Take a dip in analytical math (Zhang Kaijun)

It is said that the latest reform of the college entrance examination of the math test occupies an important share, which reminds of why the "society like Mathematics" this education problem. Since mathematics by the public "attention" to such a point, simply let us look at the advanced mathematical Kingdom of some of the first-tier city style. As long as bold and adhere to ensure that there will be a bit of harvest.

In the face of the mathematics of the peak, in fact, all the mathematical people are mathematical efforts in the process of infinite small amount. For the great mathematicians who let us worship and respect, when compared to the vast number of mathematical colleagues, they are the infinite amount of mathematical effort. To become an infinite number of mathematical talents, the first thing is to mentally remove any "celebrity unresolved, their hopeless" pessimistic research psychological barriers, so that they are in a completely unconstrained energy Free State, and then may be true to verify their own mathematical value. Non-math people can actually open up their math hearts and explore their mathematical potentials similarly. The macro-synthesis of micro-elements such as their potentials, interests, abilities, aspirations, perseverance, diligence and self-confidence is of vital importance in the education of mathematical talents. After the baptism of mathematics education psychology, the vast number of mathematics enthusiasts to enter the modern core analysis of mathematics in the ocean swimming, without losing it as a quick way to improve the mathematical literacy.

　　

The motive of writing this article is to inspire the idea of education, not the reason of the author's knowledge. Functional analysis, harmonic analysis, complex analysis, stochastic analysis, partial differential equations and large-scale analysis, such as the core of the knowledge of the mathematical science is enough to allow the generation of mathematical people to pursue forever, so personal force is a micro-gravity just. Although the mathematical male peak is difficult to shake, but through the education telescope but can appreciate its geometry education appearance, this perhaps is this article the small aspect.

　　

　　

(i) Functional analysis

Functional analysis is one of the important branches of modern analytical mathematics, and its far-reaching theoretical system and extensive application value have exerted a great influence on modern analytical mathematics and even modern science and technology fields. The functional analysis courses in the undergraduate stage are mainly based on the normed linear space in linear functional analysis and the theory of bounded operator on the basis of some basic content. The linear functional analysis of the postgraduate stage mainly introduces the contents of compact operator and Fredholm operator, Banach algebra, non-boundary operator, linear operator semigroup, generalized function, Hilbert-schmidt operator and Trace class operator. The curriculum of nonlinear functional analysis in postgraduate stage generally teaches the basic contents of calculus, implicit function theorem and bifurcation problem, topological degree, monotone operator and variational method in Banach space. The main research directions of functional analysis are: linear operator spectrum theory, function space, Banach space geometry, operator number, non-commutative geometry, applied functional analysis and related research directions of nonlinear functional analysis.

　　

Functional analysis is a mathematical abstract extension process from "ground" to "sky" through mathematical analysis, Advanced algebra and space analytic geometry. The geometric theory of finite dimensional space and the mapping theory from finite dimension space to finite dimension space are the main contents of the one or two grade of university mathematics majors. If we consider only the operational properties of linear mappings, it is linear algebra. If we consider the continuity and smoothness of nonlinear mappings, it is calculus. If the distance concept of finite dimension space is extended to infinite dimension space, then the continuity and smoothness of the corresponding linear mapping and nonlinear mapping are considered, then the boundary of functional analysis is naturally reached. Mathematical analysis, many conclusions of higher algebra and analytic geometry have the corresponding generalization conclusions at the functional analysis level. After noticing this, you can return from "heaven" to "ground". The idea of replacing a finite dimension with an infinite dimension and a European metric to an abstract metric is the same idea, but the phenomenon is a functional analysis of the fusion of topology, algebra, geometry and analysis. The blending of mathematical thinking methods of analysis, algebra, Geometry and topology is the source of the strength of functional analysis and growth. Functional analysis has become one of the necessary tools for modern analytical mathematics.

　　

The Fields Prize winner J. Bourgain,a.connes,w. Timothy gowers,a. Grothendieck, L. Schwartz and Wolf Prize winner I. The famous mathematicians, such as M. Gelfand,m G. Krein, have made great achievements in the field of functional analysis.

　　

The bibliography of functional analysis is recommended as follows:

(1) M. Reed, B. Simon, Methods of Modern mathematical Physics, I–iv, 1970 ' s.

(2) K. Yosida, functional analysis, 1980.

