Tao - How to Become a Pure Mathematician

PDF Version of http://hk.geocities.com/mathphyweb/puremath.htm Last Revised: 25/3/2008 How to Become a Pure Mathematician Page 2 of 66 It's like if you want to be a good pianist, you have to do a lot of scales and a lot of practice, and a lot of that is kind of boring, it's work. But you need to do that before you can really be very expressive and really play beautiful music. You have to go through that phase of practice and drill. - Terry Tao http://hk.geocities.com/mathphyweb/puremath.htm How to Become a Pure Mathematician Page 3 of 66 Content: About this article Stage 1 Elementary Stuff Introductory Discrete Mathematics Introductory Algebra Introductory Calculus Stage 2 Linear Algebra Introductory Higher Algebra Calculus (Introductory Real Analysis, Several Variables Calculus, Vector Calculus, etc) Complex Variables Differential Equation Probability and Statistics Stage 3 Intro. Analysis Abstract Algebra Introductory Number Theory Introductory Topology Differential Geometry Mathematical Modelling (optional) Statistical Inference (optional) Probability and Stochastic Processes (optional) Statistical Computing (optional) Stage 4 Foundations and Discrete Mathematics Analysis Algebra 5 8 8 9 10 11 13 13 14 14 15 16 17 20 20 22 24 25 26 28 28 29 30 32 32 35 38 http://hk.geocities.com/mathphyweb/puremath.htm geocities.htm .com/mathphyweb/puremath.How to Become a Pure Mathematician Page 4 of 66 Number Theory Geometry Topology Further Calculus (optional) Mathematical Physics (optional) Probability (optional) Statistics (optional) Biostatistics (optional) 42 46 50 52 53 56 58 62 http://hk. com/mathphyweb/puremath.htm • . but hopefully it will be clear as you proceed. I got the international edition.e. It relies on two factors: well written or cheap. for a full time student. perhaps. Part time students may double the time.. for some integer n in {1. It's not surprise to spend a week or more to tackle one problem. Note the price factor may by irrelevant. I treat "free" as an element in "cheap". and recommended a few notes or texts.geocities. These two factors are not mutually exclusive. But its better for anyone to understand most parts of stage n before proceeding to n+1. 3. a list for beginners. beware of the symbols from different books in such cases. I assumed you have high school mathematics background (i. Yup. etc). One doesn't need to read every listed book within a subject to master the subject. author's nationality or religion.. Some people like to consult a few books. I listed more than enough so that you can scout around to find one that you feel comfortable with. Free material excluded. Scientists in other fields and engineers may skip first or second stages and begin at later stages.phys. Euclidean geometry. usually.nl/~thooft/theorist. I shall try to keep this list up to date whenever I exist. If you decided to attend a class. with the only difference is. What's the order of courses to study within a stage doesn't really matter. You probably like to read those books again and again in your life. • Time. some Chinese press in mainland China http://hk. or even more. basic trigonometry.html). Shouting eureka is the high point of a mathematician. The aim of this page is to introduce what different branches of mathematics are.. there is a way to become a good theoretical physicist (http://www. I don't claim that this list makes you a good pure mathematician.. according to their prior knowledge. • Assumed knowledge. Things may come to your head suddenly. it often happens that you couldn't solve a problem within an hour. since I belong to the complement of good pure mathematician. This list is written for those who want to learn mathematics but have no idea how to start. My selection will not be bounded by any publication press. Moreover.. I make no attempt to define what pure mathematics is. 4}.uu. especially in higher level courses. it always happen. Note the highlighted list does NOT indicate those books are good for beginners. I list more material in each field for more advanced studies). I also highlighted several books that you would really like to keep in your own library. don't expect the professor can teach. sometimes I get a HK $2xx book and Amazon says its US $1xx (~HK $7xx). It takes approximately one year for each stage (except for stage 4.How to Become a Pure Mathematician Page 5 of 66 About this article: What for? OK. Here is a guide to study pure mathematics. Moreover. 2. • "Axiom of choice". one rated it with 5 star (with dozens of people supporting) and at the same time another rated 1 star (with dozens of people supporting again). I wonder if any well educated student has never study Shakespeare or Plato.ps] download ftp://mirror. the Principia.wisc. Michael Artin's student's student.google.How to Become a Pure Mathematician Page 6 of 66 published photocopied of English text with a relatively cheap price.geocities. the Archimedeans (http://www. why do you study? • Other resources. I should remind you that MIT's open recourse (http://ocw.adobe. It seemed to me that lots of people study these subjects because they need to. Australia. To study science. Although I'm not into reading books online.cs. whenever they exist. just because one is a good mathematician doesn't imply he's a good author or educator.com/products/acrobat/readstep2.mhprofessional. In contrast. The list below aimed to recommend books or (usually) printable notes. but I seldom include them. Dialogo sopra i due massimi sistemi del mondo. Note that the comments are sometimes quite extreme: for the same book.php?page=doc. reading the classics (the Elements. Large proportion of these readers are lack of mathematical maturity.edu/OcwWeb/Mathematics/index. Links to Amazon for most of the listed book are included.wikibooks. If they can't pass the exam. or visit http://pages. etc) is optional.exe.archim.html). Gottfried Leibniz's student Jacob Bernoulli's student Johann Bernoulli's student Leonhard Euler's student Joseph Lagrange's student Simeon Poisson's student's student's student's student's student's student's student's student's student.mit. so that you have an easy access to other users' comments.uk/notes/) and Wikibooks (http://en. I've taken all undergraduate core pure mathematics and statistics courses there.edu/pub/mirrors/ghost/ghostgum/gsv48w32. to view [. Perhaps Terry Tao is an exceptional case.pdf] get Adobe Reader (http://www. they want to apply mathematics. To view [. Max Planck's student's student's student's student and Thomas Kuhn's student's student.cs.exe and ftp://mirror.com/mathphyweb/puremath. While for literature