entitled Foliations of 3-manifolds which are circle bundles. This work showed the existence of compact leaves in foliations of 3-dimensional manifolds.After completing his Ph.D., Thurston spent the academic year 1972-73 at the Institute for Advanced Study at Princeton. Then, in 1973, he was appointed an assistant professor of mathematics at Massachusetts Institute of Technology. In 1974 he was appointed professor of mathematics at Princeton University.Throughout this period Thurston worked on foliations. Lawson ([5]) sums up this work:-It is evident that Thurston's contributions to the field of foliations are of considerable depth. However, what sets them apart is their marvellous originality. This is also true of his subsequent work on Teichmüller space and the theory of 3-manifolds.In [8] Wall describes Thurston's contributions which led to him being awarded a Fields Medal in 1982. In fact the1982 Fields Medals were announced at a meeting of the General Assembly of the International Mathematical Union in Warsaw in early August 1982. They were not presented until the International Congress in Warsaw which could not be held in 1982 as scheduled and was delayed until the following year. Lectures on the work of Thurston which led to his receiving the Medal were made at the 1983 International Congress. Wall, giving that address, said:-Thurston has fantastic geometric insight and vision: his ideas have completely revolutionised the study of topology in 2 and 3 dimensions, and brought about a new and fruitful interplaybetween analysis, topology and geometry.Wall [8] goes on to describe Thurston's work in more detail:-The central new idea is that a very large class of closed 3-manifolds should carry a hyperbolic structure - be the quotient of hyperbolic space by a discrete group of isometries, or equivalently, carry a metric of constant negative curvature. Although this is a natural analogue of the situation for 2-manifolds, where such a result is given by Riemann's uniformisation theorem, it is much less plausible - even counter-intuitive - in the 3-dimensional situation.Kleinian groups, which are discrete isometry groups of hyperbolic 3-space, were first studied by Poincaré and a fundamental finiteness theorem was proved by Ahlfors. Thurston's work on Kleinian groups yielded many new results and established a well known conjecture. Sullivan describes this geometrical work in [6], giving the following summary:-Thurston's results are surprising and beautiful. The method is a new level of geometrical analysis - in the sense of powerful geometrical estimation on the one hand, and spatial visualisation and imagination on the other, which are truly remarkable.Thurston's work is summarised by Wall [8]:-Thurston's work has had an enormous influence on 3-dimensional topology. This area has a strong tradition of 'bare hands' techniques and relatively little interaction with other subjects. Direct arguments remain essential, but 3-dimensional topology has now firmly rejoined the main stream of mathematics.Thurston has received many honours in addition to the Fields Medal. He held a Alfred P Sloan Foundation Fellowship in 1974-75. In 1976 his work on foliations led to his being awarded the Oswald Veblen Geometry Prize of the American Mathematical Society. In 1979 he was awarded the Alan T Waterman Award, being the second mathematician to receive such an award (the first being Fefferman in 1976).</DIV><!