Representation theory of artin algebras

第十二届全国代数学术会议大会报告摘要Classical Yang-Baxter equation,its extensions and some related algebraicstructures白承铭南开大学Wefirst give a brief introduction to the classical Yang-Baxter equation which emphasizes the relationship between the tensor and operator forms.Then we give two approaches to extend the classical Yang-Baxter equation,motivated by the study of different topics like integrable systems,Rota-Baxter algebras and Lie bialgebras.Moreover,there are some interesting algebraic structures behind these two approaches.Representation type of algebras:Yesterday,today and tomorrow韩阳中国科学院数学与系统科学研究院Email:***********.cnOne task in representation theory of algebras is to classify all representations of an algebra.Before this,one needs to judge whether it is hopeful for an algebra to do so or not,namely,to determine the representation type of an algebra.In this survey,some concepts,results,ideas,methods,and problems on representation type of algebras are introduced.群环的代数K理论唐国平中科院研究生院数学科学学院报告将分为三个部分：1）本质循环群(具有指数有限的循环子群的群)的整群环的K理论。

山西师范大学本科毕业论文矩阵可逆的若干判别方法郭晓平姓名院系数学与计算机科学学院专业数学与应用数学0701班班级学号0751010139指导教师宋蔷薇答辩日期成绩矩阵可逆的若干判别方法内容摘要对线性代数和代数学而言，矩阵是一个主要研究对象和重要工具，其中可逆矩阵又是矩阵运算理论的整体不可或缺的一部分。

在矩阵理论，可逆矩阵所占的地位是不可替代的，在坐标轴旋转变换公式的矩阵表示、线性变换、线性方程组等理论研究中，它均有重要意义。

而且由于在许多有关数学、物理，经济的实际问题中，常常需要通过建立合适的数学模型化为线性代数和代数学等的问题，因此可逆矩阵也是解决实际问题比较常用的工具之一。

鉴于可逆矩阵具有重要的理论和实践意义，研究矩阵可逆的判别方法也就相当有必要了。

本文结合所学知识并查阅相关资料，系统地整理并归纳总结了十一种矩阵可逆的判别方法及其证明过程。

其中，可逆矩阵判别方法主要包括定义判别法、伴随矩阵判别法、初等变换判别法、线性方程组法、矩阵向量组的秩判别法等。

另外，本文还给出了十种特殊矩阵可逆性的相关结论，最后针对这些判别方法选取了典型的例题，以便我们更好的掌握矩阵可逆的判别方法。

【关键词】矩阵逆矩阵初等变换伴随矩阵线性方程组Some Methods for Judging Invertible MatrixAbstractThe matrix is a main research subject and an important tool in linear algebra and algebra. The invertible matrix, which plays the role of the invertible number in rational numbers, is an essential part of the matrix theory. The very important status ,which the invertible matrix holds in the matrix theory ,can not be replaced. It has the important meaning for solving linear equations, linear transformation theory problems, rotating coordinate transform formula of matrix representation theory. And In solving practical problems such as mathematics, physics, economic and other fields, it is often need to establish proper mathematical models into linear algebra and algebra issues. Therefore it also is a commonly used tool, which is widely applied in practical problem. In view of the fact that the invertible matrix has important significance in both theory and practice, the study of judging invertible matrix is quite necessary.Through combining with my knowledge, referring to the relevant materials, this paper systematically organizes and summarizes eleven kinds of methods for judging invertible matrix ,which contain definition method, the adjoin matrix method, elementary transformation method, linear equations method and so on ,and the proof process. This paper also gives ten special matrix invertible conclusions. Finally, this paper selects several typical examples aiming at these discriminate methods, so that we know the methods for judging invertible matrix.【Key Words】matrix inverse matrix elementary transformation adjoin matrix Linear equations目录一、引言 (01)二、预备知识 (01)（一）基本概念 (01)（二）可逆矩阵的性质 (01)三、矩阵可逆的若干判别方法 (02)（一）定义判别法 (02)（二）行列式判别法 (02)（三）秩判别法 (02)（四）伴随矩阵判别法 (02)（五）初等变换判别法 (02)（六）初等矩阵判别法 (02)（七）矩阵向量组的秩判别法法 (03)（八）线性方程组判别法 (03)（九）标准形判别法 (04)（十）多项式判别法 (04)（十一）特征值判别法 (05)四、十种常见矩阵的可逆性 (05)五、矩阵可逆判别方法的实例 (07)六、小结 (11)参考文献 (11)致谢 (12)矩阵可逆的若干判别方法学生姓名：郭晓平 指导老师：宋蔷薇一、引言在矩阵的乘法运算中，就像理数的倒数一样，可逆矩阵是构成矩阵运算理论体系不可或缺的一部分。