(3) J. Barros-neto, an Introduction to the theory of distributions,1981.

(4) Zhang Gongqing, Linyuan Canal, functional analysis handout, book, 1987.

(5) Zhang Gongqing, Guo Zheng, functional analysis Handout, next book, 1990.

(6) W. Rudin,functional analysis,1991.

(7) Alan connes,noncommutative geometry,1994.

(8) P. Lax, Functional analysis,2002.

(9) Kung-ching chang,methods in nonlinear analysis,2005.

(ii) Harmonic analysis

Harmonic analysis is one of the core fields of modern analytical mathematics, and its brilliant achievements make generations of analysts pour and struggle. According to Mr. Hua, the operation of the known function into the Fourier series is called harmonic analysis. In fact, the harmonic analysis is from the Fourier series and Fourier transformation theory of research began to grow and develop. From the point of view of physics, harmonic analysis is to represent the signal as the super-position superposition of the basic wave "harmonic". For centuries, the harmonic analysis has formed a huge discipline system, and has important and profound application in the fields of mathematics, information processing and quantum mechanics.

　　

From the point of application, the operation of effectively determining the Fourier series problem is called practical harmonic analysis. Finite harmonic analysis is the main frame of practical harmonic analysis, that is, the Fourier method of solving practical problems from limited to limited methods is the application value of finite harmonic analysis from the angle of the most appropriate number of items to be calculated by finite data. From the point of view of physics, it can be found that the uncertainty relation in quantum mechanics has the interpretation of the harmonic analysis version, that is, the Fourier transformation of the generalized function of the non-0 compactly-supported set described by the Paley–wiener theorem has no tight-set.

　　

Abstract harmonic analysis is a branch of modern mathematics in which harmonic analysis is more thorough, that is to study the theory of harmonic analysis on topological group, especially the Fourier transformation theory. The Ponteyagin duality theory of Abel Compact Group is a suitable portrayal of harmonic analysis features in modern mathematics processing. For the general non-Abel Local compact group, the harmonic analysis is closely related to the representation theory of unitary group. The Fourier transformation of the classical convolution is the property of the product of the Fourier transformation, which can be embodied by the Peter-weyl theorem of the compact group. When the group is neither Abel nor tight group, the general theory of abstract harmonic analysis is not very well-developed. For example, it is not known if the analogues of the Plancherel theorem exist at this time. However, in many special cases, it is possible to analyze some related problems by means of the infinite dimension representation technique.

　　

The following is a brief description of the harmonic analysis of the above, in order to the harmonic analysis of the direction of research and learning a little bit of convenience.

　　

Covering technology, maximal operator, calderón–zygmund decomposition, interpolation technique and singular integral operator are the basic contents of modern harmonic analysis. Covering technology is not only an important tool of measure theory, but also one of the main methods of harmonic analysis. The establishment of Hardy–littlewood maximal operator theory is closely related to the coverage technology. The h.-l maximal operator theory mainly embodies the boundedness theory of a class of nonlinear operators and can solve many important problems of modern analysis. Calderón–zygmund decomposition technology is the key method to study the real variable analysis of singular integrals, that is, to split any integrable function into "small part" and "majority", and then treat each part separately with different techniques is the essence of its thought. The singular integral operator is produced by an integral nucleus with singularity. The problem of the boundedness of singular integral operators is one of the important research problems. The theory of singular integral operators is now very rich.

　　

From the classical Fourie analysis of Fourier series and Fourier transform to hardy–littlewood maximal operator and singular integral operator, it can be regarded as a leap of harmonic analysis. Another significant leap in harmonic analysis should be the establishment and perfection of the BMO Space and-weight theory of space (Hardy space) and bounded average oscillation function. The author thinks: The Last leap of harmonic analysis is probably the effective practice problem of the solution of the world-class mathematics conjecture of the analytic discipline.