or philosophy. Also. I was Terry Tao's teacher's student.lizardtech.htm). most pure mathematics students study because they like the subject and enjoy it.com/download/dl_options. especially for introductory discrete mathematics.org.org/wiki/Category:Mathematics) provide another great sources of materials. ask yourself. Just feel http://hk..edu/~ghost/gsview/. Disquisitiones Arithmeticae. to view [. UNSW.edu/pub/mirrors/ghost/AFPL/gs853/gs853w32. Bear in mind that. So.wisc. Schaum's Outlines (http://www.com/) allow you to preview sections from a book. Principia Mathematica.com/category/?cat=4234) series are cheap.wisc. • Me. Google books (http://books.cs.. • Comments. with all pure mathematics courses in higher level. probability and statistics books. These are excluded in the following list.htm . you know. Sydney. you may search the relevant if you like. A product of School of Mathematics and Statistics.djvu] get http://www. com/mathphyweb/). contact can be found in the main page (http://hk. just email me. http://hk. • Disclaimer. etc.geocities. • Comments/suggestion for a book.How to Become a Pure Mathematician Page 7 of 66 like I'm an idiot.com/mathphyweb/puremath.geocities. I'm not responsible for any external link.htm . org/lecture_notes/david_santos_precalculus.uky.artofproblemsolving. Ikenaga B.uea. Elementary Euclidean Geometry: An Undergraduate Introduction Gow M.artofproblemsolving.M.com/mathphyweb/puremath. Geometry and Its Applications Mildorf T.org/lecture_notes/a-course-in-arithmetic.pdf) http://hk.J.L.htm) Meyer W.mth.html) Gibson C. Precalculus I and II Lecture Notes (FREE! http://www. we'll fix things up here.M.openmathtext. A<B (FREE! http://www.au/~wchen/lnemfolder/lnem.pdf) Sharipov R.edu/ken/ma109/notes.htm . Algebra and Trigonometry Chen W.openmathtext. Olympiad Inequalities (FREE! http://www.msc.millersville. The first stage contains the basic material that is required for most scientists and engineers. Basic Mathematics Covering Algebra and Trigonometry (FREE! http://www.K. Elementary Algebra Lecture Notes (FREE! http://www.maths. Elementary Mathematics (FREE! http://www.mq.pdf) Ward T.G.net/lecnotes/algtrg.com/Resources/Papers/KedlayaInequalities.pdf) Kubota K. College Algebra (FREE! http://www.A.L.E.How to Become a Pure Mathematician Page 8 of 66 Stage 1 For many reasons.edu. high school mathematics is taught in an informal way.. Lecture Notes on Algebra and Trigonometry (FREE! http://javeeh.geocities. Basic Algebra and Trigonometry Notes (FREE! http://marauder.B.html) Joyce D.pdf) Santos D. A Course in Pure Mathematics . and Duong X.arxiv.A. Foundations of Geometry for University Students and High-school Students (FREE! http://uk.com/Resources/Papers/MildorfInequalities.edu/~bikenaga/basicalg/balgnote.openmathtext.uk/~h720/lecturenotes/OB81lectures. Higher Algebra Beecher J.clarku.W. and Bittinger M.org/lecture_notes/elementary_algebra_book. Barnard S.Contains exercises taken from British Universities exams.A. and Child J.T. Elementary Stuff: Here is a list that helps you to refresh and enrich your high school mathematics.org/abs/math/0702029/) Veeh J.A. discrete mathematics is particularly meant for computer scientists.ac. A Short Course on Trigonometry (FREE! http://www.pdf) Santos D. Arithmetic Lecture Notes (FREE! http://www.pdf) Santos D.edu/~djoyce/trig/) Kedlaya K. Penna J. functions (injective. Logic for Everyone (FREE! http://arxiv. The concepts you learnt here will play a key role in your later studies and improve your thinking skill.trentu.edu.ndml/1175197470) Suppes P. Notes on Math Proof (FREE! http://marauder. introductory graph theory and number theory.edu/~ec/book/ch01.A short chapter from the author's algebra book. Daoud A. Halmos is one of the better expositors of mathematics.ca/~duentsch/papers/methprimer1. part in higher mathematics.org/euclid. Abstract Set Theory (FREE! http://projecteuclid.html) Enderton H.R. Proof in Mathematics: An Introduction Düntsch I. Only for Proof. Logic and Set Theory: Bilaniuk S.ca/math/sb/pcml/) Connell E.edu/~bikenaga/mathproof/mathproofnotes.tw/~htchen/dm/index. Discrete Mathematics (see the Lecture Notes section) (FREE! http://mcu. Basic Concepts of Mathematics (FREE! http://www. bijective inverse).brocku.html) Skolem T. nPr) etc in this course. and Givant S. surjective.Let me warn you: proof is a crucial. crucial. Sets. Set theory plays a significant role in mathematics. Logic as Algebra Herrmann R. Relations.math.au/~wchen/lndmfolder/lndm. Halmos P. How to Prove It: A Structured Approach . Introduction to Logic Velleman D. nCr. discrete probability (counting.edu.Don't misinterpret the word naive.B. I repeat.A.maths.org/abs/math.W.J.L.A. This is not those books for dummies nor complete idiot's guide. Euclid's algorithm. Functions (FREE! http://www. Velleman did a good job for beginners here. In general: Anderson I.H. A First Course in Discrete Mathematics Chen H.trillia. though he propose the honest title as An Outline of the Elements of Naive Set Theory. and Franklin J.mq.GM/0601709) Ikenaga B. Elements of Set Theory Halmos P.html) Introductory Discrete Mathematics: You will study logic and set theory in an introductory discrete mathematics.com/zakon1. Discrete Mathematics (FREE! http://www.T. Background and Fundamentals of Mathematics (FREE! http://www.html) http://hk.miami.html) 陳旭東：離散數學 Chen W.R.com/mathphyweb/puremath.htm . You will also study proofs. if you are serious about it.pdf) . A Problem Course in Mathematical Logic (FREE! http://euclid.How to Become a Pure Mathematician Page 9 of 66 Zakon E.millersville.cosc.geocities. Naive Set Theory . L.L.edu/~doyle/docs/finite/cover/cover. Ross S. Discrete Maths Lecture Notes (FREE! http://www.html) Lerma M.Read Chapter n's.com/mathphyweb/puremath. and Vesztergombi K.How to Become a Pure Mathematician Page 10 of 66 Epp S.geocities..openmathtext. and Patashnik O. See the companion website (http://www. Snell J.maths. Lectures Notes in Discrete Mathematics (FREE! http://syssci. Concrete Mathematics: A Foundation for Computer Science Kemeny J. Aposotol's Calculus Vol.edu. In order to have fun with linear algebra and complex variables in stage 2. Chen W.cs. polynomials.org/lecture_notes/discrete_math_lecture_notesII. At this stage.ca/faculty/gmacgill/guide/) Rosen K.northwestern. Discrete Mathematics Lovasz L. Discrete Mathematics and Its Applications .S. Fractals.dartmouth.pdf) MacGillivray G UVic Discrete Mathematics Study Guide (FREE! http://www. etc. Linear Algebra (FREE! http://www.com/rosen).pdf Santos D. Genetics. Chen W.atu. Topics in Finite and Discrete Mathematics Santos D.tau. Computer Graphics. Gaussian elimination. for