Author | Ho Tong wood
Source | know almost
Hello everyone, I am from prospective senior Department of Mathematics, Tsinghua Tong Ho wood. Modern mathematics course for three years, I want to record some of his insights. Looking back three years, that take a lot of detours, a lot of sense to do a little thing, presumably with seniors, fewer teachers in-depth communication, so I want to use analyze their own experiences, strengths and weaknesses of the way, to you a student talent show ordinary course of undergraduate study, I hope that those who are better able to take advantage of this three-year period.
For the students do not want to start to finish, you can choose to see the part according to the directory, you can also look at Section VIII: attend the recommended order. The idea that only a personal point of view, we welcome the discussion!
table of Contents
I. Guidelines
Second, the basic language: number of points, lines generation smoke generation topology manifold
Third, inspirational and intuitive: Riemann surfaces, differential topology, differential geometry
Fourth, the unified theory: algebraic topology, algebraic geometry
Five complementary tools: homological algebra, commutative algebra
Sixth, the queen of mathematics: algebraic number theory
Seven mound ready to race
Eight, to attend the recommended order
IX Appendix: Course Outline
First, the guiding ideology: breadth-first
Why am I the end of the junior year to write this recommendation, because the senior people to have to start preparing their own thesis that a little direction, and the first three years are the basic foundations of mathematics learning phase. Teachers say, at the undergraduate time to learn a lot; Mr Yau also often said to be proficient in at least two directions mathematician is it possible to see connections in different directions, in order to make a big achievement. "Found links different disciplines," I gradually realized that the efforts to target, its essence is a better understanding of mathematics, but also the redundant stuff and shrink up of on his own knowledge of the original system.
So this proposed (from me) limitation is that "breadth-first" guiding ideology, I can not understand a lot of students (they are often gifted) aim at the very early algebraic geometry or algebraic topology or analysis the depth of this behavior has been learned, and I tried to pick up a book to learn from scratch tail, but tend to be thwarted by the sudden appearance of a new concept, really do not understand the motivation to study it, learn it in order to go down deep It became some kind of mechanically forced sex (but I think they definitely see through this motivation). Another limitation is that I analytics badly.
My freshman Tai Third, three years retreats the 31 math:
Analysis categories: Mathematical Analysis (1), mathematical analysis (2), mathematical analysis (3), real analysis, complex analysis (1), complex analysis (2), functional analysis, ordinary differential equations, partial differential equations (1) , partial differential equations (2), analytical mechanics, probability theory
Geometry classes: differential manifold, topology, differential topology, algebraic topology, differential geometry, Riemannian geometry, Riemann surface, complex geometry
Algebra class: Linear Algebra (1), Linear Algebra (2), leading basic algebra, abstract algebra (1), abstract algebra (2), algebraic number theory (1), algebraic number theory (2), algebraic geometry (1), algebraic geometry (2), Lie Lie algebra, group representation
Attendance is chronologically:
Freshman: Mathematical Analysis (1), Linear Algebra (1)
Large about: Mathematical Analysis (2), Linear Algebra (2), complex analysis (1), leading basic algebra
Sophomore: Mathematical Analysis (3), ordinary differential equations, topology, abstract algebra (1)
Sophomore: real analysis, mechanical analysis, probability theory, differential topology, algebraic number theory (1), Riemann surfaces
Junior year: functional analysis, partial differential equations (2), differentiable manifolds, algebraic number theory (2), differential geometry, algebraic geometry (1), Li Lie algebra, abstract algebra (2), complex analysis (2)
Under junior: Partial Differential Equations (1), Riemannian geometry, complex geometry, algebraic topology, group representation, algebraic geometry (2)
(Estimated number of people would be surprised at my elective much, this is actually a double-edged sword)
(Non-school students learn the syllabus may see Appendix)
Second, the basic language: number of points, lines generation smoke generation topology manifold
We will listen to a school teacher said the number of points and lines on behalf of the other is that you learn the foundation of all mathematics, I think this sentence is "all mathematics" must include probability, statistics and application, but if the limitations on the foundations of mathematics, then , be sure to add smoke-generation and topology.