美国数学本科生、研究生基础课程参考书目在网上找书的时候恰好看到这个，看着觉得的确是经典书目大全，贴在这里供学弟学妹们参考：）其中所谓第几学年云云，各校要求不同，像我所在的学校，一般学生第一年选三到四门基础课（代数、分析、几何三大类中至少各挑一门），学年末进行qualifying笔试。

第二年开始选自己喜爱方向的高级课程，并通过qualifying口试。

第三年开始做research，并通过第二语言考试（法语或德语或俄语，一般人都选法语，因为代数几何经典大作都是法语的）. 而Princeton 就没有基础课，只有seminar类型的课。

第一学年几何与拓扑：1、James R. Munkres, Topology：较新的拓扑学的教材适用于本科高年级或研究生一级；2、Basic Topology by Armstrong：本科生拓扑学教材；3、Kelley, General Topology：一般拓扑学的经典教材，不过观点较老；4、Willard, General Topology：一般拓扑学新的经典教材；5、Glen Bredon, Topology and geometry：研究生一年级的拓扑、几何教材；6、Introduction to Topological Manifolds by John M. Lee：研究生一年级的拓扑、几何教材，是一本新书；7、from calculus to cohomology by Madsen：很好的本科生代数拓扑、微分流形教材。

代数：1、Abstract Algebra Dummit：最好的本科代数学参考书，标准的研究生一年级代数材；2、Algebra Lang：标准的研究生一、二年级代数教材，难度很高，适合作参考书；3、Algebra Hungerford：标准的研究生一年级代数教材，适合作参考书；4、Algebra M,Artin：标准的本科生代数教材；5、Advanced Modern Algebra by Rotman：较新的研究生代数教材，很全面；6、Algebra：a graduate course by Isaacs：较新的研究生代数教材；7、Basic algebra Vol I&II by Jacobson：经典的代数学全面参考书，适合研究生参考。

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。

研究生课程《Artin代数表示论》教学大纲课程编号：Math2117课程名称：Artin代数表示论英文名称：Representation Theoy of Artin Algebras开课单位：数学科学学院开课学期：秋课内学时：36教学方式：讲授适用专业及层次：代数方向硕士考核方式：考试预修课程：高等代数、近世代数一、教学目标与要求本课程较全面、系统地介绍Artin代数与有限维代数的表示理论，重点是箭图与路代数，箭图上的表示，Auslander-Reiten理论，倾斜理论等，难点是理解Gabriel定理、箭图上的表示、几乎可列序列，Auslander-Reiten箭图，倾斜模等。

通过本课程中基本概念和基本定理的阐述和论证，培养研究生的抽象思维与逻辑推理能力，提高研究生的数学素养。

在重视数学论证的同时，强调Artin代数表示理论研究的研究方法。

通过本课程的学习，要求研究生掌握Artin代数表示论的基本理论和方法，为学习后开展科学研究打好基础。

二、课程内容与学时分配Chapter 1. Quivers and Algebras （8学时）Quivers and path algebrasAdmissible ideals and quotients of path algebrasThe quiver of a finite dimensional algebraChapter 2. Representations and Modules （10学时）2.1 Representations of bound quivers2.2 The simple, projective and injective modules2.3 The dimension vector of a module and the Euler characteristicChapter 3. Auslander-Reiten Theory （10学时）3.1 Irreducible morphisms and almost split sequences3.2 Auslander-Reiten translations3.3 Existence of almost split sequences3.4 Auslander-Reiten quiver of an algebraChapter 4. Titling Theory （8学时）4.1 Torsion pairs4.2 Partial tilting modules and tilting modules4.3 Tilting theorem of Brenner and Butler4.4 Torison pairs induced by tilting modules三、教材I. Assem, D. Simon and A. Skowroński, Elements of the Representation Theory of Associative Algebras, 1: Techniques of Representation theory, London Math. Soc. Stud. Texts 65, Cambridge, New York, 2006.主要参考书M. Auslander, I. Reiten and S. SmalØ, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, Cambridge, New York, 1995.。