all positive integer n<8.math. Finan M. etc. system of equation. Discrete Mathematics with Applications .L.openmathtext. Anton H. T.I enjoy reading Epp's text when I first learn this topic.au/~wchen/lnlafolder/lnla.edu/~sepp/DMwA3e.edu/math/faculty/finan/main2.pdf) Lovasz L. vector space. Chaos. it's better to learn complex number now.html) .Another nice introductory text on this topic.H.htm . linear transformation.L. matrix. Introduction to Finite Mathematics (FREE! http://www. Discrete Maths II (FREE! http://www. and Rorres C.org/lecture_notes/discrete_math_lecture_notes. Miscellaneous Topics in First Year Mathematics (FREE! http://hk.math. Pelikan J.The application version introduced how linear algebra can be applied to Markov Chains. A few Americans (exchanged to Sydney) in my third year abstract algebra class told us they couldn't follow because they haven't learnt much complex number. Don't forget to check out the errata (http://condor.depaul. and Thompson G. Knuth D.htm).M.ac. Games of Strategy. some author makes good effort to spark student's interest.mhhe.L.pdf) Graham R.pdf) Introductory Algebra: Complex number.il/~odedr/teaching/discrete_math_fall_2005/dmbook. 1 in the calculus sections also introduce topics that is relevant to this area.E.A.edu/~mlerma/papers/discrete_mathematics-2005.mq.B. Discrete Mathematics (FREE! http://www. and Vesztergombi K. Cryptography. Notes on Discrete Mathematics (FREE! http://www.uvic.math.L.G... Graph Theory.W. Elementary Linear Algebra .M.W. inverse trigonometric functions.etsu.maths.openmathtext. that's fine.edu. Matthews K. First Year Calculus (FREE! http://www.edu/classes/calcII/calcII.org/book/) Santos D. basic differential equations.aspx and http://tutorial. say vector calculus.edu. Numbers and Functions: Steps into Analysis Chen W.One of the better calculus text in print.html) Dawkins P. P Calculus I and Calculus II (FREE! http://tutorial.au/~wchen/lnmtfymfolder/lnmtfym. indeterminate form. Lectures on Matrices (FREE! http://www.How to Become a Pure Mathematician Page 11 of 66 http://www. and John F.edu/classes/calcI/calcI. you would like to leave a few section. Introduction to Calculus and Analysis I . hyperbolic functions.lamar.au/~wchen/lnfycfolder/lnfyc.B. This is a standard modern text. I would suggest a few non-standard texts and notes.ac.lamar. Elementary Linear Algebra (FREE! http://www. learn the epsilon and delta proof. line and surface integrals.in/~srg/acicara/) for errata. fundamental theorem of calculus. basic idea of several variables. Linear Algebra and Its Applications Wedderburn J. infinite series. Calculus (FREE! http://www. to the next stage. calculus is crucial.numbertheory.edu/math/gardner/2010/notes.aspx) Garrett P.mq.htm) Leave Chapter 6 and 7 to the next stage.edu/~djoyce/complex/) Lang S.P.org/lecture_notes/new_linearalgebra.clarku.lamar. Algebra (FREE! http://tutorial.mq. Linear Algebra (FREE! http://www. logarithm and exponential functions. Linear Algebra Notes (FREE! http://www. Apostol T.M. limit of sequences.geocities. but you should get use to it.math. Stewart's Calculus. If you learn first year calculus in a college.H. The first thing we need to fix up is the definition of limit.umn. A Course in Calculus and Real Analysis . Crowell B.lightandmatter. your suggested text is probably J.edu/classes/alg/alg. it is colorful and quite detailed. Joyce D.L. A Short Course on Complex Numbers (FREE! http://www.math.math. Not friendly for beginners.com/calc/) Dawkins.ams.aspx) Gardner R.edu/~garrett/calculus/) Ghorpade S.iitb.math.html) Courant R. triple.htm .He's a member of the Bourbaki and wrote many books.pdf) Strang G. double.org/online_bks/coll17/) Introductory Calculus: As you may have seen. Undergraduate Algebra . Other key concepts are the mean value theorem. It is not only important in advanced mathematics but also can be applied to many other fields.E. This book is thick. 1 Burn R.com/mathphyweb/puremath. Calculus (FREE! http://www.W. If you prefer it. Calculus Vol.V.See their web (http://www.M. http://hk.maths.R.math. and Limaye B. Notes on Calculus (FREE! http://marauder.org/lecture_notes/jose_diff_calc.Contains Cambridge Mathematical Tripos' exam paper. A Course of Pure Mathematics .math.L.J.H.html) Santos D.J.pdf) Shapiro B.com/math150B/spring2002/) Spivak M.E. Strang G. Calculus Bible (FREE! http://www.How to Become a Pure Mathematician Page 12 of 66 Gill G.openmathtext.R. Lectures Notes on Calculus (FREE! http://javeeh.com/mathphyweb/puremath. Advanced Level Pure Mathematics Tranter C. Spivak presents a first course in real analysis or a course in calculus that has been carefully developed with attention given to the real analysis foundations with an eye on rigour.B.pdf) Ikenaga B. and Physicists: An Introduction to Abstract Mathematics (FREE! http://mathcs.edu/ans7870/resources/Strang/strangtext.edu/~ahwang/print/calc.geocities.S.htm . Techniques of Mathematical Analysis Veeh J. and Finney R. Calculus . Giles.To paraphrase J.net/lecnotes/calc1. Computer Scientists. have fun! Hwang A. Calculus and Analysis 2 (FREE! http://www.holycross. Calculus and Analytic Geometry Tranter C.htm) Thomas G.bruce-shapiro.A.edu/~bikenaga/calculus/calculusnotes.D. Calculus (FREE! http://ocw.millersville.pdf) 張筑生： 《數學分析新講》第一、二冊，北京大學出版社 http://hk. Differential Calculus (FREE! http://www.byu.mit. Calculus for Mathematicians.edu/Math/CalculusBible/) Hardy G. pdf) Lang S.edu/) Bowen R.M.smcvt.html) Kuttler K.matrixanalysis.mengr. As a consequence. beware of it. minimal and characteristic polynomials.V.mq. and many areas in natural sciences and social sciences.A. etc). direct sum decompositions. Several authors focus on the application and often do calculation without justifying. Linear Algebra .math. Chemistry. Engineering. Meyer C.Not excellent for a first course. Linear Algebra Problem Book Hefferon J.millersville.lamar. Dawkins P.tamu. and Beauregard R.edu/~klkuttle/Linearalgebra.R. Finite-Dimensional Vector Spaces Halmos P. Biology. textbooks are written and catalog like the following way. vector spaces. An Introduction To Linear Algebra (FREE! http://www.com/mathphyweb/puremath.B.L. and Wang C-C. Jordan canonical forms.math. functions of matrices. and Maple (FREE! http://www. least square. multilinear map.math. Such material can be applied to linear programming. fractals.edu/linalg. Linear Algebra . Linear Algebra.edu/classes/linalg/linalg. generalised eigenspaces. for integer n>7. orthogonality. mathematical analysis and differential equation. Matrix Analysis and Applied Linear Algebra (FREE! http://www. it goes beyond standard first course of linear algebra. several variable