　　

The research of hardy space originates from the analysis of Fourier series and single complex variables, so far, it has rich connotation, especially the intervention of high-dimensional real method, which makes the space theory have the essential modern development. The bounded average oscillation function of the BMO Space, also known as John-nirenberg space, is in the Analytic master F. John and L. Nirenberg first studies the topological properties of the space and gives a precise definition. -Space, BMO Space and-Power theory is the three great inventions of modern harmonic analysis. C. Fefferman's main work in obtaining the Fields award is in L.. On the basis of Nirenberg work, it is found that BMO Space is the dual space of space. BMO Space has important applications in many fields of analytical mathematics and in probability seedling theory. Based on BMO Space, L. Nirenberg and H. Brezis also discovered that VMO space (vanishing average oscillation space) as a subspace of BMO Space, especially the mapping of topological degree theory to VMO space, makes topology marvel. -Weight theory plays an important role in the study of the boundedness of singular integral operators. R. R. Coifman and C. Fefferman have made important contributions to the establishment of the right-to-power theory.

　　

China's world-class mathematician Mr. Hua in the field of classical harmonic analysis has achieved world-leading results. His masterpiece "The Harmonic analysis on the typical domain of the theory of multi-complex function" has won the first prize of National Natural Science Award. The School of Harmonic analysis of Peking University has made great contributions to the training of Chinese people in the direction of harmonic analysis.

　　

The Master of harmonic analysis who won the Wolf and the Fields Awards has a. P. calderón,c fefferman,e M. stein,t Tao.

　　

The mathematical writings on harmonic analysis are recommended as follows:

　　

(1) E. M. Stein, Singular integrals and differentiability Properties of Functions, 1970.

(2) E. M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean Spaces, 1971.

(3) E. M. Stein, Harmonic analysis:real Variable Methods, orthogonality, and Oscillatory integrals, 1993.

(iii) complex analysis

In the disciplines of algebra and analysis, complex fields are important basic mathematical fertile soil. The calculus theory of complex variable function is one of the main contents of classical complex analysis. In the complex soil of the calculus, in addition to inherit the tradition, of course, there will certainly be a new world, for example, Cauchy integral theory, Weierstrass series theory and complex Riemann geometric theory is the unique theory of the plural domain. In the course of the complex function theory of the university, the conformal mapping theory, as a double-shot analytic mapping, should be one of the highlight parts of the course. The intrinsic properties of the analytic mappings are the infinite-minor, non-critical-point-preserving properties of the analytic mappings, and the open-mapping nature of open-set mappings of the very value maps. The Cauchy-riemann equation corresponding to the real part and imaginary part of the analytic map is the starting point of the partial differential equation in depth generalization of analytic mapping theory. The contents of the complex function of university undergraduate have been applied in the fields of engineering, electronic engineering and space engineering besides mathematics.

　　

Complex analysis can be divided into single complex variable function theory, multiple complex variable function theory, and analytic theory on complex manifold three important parts. Since the emergence of the single complex variable in the 19th century, the theory of the function theory of single complex variables has been perfected. With the development of several complex variable function theory has become one of the modern mainstream analytical mathematics field. The content of single complex complex analysis in postgraduate stage includes Riemann mapping theorem, boundary correspondence theorem of conformal mapping, single value theorem, generalized Schwarz lemma, conformal invariants (conformal mode and extremum length), quasi-conformal mapping, Riemann surface, Riemann-roch theorem, The single-valued theorem and the approximation theory of complex variable function are important contents. In view of the improvement of the theory of single complex change, the research trend in this field is developing in depth to the complex dynamical system direction. The multi-variable curriculum in postgraduate stage is mainly introduced in two aspects of classical multi-complex and modern multi-complex transformation. The former is a theory of high-dimensional complex space generalization in the theory of single-complex transformation: The algebraic domain in high-dimensional complex space and the theory of multi-complex function, while the latter is the corresponding theory of function theory on complex manifold. The multi-complex course is more difficult, so the student team is generally smaller. As the generalization theory of the single complex variable function theory, the multi-complex function theory is also confronted with the fundamental problems of inheritance and carrying forward. The two basic theorems of multi-complex change, which are different from the single complex, are "the Poincaré theorem that there is no full-body mapping of units in high-dimensional complex spaces into multi-cylinders in the same space, and that the existence of such an area in a high-dimensional complex space makes all pure functions in this region must be fully extended to larger regions" The Hartoge theorem. Poincaré theorem shows that the Riemann mapping theorem of a single complex with a number of dimensions greater than or equal to 2 o'clock is no longer valid. The Hartoge theorem produces the appropriate regional discriminant problem in the study of function theory in high dimensional complex space. The functions, upper homology, differential forms, Cauvhy integrals, and the basic theorems of dolbeault and de Rham are the core contents of modern multi-complex transformation.