Algebraic Graph TheoryChris Godsil(University of Waterloo),Mike Newman(University of Ottawa)April25–291Overview of the FieldAlgebraic graph theory comprises both the study of algebraic objects arising in connection with graphs, for example,automorphism groups of graphs along with the use of algebraic tools to establish interesting properties of combinatorial objects.One of the oldest themes in the area is the investigation of the relation between properties of a graph and the spectrum of its adjacency matrix.A central topic and important source of tools is the theory of association schemes.An association scheme is,roughly speaking,a collection of graphs on a common vertex set whichfit together in a highly regular fashion.These arise regularly in connection with extremal structures:such structures often have an unex-pected degree of regularity and,because of this,often give rise to an association scheme.This in turn leads to a semisimple commutative algebra and the representation theory of this algebra provides useful restrictions on the underlying combinatorial object.Thus in coding theory we look for codes that are as large as possible, since such codes are most effective in transmitting information over noisy channels.The theory of association schemes provides the most effective means for determining just how large is actually possible;this theory rests on Delsarte’s thesis[4],which showed how to use schemes to translate the problem into a question that be solved by linear programming.2Recent Developments and Open ProblemsBrouwer,Haemers and Cioabˇa have recently shown how information on the spectrum of a graph can be used to proved that certain classes of graphs must contain perfect matchings.Brouwer and others have also investigated the connectivity of strongly-regular and distance-regular graphs.This is an old question,but much remains to be done.Recently Brouwer and Koolen[2]proved that the vertex connectivity of a distance-regular graph is equal to its valency.Haemers and Van Dam have worked on extensively on the question of which graphs are characterized by the spectrum of their adjacency matrix.They consider both general graphs and special classes,such as distance-regular graphs.One very significant and unexpected outcome of this work was the construction,by Koolen and Van Dam[10],of a new family of distance-regular graphs with the same parameters as the Grassmann graphs.(The vertices of these graphs are the k-dimensional subspaces of a vector space of dimension v over thefinitefield GF(q);two vertices are adjacent if their intersection has dimension k1.The graphs are q-analog of the Johnson graphs,which play a role in design theory.)These graphs showed that the widely held belief that we knew all distance-regular graphs of“large diameter”was false,and they indicate that the classification of distance-regular graphs will be more complex(and more interesting?)than we expected.1Association schemes have long been applied to problems in extremal set theory and coding theory.In his(very)recent thesis,Vanhove[14]has demonstrated that they can also provide many interesting results in finite geometry.Recent work by Schrijver and others[13]showed how schemes could used in combination with semidef-inite programming to provide significant improvements to the best known bounds.However these methods are difficult to use,we do not yet have a feel for we might most usefully apply them and their underlying theory is imperfectly understood.Work in Quantum Information theory is leading to a wide range of questions which can be successfully studied using ideas and tools from Algebraic Graph Theory.Methods fromfinite geometry provide the most effective means of constructing mutually unbiased bases,which play a role in quantum information theory and in certain cryptographic protocols.One important question is to determine the maximum size of a set of mutually unbiased bases in d-dimensional complex space.If d is a prime power the geometric methods just mentioned provide sets of size d+1,which is the largest possible.But if d is twice an odd integer then in most cases no set larger than three has been found.Whether larger sets exist is an important open problem. 3Presentation HighlightsThe talks mostlyfitted into one of four areas,which we discuss separately.3.1SpectraWillem Haemers spoke on universal adjacency matrices with only two distinct eigenvalues.Such matrices are linear combinations of I,J,D and A(where D is the diagonal matrix of vertex degrees and A the usual adjacency matrix).Any matrix usually considered in spectral graph theory has this form,but Willem is considering these matrices in general.His talk focussed on the graphs for which some universal adjacency matrix has only two eigenvalues.With Omidi he has proved that such a graph must either be strong(its Seidel matrix has only two eigenvalues)or it has exactly two different vertex degrees and the subgraph induced by the vertices of a given degree must be regular.Brouwer formulated a conjecture on the minimum size of a subset S of the vertices of a strongly-regular graph X such that no component of X\S was a single vertex.Cioabˇa spoke on his recent work with Jack Koolen on this conjecture.They proved that it is false,and there are four infinite families of counterexamples.3.2PhysicsAs noted above,algebraic graph theory has many applications and potential applications to problems in quantum computing,although the connection has become apparent only very recently.A number of talks were related to this connection.One important problem in quantum computing is whether there is a quantum algorithm for the graph isomorphism problem that would be faster than the classical approaches.Currently the situation is quite open.Martin Roetteler’s talk described recent work[1]on this problem.For our workshop’s viewpoint,one surprising feature is that the work made use of the Bose-Mesner algebra of a related association scheme; this connection had not been made before.Severini discussed quantum applications of what is known as the Lov´a sz theta-function of a graph.This function can be viewed as an eigenvalue bound and is closely related to both the LP bound of Delsarte and the Delsarte-Hoffman bound on the size of an independent set in a regular graph.Severini’s work shows that Lov´a sz’s theta-function provides a bound on the capacity of a certain channel arising in quantum communication theoryWork in quantum information theory has lead to interest in complex Hadamard matrices—these are d×d complex matrices H such that all entries of H have the same absolute value and HH∗=dI.Both Chan and Sz¨o ll˝o si dealt with these in their talks.Aidan Roy spoke on complex spherical designs.Real spherical designs were much studied by Seidel and his coworkers,because of their many applications in combinatorics and other areas.The complex case languished because there were no apparent applications,but now we have learnt that these manifest them-selves in quantum information theory under acronyms such as MUBs and SIC-POVMs.Roy’s talk focussedon a recent 45page paper with Suda [12],where (among other things)they showed that extremal complex designs gave rise to association schemes.One feature of this work is that the matrices in their schemes are not symmetric,which is surprising because we have very few interesting examples of non-symmetric schemes that do not arise as conjugacy class schemes of finite groups.3.3Extremal Set TheoryCoherent configurations are a non-commutative extension of association schemes.They have played a