-- THE POST -->第2个牛人：Michael Freedman entered the University of California at Berkeley in 1968 and continued his studies at Princeton University in 1969. He was awarded a doctorate by Princeton in 1973 for his doctoral dissertation entitled Codimension-Two Surgery. His thesis supervisor was William Browder.After graduating Freedman was appointed a lecturer in the Department of Mathematics at the University of California at Berkeley. He held this post from 1973 until 1975 when he became a member of the Institute for Advanced Study at Princeton. In 1976 he was appointed as assistant professor in the Department of Mathematics at the University of California at San Diego.Freedman was promoted to associate professor at San Diego in 1979. He spent the year 1980/81 at the Institute for Advanced Study at Princeton returning to the University of California at San Diego where he was promoted to professor on 1982. He holds this post in addition to the Charles Lee Powell Chair of Mathematics which he was appointed to in 1985.Freedman was awarded a Fields Medal in 1986 for his work on the Poincaré conjecture. The Poincaré conjecture, one of the famous problems of 20th-century mathematics, asserts that a simply connected closed 3-dimensional manifold is a 3-dimensional sphere. The higher dimensional Poincaréconjecture claims that any closed n-manifold which is homotopy equivalent to the n-sphere must be the n-sphere. When n = 3 this is equivalent to the Poincaré conjecture. Smale proved the higher dimensional Poincaré conjecture in 1961 for n at least 5. Freedman proved the conjecture for n = 4 in 1982 but the original conjecture remains open.Milnor, describing Freedman's work which led to the award of a Fields Medal at the International Congress of Mathematicians in Berkeley in 1986, said:-Michael Freedman has not only proved the Poincaré hypothesis for 4-dimensional topological manifolds, thus characterising the sphere S4, but has also given us classification theorems, easy to state and to use but difficult to prove, for much more general 4-manifolds. The simple nature of his results in the topological case must be contrasted with the extreme complications which are now known to occur in the study of differentiable and piecewise linear 4-manifolds. ... Freedman's 1982 proof of the 4-dimensional Poincaré hypothesis was an extraordinary tour de force. His methods were so sharp as to actually provide a complete classification of all compact simply connected topological 4-manifolds, yielding many previously unknown examples of such manifolds, and many previously unknown homeomorphisms between known manifolds.Freedman has received many honours for his work. He was California Scientist of the Year in 1984 and, in the same year, he was made a MacArthur Foundation Fellow and also was elected to the National Academy of Sciences. In 1985 he was elected to the American Academy of Arts and Science. In addition to being awarded the Fields Medal in 1986, he also received the Veblen Prize from the American Mathematical Society in that year. The citation for the Veblen Prize reads (see [3]):-After the discovery in the early 60s of a proof for the Poincaré conjecture and other properties of simply connected manifolds of dimension greater than four, one of the biggest open problems, besides the three dimensional Poincaré conjecture, was the classification of closed simply connected four manifolds. In his paper, The topology of four-dimensional manifolds, published in the Journal of Differential Geometry (1982), Freedman solved this problem, and in particular, the four-dimensional Poincaré conjecture. The major innovation was the solution of the simply connected surgery problem by proving a homotopy theoretic condition suggested by Casson for embedding a 2-handle, i.e. a thickened disc in a four manifold with boundary.Besides these results about closed simply connected four manifolds, Freedman also proved:(a) Any four manifold properly equivalent to R4 is homeomorphic to R4; a related result holds for S3 R.