calculus.As Lang had said this book isn't aimed for introductory linear algebra.byu.aspx) Fraleigh J. inner product spaces.htm .D. Notes on Linear Algebra (FREE! http://marauder. A First Course in Linear Algebra (FREE! http://linear. linear transformations.gatech.geocities.edu/~bikenaga/linearalgebra/linearalgebranotes.com/) http://hk. normal matrices.edu/rbowen/) Chen W. Physics students have to take linear algebra. Linear Algebra (FREE! http://www.html/) Herod J.edu. Linear Algebra (FREE! http://tutorial. exponentials of matrices. Beezer R. Linear Algebra: Vectors. Gram-Schmidt algorithm. Halmos P. computer graphics.html) .html) Ikenaga B. Cayley-Hamilton theorem.edu/~herod/Hspace/Hspace. Linear Algebra (FREE! http://joshua.How to Become a Pure Mathematician Page 13 of 66 Stage 2 Mathematics program in the University are constructed in the way that courses are offered in favor of other schools (Physics. Introduction to Vectors and Tensors (FREE! http://www1. etc will be studied in this course. Infinite Dimensions.maths.W.Read Chapter n's. QR-factorisation.ups.R. Economic. Householder algorithm. For example.au/~wchen/lnlafolder/lnla. norms. Make sure you have a solid foundation before you read this. Elementary Algebra Course Notes (FREE! http://www.html) Chen W. 2 Bartle R.edu/classes/calcIII/calcIII. Smith R.Professor Shilov at the Moscow State University states that this text "considers spaces over an arbitrary field.html) Chen W.gatech." Introductory Higher Algebra: I've taken this course entitled "Finite Mathematics".math. Assuming you have learnt the very basic of number theory in the discrete mathematics section. connected set. Chapman R.E.au/~wchen/lnmvafolder/lnmva.uga. and Herod J. Several Variables Calculus. Fundamentals of Analysis (FREE! http://www.org/abs/math. Gauss' lemma. compact set.arxiv.lamar.This book isn't terribly good. Chinese remainder theorem and their applications (coding. boundary point.M. Calculus III (FREE! http://tutorial. limit point.mq. Introduction to Calculus and Analysis II/1. Things like prime numbers.If you want to get one several variables calculus text. discrete mathematics and the third stage abstract algebra. like open set.aspx) http://hk.secamlocal.uk/people/staff/rjchapma/notes/arith.maths. II/2 . and John F. Course of Linear Algebra and Multidimensional Geometry (FREE! http://uk. Fundamental Theorem of Arithmetic.edu/~cain/notes/calculus.pdf) Childs L. To get yourself ready for the stage 3 analysis. Multivariable and Vector Analysis (FREE! http://www. etc): Learn the very basic concept of real analysis. Linear Algebra .geocities. make sure you understand those stuff. A Concrete Introduction to Higher Algebra . Also learn basic Fourier series.maths.edu/~roy/) Calculus (Introductory Real Analysis. double.math. close set. gradients. software engineers and pure mathematicians (optional but useful). a course for computer scientists.math. Fermat's little theorem.L.edu. Bolzano-Weierstrass theorem. Calculus Vol. Green's theorem and divergence theorem are the key things. Euler's theorem. Dawkins P. inverse and implicit function theorems can also be studied. closure. bounded set. with the real and complex spaces being considered as closely related special cases of the general theory.W. cylindrical coordinates.edu.ex. the Lagrange multiplier.W.N.HO/0405323/) Shilov G. Next.L. Vector Calculus.How to Become a Pure Mathematician Page 14 of 66 Sharipov R. Multivariable Calculus (FREE! http://www. For the several variables section. The Elements of Real Analysis Cain C.au/~wchen/lnfafolder/lnfa.com/mathphyweb/puremath.ac.G. tests for primality. Apostol T.mq. This is a bridge that connects first stage basic algebra. you are ready to get a taste of higher algebra here. triple and surface integrals.htm . A Guide to Arithmetic (FREE! http://www.html) Courant R. RSA) etc are concerned in this section. 1.net:8090/~keisler/calc. Ikenaga B.pdf) Wachsmuth B. MIT Press. you may wonder what's the http://hk.ac. Introduction to Analysis .htm . The Calculus of Functions of Several Variables (FREE! http://math. Calculus Notes (FREE! http://marauder.edu/~wachsmut/ira/index. Analysis 1 (FREE! http://www.geocities. Problems in Mathematical Analysis Gaughan E.F. and Bruckner A. Cambridge.trillia.wisc.mth. Basic Elements of Real Analysis Rosenlicht M.B.03w/) Not only Terry is a good mathematician.openmathtext.com/zakon-analysisI.org/lecture_notes/vector_calculus_book3.Expensive.edu/~bikenaga/calc/calcnote. Multivariate Calculus in 25 Easy Lectures (FREE! http://www. Mathematical Analysis I (FREE! http://www.bruce-shapiro.E. can you believe it's free? Lewin J.P.millersville. An Interactive Introduction to Mathematical Analysis . Mass. Keisler H. Introduction to Analysis Santos D.aspx) Veeh J. I.pdf) Shilov G.edu/~dcs/mvbook/) Tao T.com/download. Have a look in the library.nyud.html) Kaplan W.H.kcl.Check it out.. Elementary Real Analysis (FREE! http://classicalrealanalysis.com/math250/math250. Bruckner J. While you're in high school.E.com/mathphyweb/puremath. Elementary Real and Complex Analysis .html) Wilde I.How to Become a Pure Mathematician Page 15 of 66 Demidovich B. Thomson B.S.Come with a CD-ROM.edu/~tao/resource/general/131ah. IRA: Interactive Real Analysis (FREE! http://pirate.pdf) Sloughter D.G.math.ucla.D. but also a good educator.html) . Elementary Calculus: An Approach Using Infinitesimals (FREE! http://www.html) 張筑生： 《數學分析新講》第三冊，北京大學出版社 Complex Variables (Introductory Complex Analysis): Another beautiful branch of mathematics. Real Analysis and Foundations .furman. Advanced Calculus Krantz S.The author is a fan of Rudin and he was trying to bridge between less rigorousX books and hard-edged classical books like baby Rudin. aimed particularly for non-mathematicians who need to study this hardest ever course (the author means analysis) in their life.Republication of Mathematical Analysis Vol.net/lecnotes/calc3. Lectures Notes on Multivariate Calculus (FREE! http://javeeh.G.uk/~iwilde/notes/jha/index. Very friendly.M. Protter M. Real Analysis: An Introduction (FREE! http://www.math.html) Zakon E.edu. Shapiro B.shu.J. Vector Calculus Notes (FREE! http://www.A. Complex Analysis (FREE! http://www.gatech.cam. Introduction to Complex Analysis (FREE! http://www.ucla.dpmms.html) Kuttler K. two points boundary problem. and Hoffman M. and Silverman H.math. basic partial differential equation. Bessel's equation. Cain G. Legendre's equation. you will see the beauty of this construction. elliptic eigen-problems.F.V.sfsu. http://hk. Fredholm alternative. Ponnusamy S.caltech. Weierstrass and Cauchy are key figures in this area.ac. Schaum's Outline of