　　

In the field of single-complex transformation and multi-complex transformation, Chinese mathematicians have achieved the world advanced level of research results. For example, the mathematicians of the Xiangqinglai School of China have made world-class research achievements in the field of the value distribution theory of meromorphic functions of single complex variation. The mathematicians of Hua School have achieved world leading achievements in many directions, such as harmonic analysis, typical domain, typical manifold, integral representation and boundary value problem, Schwarz lemma, quasi-convex domain, etc. in the theory of multi-variable function.

　　

The fields prize and the famous mathematician L of the Wolf Prize winner. V. ahlfors,l. carleon,h Cartan,kodaira kunihiko,j, Serre,c L. Siege,s, Smirnov and others have made outstanding achievements in the field of complex analysis.

　　

The introductory math book on complex analysis is recommended as follows:

　　

(1) W. Rudin, Real and Complex analysis,1966.

(2) H. Grauer, K. Fritzsche, Several Complex variables,1976.

(3) L. V. Ahlfors, Complex Analysis, 1979.

(4) Gong Sheng, concise complex analysis, 1996.

(5) Li Zhong, Complex analysis Guide, 2004.

　　　　　　　　　

(iv) Stochastic analysis

Stochastic analysis is a modern branch of mathematics in which the theory of probabilistic analysis develops in depth. On the basis of the foundation of stochastic process theory, the method of core mathematics, such as topology, Algebra, geometry and analysis, is mixed in the background of practical and applied problems, and stochastic analysis has become an important member of the Mainstream Mathematics Branch Club in the world today. The level of random analysis of mathematicians in China has entered the ranks of the world, and has been invited to do an hour report and 45 minutes report at the International Congress of Mathematicians. Our country's stochastic Mathematics research team also is famous in the world with the Chinese Academy of Sciences and some famous universities random Mathematics school.

　　

The two basic cells of stochastic mathematics should be measure theory and randomness. Randomness is a common objective phenomenon in nature, and measurement theory is an important mathematical structure of mathematical analysis. To see the world as a mathematical model is a basic portrayal of the broad devotion of mathematicians, such as the application of random mathematics. Only from the mathematical point of view random mathematics, then really do not have to mention the random word, as long as the study of the development of the measure theory can be. The theory of calculus, or the analytic theory, in the space of all finite measure, is actually the daily work of the random analysis scholar. Of course, with two different viewpoints, the measurable functional family in the sense of measure can bring about the essential difference between the two kinds of research and development in the way of thinking. For example, the thought of orbital space for the random process of a family of random variables is very important to the development of stochastic mathematics.

　　

The approximation to practical problems of the smooth vector field orbits and stochastic (ordinary) differential equation models characterized by the ordinary differential equation model is often better than the latter. Therefore, the concept of stochastic differential and random integrals is the most critical element of discipline creation. Wolf Prize winner K. The concept of random integrals defined by Ito for Brownian motion, along with the resulting Ito integral formula, makes random analysis a beautiful psalm in the Analytical Mathematics Library. Brownian motion Sample orbital function is continuous, but almost everywhere non-bounded variation and the nature of everywhere so that the usual riemann-stieltjes integral and lebesgue-stieltjes integral according to the sample orbital function can not be defined, Because the Riemann-stieltjes integral is defined in the Darboux and does not converge with probability 1. However, the foregoing Darboux and can converge in the mean square sense. It is this point that inspires the creation of Ito points and establishes the independent status of Ito integral. It is noted that the non-smooth characteristics of the sample orbits of stochastic process, the subsequent many random branches of mathematics, such as stochastic differential geometry, are thus obtained the independent position of mathematics. In the course of random differential equation, most of the courses of random analysis in undergraduate course are in the form of courses, and mainly teach the basic properties of brown motion and white noise, the random integral and Ito formula, and the solvability of stochastic differential equation. The fact that different types of stochastic processes can define corresponding random integrals in the proper sense is also often described. The postgraduate phase of the random analysis course can be "days high-flying birds, sea wide by diving". Backward stochastic differential equation, endrin type theory, large deviation theory, infinite dimension stochastic analysis, quasi-likelihood analysis, free probability theory, stochastic partial differential equation, stochastic dynamical system, stochastic differential geometry and so on are all beneficial ingredients for postgraduate random analysis courses. Of course, this phase of stochastic analysis has entered the integrated core of the home of mathematics, is not only to understand and grasp the theory of measurement as simple as the thing. The real charm of mathematics, in fact, is unification's mathematical values, the advanced level of stochastic analysis is no exception.