(b) There is a nonsmoothable closed four manifold.© The four-dimensional Hauptvermutung is false; i.e. there are four manifolds with inequivalent combinatorial triangulations.Finally, we note that the results of the above mentioned paper, together with Donaldson's work, produced the startling example of an exotic smoothing of R4.In his reply Freedman thanked his teachers (who he said included his students) and also gave some fascinating views on mathematics [3]:-My primary interest in geometry is for the light it sheds on the topology of manifolds. Here it seems important to be open to the entire spectrum of geometry, from formal to concrete. By spectrum, I mean the variety of ways in which we can think about mathematical structures. At one extreme the intuition for problems arises almost entirely from mental pictures. At the other extreme the geometric burden is shifted to symbolic and algebraic thinking. Of course this extreme is only a middle ground from the viewpoint of algebra, which is prepared to go much further in the direction of formal operations and abandon geometric intuition altogether.In the same reply Freedman also talks about the influence mathematics can have on the world and the way that mathematicians should express their ideas:-In the nineteenth century there was a movement, of which Steiner was a principal exponent, to keep geometry pure and ward off the depredations of algebra. Today I think we feel that much of the power of mathematics comes from combining insights from seemingly distant branches of the discipline. Mathematics is not so much a collection of different subjects as a way of thinking. As such, it may be applied to any branch of knowledge. I want to applaud the efforts now being made by mathematicians to publish ideas on education, energy, economics, defence, and world peace. Experience inside mathematics shows that it isn't necessary to be an old hand in an area to make a contribution. Outside mathematics the situation is less clear, but I cannot help feeling that there, too, it is a mistake to leave important issues entirely to experts.In June 1987 Freedman was presented with the National Medal of Science at the White House by President Ronald Reagan. The following year he received the Humboldt Award and, in 1994, he received the Guggenheim Fellowship Award.<DIV class=postcolor>介绍第3个牛人：Simon Donaldson's secondary school education was at Sevenoaks School in Kent which he attended from 1970 to 1975. He then entered Pembroke College, Cambridge where he studied until 1980, receiving his B.A. in 1979. One of his tutors at Cambridge described him as a very good student but certainly not the top student in his year. Apparently he would always come to his tutorials carrying a violin case.In 1980 Donaldson began postgraduate work at Worcester College, Oxford, first under Nigel Hitchen's supervision and later under Atiyah's supervision. Atiyah writes in [2]:-In 1982, when he was a second-year graduate student, Simon Donaldson proved a result that stunned the mathematical world.This result was published by Donaldson in a paper Self-dual connections and the topology of smooth 4-manifolds which appeared in the Bulletin of the American Mathematical Society in 1983. Atiyah continues his description of Donaldson's work [2]:-Together with the important work of Michael Freedman ..., Donaldson's result implied that there are "exotic" 4-spaces, i.e. 4-dimensional differentiable manifolds which are topologically