Freshman when the teacher told us not to vote for more lessons, learn to concentrate on the number of points and lines generations, as newly enrolled students a little in awe of the unknown, I will only choose the number of points and lines generations. It now appears that, for most of the students, then the teacher is right, because many students not suited to this kind of college entrance math very different way of thinking, a lot of people could not even pass the midterm exam; but it is fortunate that I get started soon, probably because I have been in high school when he saw half of this Zhuoli Qi.
Because only took two classes, after-school time to do Zhuoli Qi took exercises. It now appears that, despite Zhuoli Qi exercises many of which are in the future may learn of numerical analysis, physical content inside the university, but the resulting effect will only exercise the role. I spent a lot of time on it, often spend half a day on an entire title, not to say this is not good, but a better alternative, you can use the time to learn on behalf of smoke and topology. I later learned that there are seven words Wangzhi Han seniors class of three school brother are in big eleven enrollment to attend the topology.
Learn abstract algebra, algebraic equivalent to opening the door to the class; learn the topology and differentiable manifolds, opens the door corresponding to the geometry class.
Abstract Algebra: I am a self-winter Yaomu Sheng "abstract algebra" and Artin's "Algebra" (Algebra), is to learn to learn again do not understand and will easily forget, so I spent two weeks in the summer after freshman year read "Introduction to group Theory" (an Introduction to the Theory of groups) Rotman the previous chapters, and the exercises are done (basic courses that have to bite in earnest from the beginning to the end). I was feeling on the part of my group of smoke generation has no problem, then opened a sophomore abstract algebra class I heard one did not go, the examination took only half the time took a perfect score. Not bad but learn better relationship with test scores.
Although I Yaomu Sheng "abstract algebra" (This is our textbook) watched it three times or more, since that group theory and ring theory to master well, but did not learn to Galois theory. Xu Kai seniors also said, when he had learned Galois theory have encountered difficulties, he recommended to me Hungerford's "Algebra" (Algebra). Hungerford unavoidably, the process of establishing Galois theory clearly written.
But it is easy to make us into a misunderstanding, is the only known proof Galois theory, but will not use it. Science can not learn it again, and everything should go to school adult version. For three years, I put on the field Hungerford read a chapter at least three times, Galois theory also learn back and forth at least three times, the key is, I do not just re-look at Hungerford this book, but in the study abstract Algebra 2, when algebraic number theory, Galois theory of place with the application, until the local use Galois theory, come back around and study a second time, third time, in order to learn better. This year the class discussion hill race, based on familiar concepts, I use notes on an A4-sized established Galois theory, this is called the adult version, is Mr. Hua said, the chance to read thick , then the chance to read thin.
Topology: the topology of my sophomore scholarship. Teacher style is very elegant, many things do not give rigorous proof, but I have enough in earnest after school, often did not understand a lesson, do not go to after-school review, delayed for several weeks, you will find this door more and more difficult courses, and the final end of review when finished combing all the knowledge, but also feel that this course did not say how much.
But this is not good study habits, after all, the things they do not understand too long, there are a lot of courses I want to come all this attitude towards learning, despite the end of the review can rely on to get a high score but in fact I learned something over a semester forgotten the basic. With Professor Yu Pun chat, he said: you put things forgotten, that is not learned. Yu Pun teacher is doing partial differential equations, but his algebraic geometry of this science are not forgotten, to start the title with ease. So I think, to learn new things every week to go over, it will greatly increase the possibility of the Institute.
Differentiable manifold: mathematical analysis, they also speak manifolds, say European space surfaces, but coordinates European space can become a role too much, when I learned often confuse the concept. If this time to learn the general theory of differential manifold, these concepts will be particularly clear reason. I used the "Introduction manifold" Tu's (An Introduction to Manifolds) entry. Only to see the concepts in the previous chapters, we learn manifolds, etc. on the line.
Manifold is the geometric objects abstracted concept, is the most basic subjects, almost all of the geometry class should take a few lessons time to define science manifold, I remember my sophomore year at the Riemann geometry, Riemann surfaces, differential topology, when three courses while speaking differential manifold, heard all tired.
Third, inspirational and intuitive: Riemann surfaces, differential topology, differential geometry
We often see very surprised to see the introduction of new concepts in the book, look for the reason the definition of this new concept, you have to look at the history of the formation of its historical perspective. I always believe that the brain is not the God of the mathematician's brain, it can be understood, there must be the root of all the ideas, it is bred out of the prevailing environment.