Complex Variables Tao T.00w/) Wilkins D. Complex Analysis for Applications (FREE! http://www. Complex Variables with Applications Differential Equation: Isaac Newton originally studies behaviour of dynamical systems using differential equations.pdf) Marsden J.pdf) Beck M.html) Brown J.maths.edu/~bass/ca.edu/~cain/winter99/complex. and Snider A.math. Key concepts to learn are: analytic functions. Functions of a Complex Variable (FREE! http://www.How to Become a Pure Mathematician Page 16 of 66 point of introducing imaginary number i. reduction of order. Green's function.html) Carne T.1. Cauchy-Riemann equations. contour integral.. Frobenius normal form.R. residues and poles.J.edu. Cauchy-Goursat theorem. including physics. Complex Variables and Applications . biology. Marchesi G.mq.R.A.htm . singular).See Marsden's page (http://www. and Fokas A. and Churchill R.edu/beck/complex.S. heat equation and wave equation.edu/~marsden/books/Basic_Complex_Analysis. Gauss. Fundamentals of Complex Analysis Spiegel M.B. Mathematical models are often described with differential equations and they are widely used in many fields. Sturm-Liouville problems (inhomogeneous.byu.edu/~klkuttle/math532notes01. variation of parameters.au/~wchen/lnicafolder/lnica. chemistry.com/mathphyweb/puremath. economics etc.E. and Pixton D.uconn. does it really exists? Is it really useful? In an introductory complex analysis course.maths.L.geocities.edu/~tao/resource/general/132. Complex Variables: Introduction and Applications Bass R. complete orthogonal system.ie/~dwilkins/Courses/214/) Wunsch D. Ablowitz M.math. Mathematical methods used in solving differential equation also play an important in advanced studies. Complex Analysis (FREE! http://www. Complex Analysis (FREE! http://www.uk/site2002/Teaching/IB/ComplexAnalysis/2005-2006/Notes. Key topics to learn are: Wronskians.math.pdf) Chen W.D.J.Study up to chapter 7. series solutions.cds.W.W. Complex Analysis (FREE! http://www.K. Riemann.tcd. Basic Complex Analysis .html) for errata. A First Course in Complex Analysis (FREE! http://math. Complex Variables with Applications Saff E. First six chapters.com/math351/351notes/) Simmons G.Leave chapter 5 and 6 to next stage.aspx) Forsyth A. Differential Equations: Theory.edu/~bikenaga/diffeq/deqnote.F. Fifty Challenging Problems in Probability with Solutions .athenasc. Lectures Notes on Ordinary Differential Equations (FREE! http://javeeh.lamar. Mosteller F.pdf) Bertsekas D.uconn. linear model.M. basic hypothesis testing. ANOVA (analysis of variance) etc. Ross S. conditional probability.G.stat. convergence of r.pdf Boik R.edu/classes/de/de.C. Tutorial Notes (FREE! http://www.L. Probability (FREE! http://www. Santos D.F. Differential Equations Notes (FREE! http://marauder.Leave the last chapter. you should have basic knowledge of statistics. If you want to go a bit further.montana.math. Lecture Notes on Differential Equations (FREE! http://www. http://hk.edu/~bass/elemprob.math.edu/~chance/teaching_aids/books_articles/probability_book/book.purdue.org/lecture_notes/probability_book2.A.N.pdf) Stirzaker D. residual. maximum likelihood estimator. Solution: http://www.html) Shapiro B. The Theory of Probability .html) .How to Become a Pure Mathematician Page 17 of 66 Boyce W.math. Differential Equations (FREE! http://tutorial.pdf) Gnedenko B.E.dartmouth.V.edu/~rjboik/classes/502/notes.Both Simmons and Krantz are writers for mathematicians. Probability Lecture Notes (FREE! http://www. and Tsitsiklis J. Bayes' Theorem).First eight chapters.htm .420. expectation.pdf) Probability and Statistics: Well.Later stages topics included.Read chapter 1-6. J. Grinstead C. Schaum's Outline of Probability . and Krantz S.'s Introduction to Probability (FREE! http://www. Probability: Bass R.com/mathphyweb/puremath.E.v.net/lecnotes/odes.P. Lipschutz S. A First Course in Probability . and Practice .02. Lebanon G. Elementary Probability . categorical predictors.com/probsolved. and Snell J. p-values are the basic. A Treatise on Differential Equations Ikenaga B.Read first ten chapters. Undergraduate Probability (FREE! http://www.openmathtext. being a well educated mathematician.edu/~lebanon/notes/) .). check out linear regression. Elementary Differential Equations and Boundary Value Problems Dawkins P.geocities.R. logistic regression.millersville..Read P1 P14. and Lipson M. Things to learn: basic probability theory (independent events. Veeh J. and DiPrima R. Technique. random variables (r.bruce-shapiro. Introduction to Probability .v. htm) Second course in Statistics: Janke S.htm . Lopuhaä H.leidenuniv. Introduction to Linear Models and Statistical Inference . and Tinsley F.H.M. StatPrimer (FREE! http://www.As a reference.edu/~michael/book.montana.missouristate.geocities.pdf) Bulmer M. Introduction to Regression Models and Analysis of Variance (FREE! http://hk.L.edu/~rjboik/classes/502/notes. A Modern Introduction to Probability and Statistics: Understanding Why and How El-Taha M.com/intro.duke.M.edu/faculty/gerstman/StatPrimer/) Hodges J. Introduction to Probability and Statistics for Engineers and Scientists Simon J. Schaum's Outline of Statistics .html) Taylor J.. and Marx M. Introductory Statistics: Concepts. Introduction to Statistical Thought (FREE! http://www.com/mathphyweb/puremath.montana. Classical and Modern Regression with Applications Miller J.W.resample. Online Statistics: An Interactive Multimedia Course of Study (FREE! http://www. Introduction to Applied Statistics (FREE! http://www-stat.com/) Larsen R.stat-help.math.nl/~redig/lecturenotesstatistics.stat.R. Elementary Statistics (FREE! http://www..E.. Statistics: A Bayesian Perspective Boik R. Statistics (FREE! http://www.math.notes/) Boik R.Non-calculus based text by two renowned statisticians.L. MBA604 Introduction Pobabiltity and Statistics Lecture Notes (FREE! http://www. Resampling: The New Statistics (FREE! http://www.com/content/text/index..edu/~jtaylo/courses/stats191/index.psychstat.com/milleratotago/) Taylor J. Stockburger D.pdf) Gerstman B. Peres C. Introductory Statistics Notes (FREE! http://www. and Haden P.424. Myers R.03.J. J. interested reader may read up to chapter 16 at this stage.L. Models. J.html) Ross S.Examples illustrated with Minitab.A.edu/~rjboik/classes/216/lecture.B.onlinestatbook.L.L.stanford.G. and Meester L. Kraaikamp C. Chapters up to 12 are core.P.edu/sbk00.J. Principles of Statistics DeCoster J. Lane D. An Introduction to Mathematical Statistics and Its Applications Lavine M. and Zitek E. and Applications (FREE! http://www.geocities. Basic Concepts of Probability and Statistics .How to Become a Pure Mathematician Page 18 of 66 