but not differentiably equivalent to the standard Euclidean 4-space R4. What makes this result so surprising is that n = 4 is the only value for which such exotic n-spaces exist. These exotic 4-spaces have the remarkable property that (unlike R4) they contain compact sets which cannot be contained inside any differentiably embedded 3-sphere !After being awarded his doctorate from Oxford in 1983, Donaldson was appointed a Junior Research Fellow at All Souls College, Oxford. He spent the academic year 1983-84 at the Institute for Advanced Study at Princeton, After returning to Oxford he was appointed Wallis Professor of Mathematics in 1985, a position he continues to hold.Donaldson has received many honours for his work. He received the Junior Whitehead Prize from the London Mathematical Society in 1985. In the following year he was elected a Fellow of the Royal Society and, also in 1986, he received a Fields Medal at the International Congress at Berkeley. In 1991 Donaldson received the Sir William Hopkins Prize from the Cambridge Philosophical Society. Then, the following year, he received the Royal Medal from the Royal Society. He also received the Crafoord Prize from the Royal Swedish Academy of Sciences in 1994:-... for his fundamental investigations in four-dimensional geometry through application of instantons, in particular his discovery of new differential invariants ...Atiyah describes the contribution which led to Donaldson's award of a Fields Medal in [2]. He sums up Donaldson's contribution:-When Donaldson produced his first few results on 4-manifolds, the ideas were so new and foreign to geometers and topologists that they merely gazed in bewildered admiration.Slowly the message has gotten across and now Donaldson's ideas are beginning to be used by others in a variety of ways. ... Donaldson has opened up an entirely new area; unexpected and mysterious phenomena about the geometry of 4-dimensions have been discovered. Moreover the methods are new and extremely subtle, using difficult nonlinear partial differential equations. On the other hand, this theory is firmly in the mainstream of mathematics, having intimate links with the past, incorporating ideas from theoretical physics, and tying in beautifully with algebraic geometry.The article [3] is very interesting and provides both a collection of reminiscences by Donaldson on how he came to make his major discoveries while a graduate student at Oxford and also a survey of areas which he has worked on in recent years. Donaldson writes in [3] that nearly all his work has all come under the headings:-(1) Differential geometry of holomorphic vector bundles.(2) Applications of gauge theory to 4-manifold topology.and he relates his contribution to that of many others in the field.Donaldson's work in summed up by R Stern in [6]:-In 1982 Simon Donaldson began a rich geometrical journey that is leading us to an exciting conclusion to this century. He has created an entirely new and exciting area of research through which much of mathematics passes and which continues to yield mysterious and unexpected phenomena about the topology and geometry of smooth 4-manifolds</DIV><DIV class=postcolor>下面continue介绍第4个牛人：Robion Kirby。
831 柯尼希-艾盖尔瓦里定理
柯尼希-艾盖尔瓦里定理König-Egerváry Theorem定理1假设K为没有孤立顶点的简单图G的任意一个点覆盖集，M为G的任意一个匹配，则 |M|≤|K|。
特别是ν(G) ≤β(G)推论假设K为没有孤立顶点的简单图G的任意一个点覆盖集，M为G的任意一个匹配，若|M|=|K|，则M是一个最大匹配，K 是一个最小覆盖。
证明由|M| ≤ν(G) ≤β(G) ≤ |K|即得定理2（柯尼希-艾盖尔瓦里定理，König-Egerváry theorem）二部图中最大匹配的边数等于最小点覆盖数的顶点数。
K3?证明假设M是一个最大匹配，V *为G的一个最小点覆盖集。