Professor Zhou Jian said: Complex analysis linked with all modern mathematics, all of modern mathematics are born from inside the complex analysis.
Riemann surfaces: After studying the complex analysis, you can learn Riemann surface. Riemann surfaces is complex analysis analytic extension comes - analytic continuation, it gave birth to the concept of "layer", as well as its links with the basic group, such as function along two paths analytical continuation, extension Rio is the same value out of it? If the two paths can be continuously varied to each other, then the value of the extension is the same.
Meromorphic functions on the Riemann surface research, can set up one-Riemann surface of the function domain, which is the geometric and algebraic linked. Concepts in algebraic number theory, such as the prime ideal decomposition, you can use a geometric vision to look at. We can also be extended to them in algebraic geometry. So when you learn algebraic geometry in the finite morphism, know its geometric origin, will you accept the theory of algebraic geometry is very helpful.
Calculus on Riemann surfaces of research to prove Riemann-Roch theorem, which is also the beginning of the application of partial differential equations Hodge theory.
Differential Topology: In addition to the simple tune with the triangulation of the leads, the de Rham cohomology should be the most intuitive theory with examples tune. Homology groups there are too many opinions, and stood angle differential form of the equation is to look at this (of course, this argument bad way, because it can be said similarly complex analysis is the study of Cauchy-Riemann equations).
The de Rham cohomology, the integration of this operation brings Poincare duality. And some students are first in algebraic topology, the use Poincare on singular homology studies duality, although the approach is very natural in algebra, but if the first with the de Rham cohomology background, understanding of the operation coherent general coefficient will be good a lot of.
To things on the entire structure defining manifold, the cohomology Cech will occur naturally, accompanying, as well as spectral sequence. In short, the geometry is a natural and intuitive to our human experience, find good in geometry in mathematics, do promotion, focusing instead stuffy than the general theory of the brain to learn hard down much more comfortable.
Differential Geometry: Professor Lee said: formulas of differential geometry is that the way research with calculus on curves and surfaces. I think there is only a differential geometry: Manifold bent. Differential geometry is a relatively classic courses, but only after learning of differential geometry, do not feel Riemannian curvature tensor defined in very human.
Cartan and Chern in differential geometry in the development of a method of moving frames, in equally complex geometric calculation on vector bundles on contact, to understand them, or understand "tensor" This is a difficult concept. Although we talked about the line on behalf of the tensor differential manifold where also spoke tensor, but once in differential geometry or complex geometry in with them, especially in the physicist there with them, you will find it difficult to understand them in doing the same thing (with coordinate component with tensor language, with a moving frame method), it is possible to Sipilailian seniors clear to you, if he do not know about, then you know he did not learn this thing.
Fourth, the unified theory: algebraic topology, algebraic geometry
I think a lot of people will think I put two algebraic topology and algebraic geometry course wrote "unified theory" like a public branch dry out things that I have here, "Unity" was standing on the front of the point of view of geometry . Algebraic topology to algebraic geometry in some of the abstract operation, the coefficient becomes a factor, everything to go in the direction of universal. Algebraic geometry is the same, even though we often say that he is the study of the zero polynomial, which sounds like high school analytic geometry, but in fact, outline the concept developed in the fifties and sixties of the last century, is in the complex geometry of algebraic operations abstracted.
The layer of algebraic geometry encountered in the same tone can unify many common cohomology theory, on the constant coefficient in the topology in tune with the school, or group of the same tune, which can be used to express the language layer.
Point of view of category theory is a mathematical revolution. Sixth class school brother Wu Yuchen especially liked everything will categorization. As a result, many mathematical concepts in various fields over the centuries and it can be reduced structural point of view. Private thought "Categorization" it is possible to rewrite the history of mathematics.