　　　　　　　　　

The mathematical writings on stochastic analysis are recommended as follows:

(1) A. Friedman, Stochastic differential equations and applications, vol.1,1975.

(2) A. Friedman, Stochastic differential equations and applications, vol.2,1976.

(3) I. Karatzas, S. E. Shreve, Brownian Motion and Stochastic calculus, 1991.

(4) P. Malliavin, Stochastic analysis, 1997.

(5) Huang Zhiyuan, basic theory of Stochastic Analysis (second edition), Science Press, 2001.

　　　　　　　　　

(v) Partial differential equations

The partial differential equation can be understood as its name implies the quantitative relation of the unknown function and several partial derivatives. Unknown function is the mathematical model function of human being to the mysterious unknown nature phenomenon. The derivative is the degree of change of the target function over time. The partial derivative is the variation rate of some factors which is caused by the diversification of the influence factors in nature. From the causal relationship, the partial differential equation can be considered as the mathematical model of all the causal phenomena in nature, so the great degree of its application value beyond mathematics is conceivable.

　　

The position of the partial differential equation in the Mathematics King's country is also rich and prosperous. The Clay Mathematics Institute in Massachusetts, USA, announced a sensational media event on May 24, 2000 at the French Academy in Paris: $1 million for each of the following seven "Millennium Math Puzzles": NP complete problem, Hodge conjecture, Poincaré conjecture, Riemann hypothesis, Yang-Mills theory, Navier–stokes equation, BSD conjecture. Of these seven world-class math puzzles, at least two of the half-problems are related to partial differential equations. In addition, at least three-fourths of all academicians of the Faculty of Mathematics and Physics of the Chinese Academy of Sciences are familiar with PDEs, and nearly half of all mathematical academicians are involved in the mathematical work of the partial differential equation. One of the main reasons why the attraction of partial differential equations is so large is that the theoretical value and application value of partial differential equation are "infinity". If the whole of mathematics is likened to the universe, the Earth is likened to the mathematical field of application and mathematical analytical mathematics, the outer earth (including the atmosphere) of the cosmic part is likened to the core of mathematics, then the partial differential equation is free to travel to the Earth and the outer space of the "space-day spacecraft." The Poincaré conjecture in the thousand-year maths puzzle has been solved recently, but the topological problem seemingly unrelated to the knowledge of partial differential equation is solved by borrowing the theory and method of partial differential equation. From hard analysis to soft analysis, to modern analysis, and even to other core fields of mathematics, partial differential equations exist almost everywhere.

　　

The research level of the Chinese mathematicians ' Pde has reached world-class level, and the scientific research based on the partial differential equation School of Chinese Academy of Sciences and some famous universities has been appearing frequently in the world first-Class mathematics magazine. And the academic exchange and influence of the Chinese Pde team and the world-class international differential equations team are already in a mutually beneficial and winning state.

　　

Partial differential equations can be divided into linear and non-linear, can also be divided into first-order equations, second-order equations and high-level equations, or elliptic, parabolic, hyperbolic, and so on. Each case has a large theoretical system and research results. Unlike ordinary differential equations, the overwhelming majority of partial differential equations cannot find the analytic expression of general solution or solution, even a linear equation can not be solved. At the same time, the problems of physics and engineering technology need to consider the equation and the definite solution condition (initial condition, boundary condition, etc.). Therefore, the definite solution problem is the main object of the partial differential equation, of course, very few nonlinear partial differential equations also have the exact solution of the expression.