(1) 由定理1有|M| ≤ |V *|(2) 下面证明|M| ≥ |V *|综合(1)、(2)可得|M| = |V *|X cx1x3x2x4x5y1y4y3y1Y5Y dX cx1x3x2x4x5y1y4y3y1Y5Y dY Y d X cX cx1x3x2x4x5y1y4y3y1Y5Y d(2) 下面证明 |M | ≥ |V *|令 X c =V *∩X ，Y d =V *∩Y ，c =|X c |，d =|Y d |考虑顶点为 X c ∪(Y -Y d ) 的 G 的导出子图 G ’，它也是二部图，其中存在饱和 X c 中每个顶点的匹配 M 1同理，顶点为 (X -X c )∪Y d 的 G 的导出子图中存在饱和 Y d 中每个顶点的匹配 M 2 易见 M 1∪M 2 也是 G 的一个匹配，于是 |M |≥|M 1∪M 2|=|M 1|+|M 2|=|X c |+|Y d |=|V *|柯尼希-艾盖尔瓦里定理还可以从另一个角度来描述：所谓布尔矩阵的覆盖，是指选择了矩阵中某些行与列，这些行与列包含了矩阵中的所有非零元“1”。
称能覆盖全部非零元的最少行、列数之和为最小覆盖数，记为S对于任一个m n布尔矩阵中，如果盖住了全部m行或全部n列，就会盖住全部非零元。
奥斯特洛格拉德斯基公式
奥斯特洛格拉德斯基公式奥斯特洛格拉德斯基公式（Ostrogradsky’s formula），又称高斯公式（Gauss’s theorem）或散度定理（divergence theorem），是数学中的一个重要定理，用于计算矢量场的散度。
奥斯特洛格拉德斯基公式是由乌克兰数学家米哈伊尔·奥斯特洛格拉德斯基（Mikhail Vasilyevich Ostrogradsky）在19世纪初发现并证明的。
这个公式在数学物理、流体力学、电磁学等领域中具有广泛的应用。
散度是矢量场的一个重要概念，用于描述矢量场在某一点的流出或流入程度。
在三维空间中，给定一个矢量场F(x,y,z)=(P(x,y,z),Q(x,y,z),R(x,y,z))，其中P、Q、R是关于x、y、z的连续函数。
我们定义矢量场F的散度为div(F)=∂P/∂x+∂Q/∂y+∂R/∂z。
奥斯特洛格拉德斯基公式给出了一个计算矢量场散度的重要方法。
具体来说，该公式表明，对于一个空间闭合曲面S，其边界为曲线C，如果矢量场F(x,y,z)在曲面S上连续可微，并且C是一个简单、光滑的曲线，那么曲线C的正向与曲面S的法向量n的夹角为θ，我们有以下公式：∬S F·dS = ∭V div(F)dV其中，∬S F·dS表示矢量场F在曲面S上的通量，即矢量场F通过曲面S的流出或流入的总量；∭V div(F)dV表示矢量场F 的散度在空间闭合曲面S所围成的体积V内的总和。
这个公式的意义在于，它将计算矢量场散度的问题转化为计算矢量场在曲面上的通量的问题。
通量的计算相对容易，因为我们可以通过对曲面进行参数化，将曲面上的积分转化为参数空间上的积分。
奥斯特洛格拉德斯基公式的证明可以通过将空间划分为无穷小的立方体，并利用极限的思想进行推导。
具体来说，我们可以将空间划分为无穷小的立方体，然后对每个立方体应用高斯公式，再对所有立方体的贡献进行求和，通过取极限，便可得到奥斯特洛格拉德斯基公式。
非对易几何导论
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§2 Riemann球 面上的谱三元组 (spectral triples) §2.1 线丛和旋量丛 . . . . . . . . . . . . . . . . §2.2 球面上的Dirac算子 . . . . . . . . . . . . . §2.3 旋量调和函数(spinor harmonics)和D / 的谱 §2.4 挠曲的旋量模(module) . . . . . . . . . . . §2.5 一个可约的谱三元组 . . . . . . . . . . . . §3 实 谱 三 元 组：公理基础 §3.1 数据集合 . . . . . . . . . . . §3.2 无穷小与维数 . . . . . . . . . §3.3 阶为一(order-one)条件 . . . . §3.4 代数的光滑性 . . . . . . . . . §3.5 Hochschild闭链(cycle)和定向 . §3.6 K -闭链的有限性 . . . . . . . §3.7 Poincar´ e对偶和K -理论 . . . . §3.8 实结构 . . . . . . . . . . . . . §4 非 对 易 环 面上的几何 §4.1 Weyl算子的代数 . §4.2 非对易环面的代数 §4.3 非对易环面纲要 . . §4.4 环面上的一族几何
庞加莱猜想前言
庞加莱猜想-前言Wir m\"ussen wissen! Wir werden wissen!(我们必须知道！我们必将知道！)—— David Hilbert两年前科学版举行过一次版聚，我报告了低维拓扑里面的一些问题和进展，其中有一半篇幅是关于Poincar\'e 猜想。
版聚后，flyleaf 要求大家回去后把自己所讲的内容发在版上。
当时我甚至已经开始写了一两段，但后来又搁置了。
主要是因为自己对于低维拓扑还是一个门外汉，写出来的东西难免有疏漏之处，不敢妄下笔。
两年过去，我对低维拓扑这门学科的了解比原先多了，说话的底气也就比原先足了。
另外，由于Clay 研究所的百万巨赏，近年来Poincar\'e 猜想频频在媒体上曝光；而且Perelman 最近的工作使数学家们有理由相信我们已经充分接近于这一猜想的最后解决。
所以大概会有很多人对Poincar\'e 猜想的来龙去脉感兴趣，我也好借机一偿两年来的宿愿。
现代科学的高速发展使各学科之间的鸿沟加大，不同学科之间难以互相理解，所以非数学专业的读者在阅读本文时可能会遇到一些困难。
但限于篇幅和文章的形式，我也不可能对很多东西详细解释。
一些最基本的拓扑概念如“流形”，我将在本文的附录中解释。
还有一些“同调群”、“基本群”之类的名词，读者见到时大可不去理会它们的确切含义。
我将尽量避免使用这一类的专业术语。
作者并非拓扑方面的专家，对下面要说的很多内容都是道听途说，只知其然而不知其所以然；作者更不善于写作，写出来的东东总会枯燥无味，难登大雅之堂。
凡此种种，还请读者诸君海涵。
问题的由来Consid\'erons maintenant une vari\'et\'e [ferm\'ee] $V$ \`a trois dimensions ... Est-il possible que le groupe fondamental de $V$ ser\'eduise \`a la substitution identique, et que pourtant $V$ ne soit pas simplement connexe?—— Henri Poincar\'e在拓扑学家的眼里，篮球、排球和乒乓球并没有什么不同，它们都同胚于三维空间中的球面S^2. (我们把n+1维欧氏空间中到原点距离为1的点的集合记作S^n，称为n维球面(sphere)。
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ISSN1472-2739(on-line)1472-2747(printed)369Algebraic&GeometricTopologyATGVolume5(2005)369–378
Published:30April2005
AllintegralslopescanbeSeifertfiberedslopesforhyperbolicknots
KimihikoMotegiHyun-JongSong
AbstractWhichslopescanorcannotappearasSeifertfiberedslopesforhyperbolicknotsinthe3-sphereS3?Itisconjecturedthatifr-surgeryonahyperbolicknotinS3yieldsaSeifertfiberspace,thenrisaninteger.Weshowthatforeachintegern∈Z,thereexistsatunnelnumberone,hyperbolicknotKninS3suchthatn-surgeryonKnproducesasmallSeifertfiberspace.
AMSClassification57M25,57M50KeywordsDehnsurgery,hyperbolicknot,Seifertfiberspace,surgeryslopes
ThispaperisdedicatedtoDonaldM.Davisontheoccasionofhis60thbirthday.