sig-nificant role in work on the graph isomorphism problem but,in comparison with association schemes,they have provided much less information about interesting extremal structures.The work presented by Hobart and Williford may improve matters,since they have been able to extend and use some of the standard bounds from the theory of schemes.Delsarte [4]showed how association schemes could be used to derive linear programs,whose values provided strong upper bounds on the size of codes.Association schemes have both a combinatorial structure and an algebraic structure and these two structures are in some sense dual to one another.In Delsarte’s work,both the combinatorial and the algebraic structure had a natural linear ordering (the schemes are both metric and cometric)and this played an important role in his work.Martin explained how this linearity constraint could be relaxed.This work is important since it could lead to new bounds,and also provide a better understanding of duality.One of Rick Wilson’s many important contributions to combinatorics was his use of association schemes to prove a sharp form of the Erd˝o s-Ko-Rado theorem [15].The Erd˝o s-Ko-Rado theorem itself ([5])can certainly be called a seminal result,and by now there are many analogs and extensions of it which have been derived by a range of methods.More recently it has been realized that most of these extensions can be derived in a very natural way using the theory of association schemes.Karen Meagher presented recent joint work (with Godsil,and with Spiga,[8,11])on the case where the subsets in the Erd˝o s-Ko-Rado theorem are replaced by permutations.It has long been known that there is an interesting association scheme on permutations,but this scheme is much less manageable than the schemes used by Delsarte and,prior to the work presented by Meagher,no useful combinatorial information had been obtained from it.Chowdhury presented her recent work on a conjecture of Frankl and F¨u redi.This concerns families F of m -subsets of a set X such that any two distinct elements of have exactly λelements in common.Frankl and F¨u redi conjectured that the m -sets in any such family contain at least m 2 pairs of elements of X .Chowdhury verified this conjecture in a number of cases;she used classical combinatorial techniques and it remains to see whether algebraic methods can yield any leverage in problems of this type.3.4Finite GeometryEric Moorhouse spoke on questions concerning automorphism groups of projective planes,focussing on connections between the finite and infinite case.Thus for a group acting on a finite plane,the number of orbits on points must be equal to the number of orbits on lines.It is not known if this must be true for planes of infinite order.Is there an infinite plane such that for each positive integer k ,the automorphism group has only finitely many orbits on k -tuples?This question is open even for k =4.Simeon Ball considered the structure of subsets S of a k -dimensional vector space over a field of order q such that each d -subset of S is a basis.The canonical examples arise by adding a point at infinity to the point set of a rational normal curve.These sets arise in coding theory as maximum distance separable codes and in matroid theory,in the study of the representability of uniform matroids (to mention just two applications).It is conjectured that,if k ≤q −1then |S |≤q +1unless q is even and k =3or k =q −1,in which case |S |≤q +2.Simeon presented a proof of this theorem when q is a prime and commented on the general case.He developed a connection to Segre’s classical characterization of conics in planes of odd order,as sets of q +1points such that no three are collinear.There are many analogs between finite geometry and extremal set theory;questions about the geometry of subspaces can often be viewed as q -analogs of questions in extremal set theory.So the EKR-problem,which concerns characterizations of intersecting families of k -subsets of a fixed set,leads naturally to a study of intersecting families of k -subspaces of a finite vector space.In terms of association schemes this means we move from the Johnson scheme to the Grassmann scheme.This is fairly well understood,with thebasic results obtained by Frankl and Wilson[6].But infinite geometry,polar spaces form an important topic. Roughly speaking the object here is to study the families of subspaces that are isotropic relative to some form, for example the subspaces that lie on a smooth quadric.In group theoretic terms we are now dealing with symplectic,orthogonal and unitary groups.There are related association schemes on the isotropic subspaces of maximum dimension.Vanhove spoke on important work from his Ph.D.thesis,where he investigated the appropriate versions of the EKR problem in these schemes.4Outcome of the MeetingIt is too early to offer much in the way of concrete evidence of impact.Matt DeV os observed that a conjecture of Brouwer on the vertex connectivity of graphs in an association scheme was wrong,in a quite simple way. This indicates that the question is more complex than expected,and quite possibly more interesting.That this observation was made testifies to the scope of the meeting.On a broader level,one of the successes of the meeting was the wide variety of seemingly disparate topics that were able to come together;the ideas of algebraic graph theory touch a number of things that would at first glance seem neither algebraic nor graph theoretical.There was a lively interaction between researchers from different domains.The proportion of post-docs and graduate students was relatively high.This had a positive impact on the level of excitement and interaction at the meeting.The combination of expert and beginning researchers created a lively atmosphere for mathematical discussion.References[1]A.Ambainis,L.Magnin,M.Roetteler,J.Roland.Symmetry-assisted adversaries for quantum state gen-eration,arXiv1012.2112,35pp.[2]A.E.Brouwer,J.H.Koolen.The vertex connectivity of a distance-regular graph.European bina-torics30(2009),668–673.[3]A.E.Brouwer,D.M.Mesner.The connectivity of strongly regular graphs.European binatorics,6(1985),215–216.[4]P.Delsarte.An algebraic approach to the association schemes of coding theory.Philips Res.Rep.Suppl.,(10):vi+97,1973.[5]P.Erd˝o s,C.Ko,R.Rado.Intersection theorems for systems offinite sets.Quart.J.Math.Oxford Ser.(2),12(1961),313–320.[6]P.Frankl,R.M.Wilson.The Erd˝o s-Ko-Rado theorem for vector binatorial Theory,SeriesA,43(1986),228–236.[7]D.Gijswijt,A.Schrijver,H.Tanaka.New upper bounds for nonbinary codes based on the Terwilligeralgebra and semidefinite binatorial Theory,Series A,113(2006),1719–1731. [8]C.D.Godsil,K.Meagher.A new proof of the Erd˝o s-Ko-Rado theorem for intersecting families of per-mutations.arXiv0710.2109,18pp.[9]C.D.Godsil,G.F.Royle.Algebraic Graph Theory,Springer-Verlag,(New York),2001.[10]J.H.Koolen,E.R.van Dam.A new family of distance-regular graphs with unbounded diameter.Inven-tiones Mathematicae,162(2005),189-193.[11]K.Meagher,P.Spiga.An Erdos-Ko-Rado theorem for the derangement graph of PGL(2,q)acting onthe projective line.arXiv0910.3193,17pp.[12]A.P.Roy,plex spherical Codes and designs,(2011),arXiv1104.4692,45pp.[13]A.Schrijver.New code upper bounds from the Terwilliger algebra and semidefinite programming.IEEETransactions on Information Theory51(2005),2859–2866.[14]F.Vanhove.Incidence geometry from an algebraic graph theory point of view.Ph.D.Thesis,Gent2011.[15]R.M.Wilson.The exact bound in the Erds-Ko-Rado binatorica,4(1984),247–257.。