But for the sort of person I ordinary talent, the algebraic geometry and algebraic topology is not good science. Sophomore year, I tried it myself to see Hartshorne's "algebraic geometry" (Algebraic Geometry), but can not grasp the overall framework, but also prove prohibitive for those who save a lot of detail. Xu Kai seniors recommended I see Professor Fu Lei, "Algebraic Geometry" (Algebraic Geometry), but one by one proposition Organization heap together, I do not know algebraic geometry is like doing. But now it seems, these books are the most basic algebraic geometry language, so it is like a dictionary. But I also algebraic geometry is too little, not to say more other words. (If you want to get more advice on algebraic geometry, recommendations or ask other seniors it!)
This semester Sun Sheng Hao 2 of algebraic geometry teacher of this class, Sun Sheng Hao teacher will completely fill in the details of Hartshorne second chapters, treated like algebraic white as taught us, even more than the book Fu Lei teacher but also patience, that when I became aware that proved less than half a page of eliminating the inconvenience of how many books written on the details. A semester off, it is only know the rough form, the layer of tune with the concept, the teacher said, though it was the end of algebraic geometry class, but it is only the beginning of algebraic geometry, I do not know where to go next .
Algebraic topology is also very similar, I like most people, learn first singular homology. But in addition to outside with a long exact sequence proved homology theory, architecture is a blur for me, in other words, is the book by the same dictionary to see Mongolia. In the beginning, I used the long exact sequence, do not look at each arrow is in the end is how to map out, if later found not to speak isomorphic isomorphism is how to say the words out, a lot of geometry was abandoned a. It can be seen from the Whitehead theorem and Hurewicz theorem, if a single continuous communication between the map f CW complex induces isomorphism between the integer coefficient of homology, then f is homotopic, and this is isomorphic to f crucial, because many of the same homology groups but different ethical equivalent of two examples of space.
This semester I went to Professor Zhou Jian Bott-Tu of "differential form of algebraic topology" (Differential Forms in Algebraic Topology), although this book has been recommended Wangzhi Han seniors from my freshman year I began to read, but because I freshman sophomore math when maturity is not high, coherent theory are not familiar with, no one to coaching, I could not even lead to a written textbooks are so approachable not go on reading. But this semester more than a year after the influence of homology theory, this book finally able to read on.
A few days ago while preparing hill race, professors taught in products we use adult eyes, or the de Rham cohomology with real understanding, look. Although previously I who Poincare duality related theorems are back in familiar, because I can see that it is an integral operation of the induction coupling, and the integration of this operation can be done, we need those lengthy terms, but do not know Poincare duality What is the use. The teacher took us to the product do question, he taught us to calculate the Poincare dual cohomology ring, the outer product of differential forms will correspond submanifolds intersect. I suddenly see the light at once! Although previously I knew these conclusions and will permit, but there is a layer of paper windows that are not point out. I also retrospect, this semester Professor Zhou Jian open complex geometry class, teacher Zhou Jian Lefshchetz hyperplane intersection theory is concerned with complex geometry, then learning attitude is not serious because not understand why Zhou Jian said this called geometry, now it seems we are all Poincare duality ah!
Five complementary tools: homological algebra, commutative algebra
After the first year only took two classes free, I look at the big election on behalf of the leading basic math this course, it is about the module theory, category theory and homological algebra. At that time only for self-study of abstract algebra me, to keep up in the second half of the semester the course is very difficult, not to say that the content is very difficult - the most basic category theory, homological algebra, but the lack of motivation to learn , sometimes the teacher said the pull-back box is fiber bundle that kind of thing, I do not know. The last semester, it felt like a darkness lit a solitary small on the map, as time goes on, I forgot.
Commutative algebra is the same, because they say school seniors must first know commutative algebra algebraic geometry, but after I entered the second year, abstract algebra and learned a lot, so I began to see himself Atiyah and Macdonald's "commutative algebra guide" (Introduction to commutative Algebra). Can be said to see it again forget again, just like my real analysis, complex analysis as well as probability theory, not to use them, stupid brain would not remember them.
So now I'm tools for such subjects tend to have learning purpose, rather than a big bang to see systematic study of the entire book. To use the time, we went to see the corresponding chapter; wait until mastery are similar, then you can consider re-read through the whole sort of knowledge. And everyone says look algebraic geometry, commutative algebra to look at, but actually in our school two algebraic geometry class, the teacher will help fill that knowledge of commutative algebra, and the lesson will be used only occasionally commutative algebra, so I think Our school students can first learn algebraic geometry, learn while up corresponding commutative algebra. After all, mathematics was happy to learn, to force myself to go to the dictionary, if not liking, then do not look.