Statistics: Berry D. and Lehmann E.pdf) Dekking F. Lu J.math. Statistical Analysis with the General Linear Model (FREE! http://www.shtml) Spiegel M.sjsu. com/mathphyweb/puremath.geocities.html) http://hk.How to Become a Pure Mathematician Page 19 of 66 http://www-stat.edu/~jtaylo//courses/stats203/index.htm .stanford. In the analysis class. Kolmogorov and Fomin's text fits this stage http://hk. '[i]t is a basic principle in the study of mathematics. compactness.F. much more is necessary than merely following the individual steps in the reasoning.com/mathphyweb/puremath. and this process of assimilation should not be abandoned until it yields a full comprehension of the overall pattern of though. that a proof is not really understood until the stage is reached at which one can grasp it as a whole and see it as a single idea.F. continuity and convergence.' Also get maturity and learn how to write mathematics. This is only the beginning. every steps in a proof should be logically related and everyone can go easily from the previous step to the next. It often happens in advanced text that the author skips some steps in a proof or calculation. and Metrens J.htm . As you may see. they can be applied to other graduated level science subjects.e. No. I don't think Gauss or Galois can get full mark in mathematics assignments these days. swallowed. weak topology and stuffs like that. Ian Doust even showed us an article in the Econometrica (Hildenbrad W. and one too seldom emphasized. measure. Don't expect you can solve a problem by plugging numbers into formulae. If you insist. Introductory Analysis: At this stage you should have learnt limits. and 2) they don't bother to explain to others. while elementary text gives detail explanations. you begin to learn modern Pure Mathematics. '[t]he serious student will train himself to look for gaps in proofs. connectedness. one is just ignorant. Focus on the proof. In achieving this end. and should regard them as tacit invitations to do a little thinking on his own. (Of course not done by me!) I had tried to list more books so that you can compare and choose one or two that fit you. Despite that most of following courses are 'pure'. and digested. Contents of analysis courses varies almost everywhere (a. But as G.geocities. topological space. If one insists that the pure stuff is useless. the term analysis appear in the first and second year calculus to forth year analysis. You will study the idea of several spaces (metric space. etc. Hilbert). Vol. This is not the case for non-pure mathematics. linear operators. for 1) something obvious to them may not be obvious to everyone else. I don't bother to argue with those people anymore. 1) talks about liminf. over the reals.). Simmons has said. elementary Lebesgue theory and measure theory. I can show you a copy of full mark statistics assignment consists work of mere calculation and computation. The point is. Upper Hemi-continuity of the Equilibrium-set Correspondence for Pure Exchange Economies. limsup. A proof should be chewed. by reading good books (?). These are central concepts of calculus in both one and several variables. Banach. These can be generalised. Yup. this is the beginning. 40.How to Become a Pure Mathematician Page 20 of 66 Stage 3 At this stage.' Moreover. and Hoffman M.J. Banach spaces and Hilbert spaces.edu/~edgar/752_06/errata. Analysis: An Introduction . One feature is that proofs are often left behind examples. Bruckner A.N. Introduction to the Analysis of Metric Spaces . continuity. gives nice discussion on limit processes.ohio-state. Bruckner J.This book sits somewhere between the stage 2 calculus and stage 3 analysis in this list.htm . Real Analysis (FREE! http://classicalrealanalysis.A.edu.L.pdf). Marsden and Hoffman. and Burkill H. Elements of the Theory of Functions and Functional Analysis .html) Cohen G.V. A Course in Modern Analysis and its Applications Davidson K.edu/~bass/meas.. maybe. Introduction to Metric and Topological Spaces Also deals with Lebesgue theory. Linear Functional Analysis (FREE! http://www.C.com/download.and Thomson B.uconn.au/~wchen/lnilifolder/lnili. Apostol. Stay away from Mathematical analysis for business and finance or the like. It seems that this is a more popular edition. 2 in stage 2. Silverman. Treat this as a reference.N.This version is translated and edited by R. Kolmogorov A. and Fomin S. Searcóid M. Real Analysis with Real Applications Jost J. and Simmons make good references.mq. they cover most of the above topics.B.maths. Elementary Classical Analysis .It covers a lot of stuff and hence too concise for an introductory text. Several books entitled analysis are indeed advanced calculus (stage two calculus) or not aimed for mathematicians. and Donsig A. See the typo by Edgar (http://www.maths.S.V.W.aspx) Chen W. http://hk. A Second Course in Mathematical Analysis Giles J. Real Analysis (Measure Theory) (FREE! http://www. Only for metric spaces and topological spaces (also consult the list of Topology below): Burkill J.If you enjoy his Calculus Vol. then you should go for this.F.W.Ó.P. and Fomin S. Bass R.The material is based on lectures given by the authors in the Moscow State University.com/mathphyweb/puremath. compactness and metric topology.L.pdf) Beals R.One of the Australian Mathematical Society Lecture Series.How to Become a Pure Mathematician Page 21 of 66 well. etc: Apostol T.math. 1 in stage 1 and Calculus Vol.math. Introduction to Lebesgue Integration (FREE! http://www.mq.geocities. Mathematical Analysis .R. Introductory Real Analysis . Metric Spaces Sutherland W.M.edu.M.E.au/~wchen/lnlfafolder/lnlfa. Solution by Shin-Yi Lee can be found here.R.html) Chen W. Marsden J.A. Postmodern Analysis Kolmogorov A.L. ' said the author.pdf) In general: http://hk. Groups.pdf) Tao T.hndts/m104_Rudin_notes.jmilne. ring and field theory. Topics to be covered are basic group.txt).R.uk/~ajb/2q/index.edu/~gbergman/ug.ucdavis. See also George M.G.org/math/CourseNotes/math594g.mcs.com/mathphyweb/puremath.edu/~gbergman/ug.ucla.edu/~gbergman/ug.hndts/06x2+03F_104_q+a.math.maths.ac.Large proportion of this book can be studied at this stage. like Platonic solid and wallpaper patterns.htm .Nice mathematical writing.How to Become a Pure Mathematician Page 22 of 66 Rudin W. groups measure symmetry. Introduction to Topology and Modern Analysis .openmathtext.math. Symmetry and Fractals (FREE! http://www. Rings and Fields Mitchell J.ma. Group Theory (FREE! http://www.03s/) Wilkins D.ps). Groups and Symmetry . Rings and Fields Lecture Notes (FREE! http://www.hndts/m104_Rudin_exs.uk/~anskor/notesM2P4.maths. Baker A. General Topology