　　

The course of partial differential equation in undergraduate stage mainly teaches the linear first-order equation and the quadratic equation, especially the proper solution problem of the corresponding equation: the existence, uniqueness and stability of the solution, in which the existence part is confined to the concrete solution. The course of the partial differential equation in postgraduate stage mainly studies the qualitative theory of solution and the problem of the solution in different meanings. The preferred teaching content is a modern approach to generalized functions, Sobolev spaces, advanced courses in functional analysis, and partial differential equations. There are four magic weapons in the field of partial differential equations: micro-local analysis theory, prior estimation technique, harmonic analysis method and weak convergence method. The theory of micro-local analysis originates from the study of general linear partial differential equations, utilizing some modern analytic mathematics tools such as the wave-front set of generalized function, quasi-differential operator, Fourier integral operator, differential operator and super function. The application of micro-local analysis has also been found in the recent study of nonlinear partial differential equations. The priori estimation is an effective information estimation of the solution established under the premise of the existence of the hypothesis solution, which is very important in solving the existence problem of solution, especially for the research of nonlinear partial differential equations. The most famous priori estimates are Schauder estimates, estimates, De Georgi-nash estimates, and Krylov-safanov estimates for second-order elliptic and parabolic equations. For the general nonlinear partial differential equation, the possible norm modulus estimation of the solution itself and its derivatives is a very essential solvable factor. Harmonic analysis method and weak convergence method have shown great vitality in the research of some famous partial differential equations.

　　

The well-known mathematician J of the fields and the Wolf Prize winners. Bourgain,de giorgi,l.v h?rmander,p. D. Leray,h lewy,p. Lions,t, tao,c etc. have made outstanding contributions to the study of partial differential equations.

　　

The mathematical writings on partial differential equations are recommended as follows:

(1) A. Friedman, Partial differential equations, 1969.

(2) J. Smoller, Shock Waves and Reaction–diffusion equations, 1983.

(3) L.c.evans, Weak Convergence Methods for nonlinear Partial differential equations, 1990.

(4) m. Taylor, Partial differential Equations, vol. 1-3, 1996.

(5) L.c Evans, Partial differential Equations, 1998.

(6) Miaochangxing, Zhang Bo, partial differential equation harmonic analysis method, 2008.

　　　　　　　　　

　　　　　　　　　

(vi) Large-scale analysis

　　　　　　　　　

Mathematical analysis, Advanced algebra and spatial analytic geometry are regarded as the "first high" of modern mathematical foundations, while functional analysis, general topology and abstract algebra are considered "second-highest". The "third High" of modern mathematics is the micro-manifold. As one of the modern core mathematics large-scale analysis (also called manifold analysis) is the "third high" on the basis of the fusion of topology, algebra and geometry of the way of thinking to form a high-level analytical mathematics field.

　　

Micro-manifold is a Hausdorff topological space with "calculus structure", which contains geometric objects such as canonical curves, surfaces and regions in "first high" as special examples. Micro-manifolds are not too much to be said to be born for calculus. On the stage of differential manifolds, it is possible to consider topological problems (differential topologies), geometrical problems (differential geometry) and analytical problems (large-scale analysis), and free and full use of algebraic, geometrical, topological and analytical methods and theories to study the corresponding profound mathematical problems. Assuming that wushu is "esoteric and isomorphic" to mathematics, and martial arts "the first High", "Second High" and "third High" are "the ground of Teng, move, jump, Leap", "quincuncial pile on the teng, move, jump, leap" and "Air teng, move, jump, leap", then the micro-manifold on the modern mathematical theory is equivalent to the "third high" martial Arts. Thus, it can be seen that the fundamental importance of large-scale analysis in modern core mathematics.

　　

The wide range of analytical courses mainly teaches the mapping between manifolds and manifolds, the embedding and immersion properties of manifolds, the threshold and cross-cutting, sard theorems, tangent and vector bundles, calculus on manifolds, differential operators on manifolds, infinite dimensional manifolds, Morse theory and applications, Lie groups, dynamical systems, Singular point theory and geometric analysis and other important content. One point that should be illustrated is the importance of algebraic topological knowledge to the study of a wide range of analytic studies.

　　

The mathematicians of the School of Nonlinear analysis of Peking University have made important research achievements in the world advanced level in the theory and application of infinite dimension morse.

　　

The following is a brief introduction to the two directions of the dynamical system and geometrical analysis.