Sixth, the queen of mathematics: algebraic number theory
I came into contact with number theory is a sophomore at Fu Lei, a teacher last semester algebraic number theory 1, but Fu Lei teacher first class at Tsinghua number theory to teach undergraduates, so they talk about the proof of the prime number theorem and assignment theory, about the ring of algebraic integers , involving very little prime ideal. So in order to Qiu game, I went to the first part of self-study of Feng Keqin "algebraic number theory", put this part of Fu Lei classical than many teachers, but Qiusai Te do not like to test, it is actually very important, since the beginning of the development of algebraic number theory is looking at a larger number of primes is how domain decomposition. Xu Kai seniors recommended I see Neukirch of "algebraic number theory" (Algebraic Number Theory) in the first two chapters, but I simply can not understand the details of algebraic proof in Neukirch, feel inscrutable, it seems indeed exchange maturity algebra is not high enough, some algebraic operations can not understand it, that seniors are looking for a good teacher or buttoned details, progress will be great.
The junior semester I learned algebraic number theory teacher Chenzong Bin 2, Chenzong Bin Beijing University teacher was born, thinking fast, algebraic number theory covers a full semester of local field theory, higher-order group differences, class field theory, Tate thesis of. I totally can not keep up in class, can only take notes, Chen omitted details are particularly large, the last class only five people. The most regret is that I did not go to after-school promptly fill up, that I chose ten semester math class, a lot of after-school classes are no tubes, it does not learn to lead. I still feel that a maximum of four semester math class, so have after-school study time.
Class failed to grasp, it is winter again saw his Serre "local domain" (Local Fields) and Serge Lang's "algebraic number theory" (Algebraic Number Theory), before finally feeling a little bit. Later in the article the teacher said: you do not know and had to ask him, I said: feel the problem is too simple, or is the problem everywhere, to product said: What do you call him again to say again! Indeed, the ability to ask questions of feel a lot worse, a lot of questions they want to know, but do not want to ask the teacher to know the answer. This superficial understanding of the style may be derived from the answer to our problems and not only high school mathematics competition announced topic of the practice, still not good.
I also followed Chenzong Bin junior teacher, Zhang Zhiyu seniors, who attended the local Langlands correspondence on the seminar, but in addition I started talking about representation, the other I totally failed to understand. The case of the more difficult things the more you should spend more time, I should not have picked nine in math class also participated in this discussion class.
In my junior year that year, Tsinghua University Department of Mathematics launched a new program: Mathematics School class research training in basic plan (Junior Thesis). I chose Chenzong Bin teacher "with complex multiplication of elliptic curve arithmetic" (Arithmetic of Elliptic Curves with Complex Multiplication). I use the winter holiday to Silverman's "arithmetic on an elliptic curve" (The Arithmetic of Elliptic Curves) to copy it again, Silverman's argument algebraic geometry does not need to know, for me, very friendly. After entering junior next semester, I began to look at Silverman's second book, "Higher elliptic curve arithmetic topics" (Advanced Topics in the Arithmetic of Elliptic Curves) Chapter II: The theory of complex multiplication. But I found the second book familiar with the elliptic curve a lot higher, and I had to see it again from scratch, the first book, watching knock latex, thus forcing every step I do understand the proof, though a silly way, but better than Yimushihang to read but can not absorb much better, because it takes time to learn things.
When I knocked 70 latex later, I have come to understand the complex multiplication theory of a deep place. But I think if I went to when they talk about when junior thesis defense of this complex multiplication theory will not be too easy, after all, what the paper did not look, do not lose the momentum to learn good brother! So I began to look at Coates and Wiles proved a special case with a complex multiplication on the elliptic curve weak BSD conjecture in 1977. Really hard to read the original paper, I found Karl Rubin wrote in 1995, a note, and began eating, combined with more than a dozen papers began eating together. These papers mastery of elliptic curves requirements too high, and very difficult to read. Once I thought only want to talk about complex multiplication theory, but in the end I was planning to Chenzong Bin contents of reply when asked me to guess on a BSD verify the elliptic curve. I was thinking, if I could not even guess BSD on this special elliptic curves are verified finish, then I can not really something to talk about it. So I worked day and night to study those three weeks before the thesis defense, all they have to promote that to express extremely complex theorems all restrictions on me this special elliptic curves. Slowly brightened things up, in order to promote and make a lot of technical operations have become commonplace, and I can catch the main idea - that is, Birch and Swinnerton-Dyer made in that paper in the fifties calculations - and I finally understand why the teacher said Chenzong Bin BSD guess work out of the. In turn, we know the most important part, those who understood the technical promotion has become. Last June 10, the day, I made a very satisfactory report on the elliptic curve on a special piece of BSD guess how the cards, despite the presence of almost failed to follow after listening to the whole.