and Real Analysis (FREE! http://www. Bergman's supplement exercises (http://math.html).ie/~dwilkins/Courses/221/) Abstract Algebra: One of the beautiful branches of mathematics. Santos D.ps) and q&a (http://math. Advanced Calculus (FREE! http:/www. which introductory abstract algebra texts usually skip.N. A Field Guide to Algebra Ellis G.tcd. Analysis 2 (FREE! http://www.harvard.gla.S.st-and. Beginning Functional Analysis .geocities.phtml) Milne J. You will see how we study symmetry.math.Baby Rudin.berkeley.edu/~emsilvia/math127/math127.1.uk/~jamesm/teaching/MT4517/) Skorobogatov A. notes (http://math. Rings and Fields (FREE! http://www-history. Only discuss Groups: Armstrong M.ic.edu/~shlomo/docs/Advanced_Calculus. A modern classic. and Evelyn Silvia's Companion Notes (http://www.berkeley.html) Rings and Fields: Chambert-Loir A. Simmons F.'Numbers measure size. Principles of Mathematical Analysis . This book deals with several topics.ac.A.org/lecture_notes/david_santos_calculus1.pdf) Saxe K.berkeley. Sternberg S. Real Analysis/Advanced Calculus (FREE! http://www.ac.edu/~tao/resource/general/131bh. Get a copy and play around with it. Abstract Algebra Done Correctly (FREE! http://www. Clark A. Elements of Abstract and Linear Algebra (FREE! http://www. Abstract Algebra (FREE! http://marauder. Algebra: Abstract and Concrete (FREE! http://www. it's best to attend Daniel's lecture.math. Connell E.D. Read sections recommended by the authors in the preface.pdf) Artin M.html) Birkhoff G.se/users/m/mauritz/math/alg/index. quotations.edu/~goodman/algebrabook.torget. Abstract Algebra .Kind of handbook.As the author have said. dozens of photographs.M.au/~danielch/algebra07a/qinnotes. leave others to the next stage.com/mathphyweb/puremath. you may simply follow his list at this stage.umn. Artin. Artin states in "A Note for the Teacher" that several sections would make a coherent course.New edition of his Lectures in Abstract Algebra.H.Not aimed at this level. Intro to Abstract Algebra (FREE! http://www.If you could afford this. Modern Algebra with Applications Gillian J. Goodman F. Fraleigh J.d. Abstract Algebra Online Study Guide (FREE! http://www. and Blair W. I recommend this but I have no direct relationship with M.edu/~dvb/algebra/algebra. Abstract Algebra Ikenaga B.pdf). and Feil T. Algebra . Artin was his supervisor. See also Afra's incomplete notes (http://www.umn.H.purdue.pdf) Gilbert W.miami.uvm.math. A First Course in Abstract Algebra: Rings.J. Beachy J.math. My lecturer recommended this partly because M.edu/~beachy/abstract_algebra/study_guide/contents.Fraleigh made it.B.M. and Foote R." If you prefer those things. hundreds of figures. and Fields Arapura D.maths. and numerous tables and charts. Standard recommended text in the U.html) Herstein I. but this is the one you like to keep in your own library.htm . Elements of Abstract Algebra .uiowa.cs. See the author's website (http://www. historical notes. Higher Algebra Lecture Notes (FREE! http://web. See Foote's web (http://www. Basic Algebra I .Diagram omitted. http://hk. Ideal for a first coures.edu/~afra/goodies/abstract. A First Course in Abstract Algebra .edu/~jgallian/) for resources.htm) Chan D.A.math.math. he gives a good introduction abstract algebra for newbie. A Survey of Modern Algebra Blomqvist C.niu.pdf) . Algebraic Systems (FREE! http://hemsidor. poems.dartmouth.edu/~garrett/m/intro_algebra/notes.emba.edu/~bikenaga/abstractalgebra/abstractalgebranotes.edu/~foote/) for errata. Groups. Garrett P.S.millersville.edu. this book includes 'lines from popular songs.A.html) Jacobson N.unsw.S. As M.geocities.How to Become a Pure Mathematician Page 23 of 66 Anderson M. and Mac Lane S. Contemporary Abstract Algebra .dir/algebrabook. biographies.edu/~ec/book/) Dummit D. How to Become a Pure Mathematician Page 24 of 66 Knapp A.W. Basic Algebra - See the web (http://www.math.sunysb.edu/~aknapp/books/b-alg.html) for correction. Rotman J.J. A First Course in Abstract Algebra - Check out Rotman's page (http://www.math.uiuc.edu/~rotman/) for errata. Shahriari S. Lectures on Algebra I (FREE! http://pages.pomona.edu/~sshahriari/shahalg1-16.pdf) 李華介：大學基礎代數 (FREE! http://math.ntnu.edu.tw/~li/algebra-html/) - The best free material on introductory abstract algebra I have seen so far, provided you can read Chinese. Introductory Number Theory: Gauss once said "mathematics is the queen of sciences and number theory the queen of mathematics." G..H. Hardy once said in A Mathematician's Apology that, "If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications. But science works for evil as well as for good (and particularly, of course, in time of war); and both Gauss and less mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean." He wrote it on November 1940. It turned out coding theory, where algebra and number theory are applied, played a key role in the World War II. Adler A. and Cloury J.E. Theory of Numbers: A Text and Source Book of Problems Baker A. Algebra and Number Theory (FREE! http://www.maths.gla.ac.uk/~ajb/dvi-ps/3q-notes.pdf) Burton D.M. Elementary Number Theory - One of the better books for an introduction. Chen W.W.L. Elementary Number Theory (FREE! http://www.maths.mq.edu.au/~wchen/lnentfolder/lnent.html) Clark W.E. Elementary Number Theory (FREE! http://www.math.usf.edu/~eclark/elem_num_th_book.pdf) Coppel W.A. Number Theory: An Introduction to Mathematics: Part A, B - You may leave Part B to the next stage. Dantzig T. and Mazur J. Number: The Language of Science - Non-mathematical treatment indeed. Dirichlet P.G.L. Supplements by Dedekind R. Lectures on Number Theory Erdos P. and Suranyi J. Topics in the Theory of Numbers Everest G. and Ward T. An Introduction to Number Theory Fine B. and Rosenberger G. Number Theory: An Introduction via the Distribution of Primes Ikenaga B. Number Theory (FREE! http://marauder.millersville.edu/~bikenaga/numbertheory/numbertheorynotes.html) http://hk.geocities.com/mathphyweb/puremath.htm How to Become a Pure Mathematician Page 25 of 66 Joens G.A. and Jones J.M. Elementary Number Theory Moser L. An Introduction to the Theory of Numbers (FREE! http://www.trillia.com/moser-number.html) Nathanson M.B. Elementary Methods in Number Theory Niven I., Zuckerman H.S. and Montgomery H.L. An Introduction to the Theory of Numbers Parent D.P. Exercises in Number Theory Redmond D Number Theory Rosen K.T. Elementary Number Theory - See the website (http://www.aw-bc.com/rosen/) for supplementary material. Santos D. Number Theory Notes (FREE! http://www.openmathtext.org/lecture_notes/number_theory_book.pdf) Sato N. Number Theory (FREE! http://www.artofproblemsolving.com/Resources/Papers/SatoNT.pdf) - IMO