(0,2) 插值||(0,2) interpolation0#||zero-sharp; 读作零井或零开。

0+||zero-dagger; 读作零正。

1-因子||1-factor3-流形||3-manifold; 又称“三维流形”。

AIC准则||AIC criterion, Akaike information criterionAp 权||Ap-weightA稳定性||A-stability, absolute stabilityA最优设计||A-optimal designBCH 码||BCH code, Bose-Chaudhuri-Hocquenghem codeBIC准则||BIC criterion, Bayesian modification of the AICBMOA函数||analytic function of bounded mean oscillation; 全称“有界平均振动解析函数”。

BMO鞅||BMO martingaleBSD猜想||Birch and Swinnerton-Dyer conjecture; 全称“伯奇与斯温纳顿－戴尔猜想”。

B样条||B-splineC*代数||C*-algebra; 读作“C星代数”。

C0 类函数||function of class C0; 又称“连续函数类”。

CA T准则||CAT criterion, criterion for autoregressiveCM域||CM fieldCN 群||CN-groupCW 复形的同调||homology of CW complexCW复形||CW complexCW复形的同伦群||homotopy group of CW complexesCW剖分||CW decompositionCn 类函数||function of class Cn; 又称“n次连续可微函数类”。

Cp统计量||Cp-statisticC。