　　

The dynamical system originates from the mathematical model of classical mechanics. After the evolution of the finite and infinite dimensional systems characterized by ordinary differential equations and partial differential equations, and then to the abstract topological dynamical systems and stochastic dynamical systems, the dynamical systems have already established the important position in the field of modern core mathematics. The concept of nonlinear semi-group in nonlinear functional analysis is a dynamical system concept. The semi-group properties of algebraic operations are the main mathematical structures characterizing dynamical systems. As a qualitative research of abstract system, dynamical system is mainly characterized by topological or ergodic integrity study. The perturbation theory of Hamilton system, the ergodic theory of the Kolmogorov system and the Kam theorem are all the highlights of the dynamical system theory. J. H. Poincaré 's qualitative theory of ordinary differential equations, such as stability, orbital periodicity and regression, is the basis of the research method of dynamical systems. Based on G. D.birkhoff the study of the ergodic theorem of the three-body problem, the Kam theory, which is found to describe the stability of the Hamiltonian system, is one of the milestone works of the dynamical system theory. In the system of partial differential equations with infinite dimensions, the Kam theory has been deeply studied.

In the normal course of dynamical system, we mainly teach the following basic contents: Chaotic attractor of nonlinear differential equation system, mapping iteration and invariant set, fractal, topological dynamical system, structural stability, Hartman theorem, stable manifold theorem, double album, Markov segmentation, etc.

　　

China's Peking University Power System School, as well as some other famous universities and research institutions of mathematicians in the field of power system has achieved a world-class pioneering work. The fields Prize and the Wolf Prize winner of V. I. arnold,a. N. Kolmogorov,elon lindenstrauss,c. Mcmullen,j. K. Moser, S.P. Novikov,y. Sinai,j. Famous mathematicians such as yoccoz in the field of power systems Make a huge contribution.

　　

Geometric analysis is an important branch of large-scale analysis, and it is well known in the mathematical world to study the geometric background of the analytic method of geometrical problems. In particular, a series of brilliant achievements in the field of geometric analysis has made geometrical analysis a special academic position independent of a wide range of analysis. The research work of the world famous Chinese mathematician, the fields Prize and the Wolf Prize winner, Professor Shing, has laid a fundamental academic position in geometrical analysis, and has opened a precedent for solving the problem of world-class conjecture of topological and geometrical problems in the method of partial differential equations. "The Millennium Math Puzzle" the solution of Poincaré conjecture, one of the seven major problems of the million dollar problem, is attributed to the infinite power of geometric analysis.

　　

Poincaré conjecture is a famous topological problem proposed by French mathematician Poincaré in the early 20th century: if all the closed curves in a closed space can be shrunk to a point, then the enclosed space must be a three-dimensional sphere. In view of the great contribution of Russian mathematician Perelman to solving Poincaré conjecture, the 2006 International Mathematician Congress awarded the prize for the Mathematics prize Fields award to Perelman, but faced an embarrassing situation of refusal.

In the nearly hundred years of the topological method is hopeless to solve three-dimensional Poincaré conjecture, the Fields Prize winner Thurston (Thurston) at that time introduced a geometric structure method to cut three-dimensional manifold, so that the resolution of Poincaré conjecture appeared hope of Dawn. Later American mathematician Richard Hamilton, inspired by Shing's approach to solving Krabi conjecture by using nonlinear partial differential equations, uses the Ricci flow equation named Gregorio Ricci, a topological procedure for constructing geometric structures of three-dimensional manifolds, So that the process of solving three-dimensional Poincaré speculation is more essential to move forward. In close proximity to the solution to the Poincaré conjecture, a major obstacle to the resolution of the Poincaré conjecture arises from the singularity of Ricci flow in the spatial transformation. At the critical moment, the Russian mathematician Perelman volley, with eight years of unique skills, with three informal journal papers, one at a time to shake the hundred-year puzzle Poincaré conjecture, with the world-class related mathematicians "all the material", so that the total declaration Poincaré conjecture was formally resolved. It is obvious that the scientific significance of solving the hundred-century difficult problem Poincaré conjecture is extremely significant. The geometrical analysis method is also extremely glorious.

The research work and research team of Chinese geometric analysis mathematicians have had a positive influence on the international counterparts.

The mathematical writings on large-scale analysis are recommended as follows:

(1) D. W. Kahn, Introduction to Global analysis, 1980.

(2) Zhang Jintin, Qian min, Differential dynamical system guidance, 1991.

(3) R.clark Robinson, an Introduction to dynamical system:continuous and discrete, 2004.

(4) R.schoen and s.-t. Yau, Lectures on differential Geometry, 1994.

(5) D. W. Stroock, an Introduction to the analysis of Paths on a Riemannian manifold, 2000.

Take a dip in analytical math (Zhang Kaijun)