1IntroductionLetKbeaknotinthe3-sphereS3withatubularneighborhoodN(K).ThenthesetofslopesforK(i.e.,∂N(K)-isotopyclassesofsimpleloopson∂N(K))isidentifiedwithQ∪{∞}usingpreferredmeridian-longitudepairsothatameridiancorrespondsto∞.Aslopeγissaidtobeintegralifarepresentativeofγintersectsameridianexactlyonce,inotherwords,γcorrespondstoanintegerundertheaboveidentification.Inthefollowing,wedenoteby(K;γ)the3-manifoldobtainedfromS3byDehnsurgeryonaknotKwithslopeγ,i.e.,byattachingasolidtorustoS3−intN(K)insuchawaythatγboundsameridiandiskofthefilledsolidtorus.Ifγcorrespondstor∈Q∪{∞},thenweidentifyγandrandwrite(K;r)for(K;γ).
WedenotebyLthesetoflensslopes{r∈Q|∃hyperbolicknotK⊂S3suchthat(K;r)isalensspace},whereS3andS2×S1arealsoconsideredaslensspaces.Thenthecyclicsurgerytheorem[7]impliesthatL⊂Z.A
cGeometry&TopologyPublications370KimihikoMotegiandHyun-JongSong
resultofGabai[10,Corollary8.3]showsthat0∈L,aresultofGordonandLuecke[14]showsthat±1∈L.In[19]KronheimerandMrowkaprovethat±2∈L.Furthermore,aresultofKronheimer,Mrowka,Ozsv´athandSzab´o[20]impliesthat±3,±4∈L.Besides,Berge[4,TableofLensSpaces]suggeststhatifn∈L,then|n|≥18andnoteveryintegernwith|n|≥18appearsinL.FintushelandStern[9]hadshownthat18-surgeryonthe(−2,3,7)pretzelknotyieldsalensspace.
Whichslope(rationalnumber)canorcannotappearinthesetofSeifertfiberedslopesS={r∈Q|∃hyperbolicknotK⊂S3suchthat(K;r)isSeifertfibered}?ItisconjecturedthatS⊂Z[12].
Thepurposeofthispaperistoprove:
Theorem1.1Foreachintegern∈Z,thereexistsatunnelnumberone,hyperbolicknotKninS3suchthat(Kn;n)isasmallSeifertfiberspace(i.e.,aSeifertfiberspaceoverS2withexactlythreeexceptionalfibers).
RemarkSinceKnhastunnelnumberone,itisembeddedinagenustwoHeegaardsurfaceofS3andstronglyinvertible[26,Lemma5].See[22,Question3.1].
Theorem1.1,togetherwiththepreviousknownresults,shows:Corollary1.2LZ⊂S.Remarks(1)ForthesetofreducingslopesR={r∈Q|∃hyperbolicknotK⊂S3suchthat(K;r)isreducible},GordonandLuecke[13]haveshownthatR⊂Z.Infact,thecablingconjecture[11]assertsthatR=∅.
(2)ForthesetoftoroidalslopesT={r∈Q|∃hyperbolicknotK⊂S3suchthat(K;r)istoroidal},GordonandLuecke[15]haveshownthatT⊂Z/2(integersorhalfintegers).In[28],TeragaitoshowsthatZ⊂TandconjecturesthatTZ/2.
AcknowledgementsWewouldliketothanktherefereeforcarefulreadingandusefulcomments.ThefirstauthorwaspartiallysupportedbyGrant-in-AidforScientificResearch(No.15540095),TheMinistryofEducation,Culture,Sports,ScienceandTech-nology,Japan.
Algebraic&GeometricTopology,Volume5(2005)AllintegralslopescanbeSeifertfiberedslopesforhyperbolicknots3712HyperbolicknotswithSeifertfiberedsurgeriesOurconstructionisbasedonanexampleofalongitudinalSeifertfiberedsurgerygivenin[17].
Letk∪cbea2-bridgelinkgiveninFigure1,andletKnbeaknotobtainedfromkby1
2,−1,3n−11372KimihikoMotegiandHyun-JongSong
3n11n+4
-121
t1t
2
ck1
Lb-1213n11n+4----
--
-12-1
21-1-
3n11n+4--3n11
n+4--
3n11n+4--
3n11
n+4--
-12-121-1-
stronginversion
quotient
isotopyisotopyisotopyisotopyFigure3kintheupstairscorrespondto−1−n+4and1-untanglesurgeryonb′inthedownstairs,whereanr-untanglesurgeryisareplacementof1
5,−24n−15).Since(Kn;n)isthedoublebranchedcoverofS3branchedovertheMontesinos
Algebraic&GeometricTopology,Volume5(2005)