Seven mound ready to race
In recent years, more and more attention to the major universities mound race, admittedly hill race results are very important for each of the schools, but for our students, preparing for the race hill is the most important.
As Mr. Yau has been said on the mound tournament awards ceremony, before our students go to college to test the United States but they qualify, this hill race is to train students in basic skills. I have always thought that, as we have this talented students generally learn something is to learn again do not understand, do questions, to tease out the adult version of the knowledge of the context, be possible to learn to understand.
Had never been seen just entering college, high school competition just ended, problem-solving thinking very active on Zhuoli Qi almost did not not make to the topic. But with math homework became accustomed to deal with the university, others do not want to give serious consideration to the issue projects, problem-solving ability is greatly decreased, the brain becomes dull lot. Sophomore mound after losing game, I started doing Zhenti hill race in previous years, can be reliably feel his attention increasingly focused, thinking faster and faster. So music.brothers do not resist something of this examination, it is not like college entrance math, its title can live it!
Eight, to attend the recommended order
I have the following suggestion (which is close to the most basic of modern mathematics context) tend to algebraic geometry or ordinary students:
Concepts mathematical analysis, linear algebra, abstract algebra, topology self manifolds +: freshman
Sophomore: complex analysis, Riemann surfaces, differential topology + Bott Tu of "differential form of algebraic topology"
Junior: complex geometry, algebraic topology, algebraic geometry
Exaggeration to say that this should be every want to learn basic math students must master something.
(Welcome to the discussion)
IX Appendix: Course Outline
Analysis categories:
Mathematical Analysis (1): real number theory, limit, single-variable calculus
Mathematical Analysis (2): multivariable calculus, Calculus on Surfaces
Mathematical Analysis (3): series theory, Fourier analysis
Real Analysis: measure theory and Lebesgue integration
Complex Analysis (1): The most basic complex analysis
Complex Analysis (2): Content is not necessarily every year, iterative problem may speak rational function, hyperbolic metric
Functional Analysis: The most basic functional analysis
Ordinary Differential Equations: existence, uniqueness, continuation theorems
Partial differential equation (1): The existence and uniqueness of the wave equation, heat equation, Poisson equation
Partial differential equation (2): the presence of a unique canonical elliptic equations, equation hyperbolic, parabolic equation
Analysis of mechanics: Lagrange mechanics and some metaphysics
Probability theory: basic probability theory
Geometry categories:
Differentiable manifold: the common manifold and manifold concept study
Topology: point set topology, the basic group, cascade space cohomology theory
Differential Topology: cross manifold intersects approximation, Stokes theorems, Poincare-Hopf Theorem
Algebraic topology: homotopy theory, homology theory
Differential geometry: Theory of curves and surfaces
Riemann geometry: the basic Riemann geometry
Riemann surfaces: Algebraic Geometry and curved portion Riemann
Complex geometry: on complex manifolds homology, Hodge Theory
Algebra class:
Linear Algebra (1): matrix determinant
Linear Algebra (2): matrix diagonalization
Algebra frontier basis: module theory, category theory, homological algebra
Abstract algebra (1): the basic domain ring group
Abstract algebra (2): Galois theory of finite groups may also speak linear representation
Algebraic number theory (1): assignment theory, the prime number theorem
Algebraic number theory (2): Not necessarily every year
Algebraic Geometry (1): classical variety
Algebraic Geometry (2): On the layer outline cohomology
Lie Lie algebra: complex semi-simple Lie algebra representation theory
Group representation: complex representations of finite groups, modular representation