level. Shoup V. A Computational Introduction to Number Theory and Algebra (FREE! http://www.shoup.net/ntb/) Silverman J.H. A Friendly Introduction to Number Theory - See the web (http://www.math.brown.edu/~jhs/frint.html) for extra material. Stein W. An Explicit Approach to Elementary Number Theory (FREE! http://modular.fas.harvard.edu/edu/Fall2001/124/lectures/) Stein W. Elementary Number Theory (FREE! http://modular.math.washington.edu/ent/) 李華介：基礎數論 (FREE! http://math.ntnu.edu.tw/~li/ent-html/) Introductory Topology: Topology is an extension of geometry and built on set theory, it's about continuity. There is a joke that topologists are those who cannot distinguish between a doughnut and a coffee or tea cup. Wikipedia has a [.gif] to demonstrate this (http://en.wikipedia.org/wiki/Image:Mug_and_Torus_morph.gif). You may have studied open sets, open neighbourhood, interior, closure, boundary, basis, continuity, compactness and connectedness etc. in the analysis course, move on to homemorphism, homotopy and fundamental group. As in abstract algebra and analysis, several selected books or notes cover more than enough. You may leave a few sections to study at the next stage. Crossley M.D. Essential Topology - Undergraduate introductory level, detailed explanation. Davis S.W. Topology - It's like lecture notes, many proofs are left as exercises and many exercises are proofs. Dugundji J. Topology - Another topology text with quality. Engelking R. General Topology Gemignani M.C. Elementary Topology Henle M. A Combinatorial Introduction to Topology - Plenty of illustrations. Indeed many http://hk.geocities.com/mathphyweb/puremath.htm How to Become a Pure Mathematician Page 26 of 66 concepts in this subject can and should be visualised. Ikenaga B. Notes on Topology (FREE! http://marauder.millersville.edu/~bikenaga/topology/topnote.html) McCluskey A. and McMaster B. Topology Course Lecture Notes (FREE! http://at.yorku.ca/i/a/a/b/23.htm) Mendelson B. Introduction to Topology - Very nice introduction to the metric spaces, topological spaces, connectedness and compactness. Recommended for first reading. Moller J.M. General Topology (FREE! http://www.math.ku.dk/~moller/e03/3gt/notes/gtnotes.pdf) Morris S.A. Topology Without Tears (FREE! http://uob-community.ballarat.edu.au/~smorris/topology.htm) Munkres J.R. Topology - Another nice introduction. Go further than Mendelson's one. Also cover topics like the Tychonoff theorem, metrization theorems, complete metric spaces and function spaces, the fundamental group and covering spaces. Singer I.M. and Thorpe J.A. Lecture Notes on Elementary Topology and Geometry Steen L.A. and Seebach J.A. Counterexamples in Topology Strickland N. Topology Notes (FREE! http://neil-strickland.staff.shef.ac.uk/courses/topology/) Thurston W.P. The Geometry and Topology of Three-Manifolds (FREE! http://msri.org/publications/books/gt3m/) Ward T.B. Topology Lecture Notes (FREE! http://www.mth.uea.ac.uk/~h720/teaching/topology/materials/topology.pdf) Wilkins D.R. Topology (FREE! http://www.maths.tcd.ie/~dwilkins/Courses/212/) Differential Geometry: You may wonder, geometry takes up a large portion in high school mathematics, why isn't there any geometry course in the first two stages? In fact, geometry is kind of imbedded in stage two calculus (several variables) and linear algebra courses, they are usually assumed and will be used for this course. Here differential calculus is used to study geometry. Key things to study: multilinear algebra, curvature and torsion, Serret-Frenet equation, fundamental theorem of curves, Poincaré Index theorem (plane and surface), exterior calculus, Gauss' theorema egregium, geodesics, Gauss-Bonnet theorem. Moving frame, due to Élie Cartan, is an approach to geometry of surface. According to my lecturer John Steele, it is "computationally the easiest, notationally the neatest, aesthetically the best, makes the definitions more natural and the proofs of the two major theorems easier. The moving frame method also points the way towards several important ideas in modern differential geometry and theoretical physics. The downside (if there is one) is the reliance on exterior calculus of differential forms." O'Neill, for example, uses this approach and he manages to prove Gauss' theorema egregium in half page, see p.281. In general: http://hk.geocities.com/mathphyweb/puremath.htm get a taste of the so called French style.P.tcd. Sharipov R.wisdom.org/abs/math/0412421) Struik D.Quite popular for introductory level.R.math. Differential Geometry and Topology of Curves Csikós B. A Geometric Approach to Differential Forms Cartan H. which is nice. Check out the errata list by Bjorn Poonen (http://math.org/abs/math/0403252/) Spivak M.htm .P.cs.. Tensor Analysis Munkres J.pdf) Kreyszig E.I.edu/oldweb/courses/2002-2003/cds202/textbook/) Bishop R. Elementary Differential Geometry . Differential Geometry of Curves and Surfaces .R.Not as excellent as his books on single variable calculus and the five volumes differential geometry. Beware that ^ means cross product. Differential Forms .ie/~zaitsev/ln.cds. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus .Member of Bourbaki.S. Analysis on Manifolds Sharipov R. Differential Geometry (FREE! http://www.arxiv.purdue.L. Differential Geometry: Lecture Notes (FREE! http://www. and Goldberg S. Tensor Analysis on Manifolds Lebedev L.Solution at the back. b> means a dot b or inner product in this text. Course of Differential Geometry (FREE! http://arxiv. still better than many authors. Differential Forms with Applications to the Physical Sciences http://hk.J. and <a.maths.E. Hicks N. Notes on Differential Geometry (FREE! http://www. Kreyszig does that. Differential Forms: Arapura D. Millman R. Differential Forms and Applications Flanders H. Differential Forms and Connections do Carmo M.geocities.edu/~poonen). and Parker G.html) do Carmo M. Introduction to Differential Forms (FREE! http://www. Analysis and Applications (FREE! http://www.com/mathphyweb/puremath.il/~yakov/scanlib/hicks. Elementary Differential Geometry Pressley A. Quick Introduction to Tensor Analysis (FREE! http://uk. Lectures on Classical Differential Geometry Zaitsev D.How to Become a Pure Mathematician Page 27 of 66 Animov Y.D.P.pdf) Bachman D.J.weizmann.hu/geometry/csikos/dif/dif. Manifolds. and Ratiu T.ac.W. Tensors.Neither do Carmo nor O'Neill introduce the matrix notation when they first discuss the Frenet formulae.J. Darling R. Elements of Differential Geometry O'Neill B.berkeley. Marsden J. and Cloud M.pdf) Tensor Analysis and Manifolds: Abraham R.caltech.elte. Differential Geometry .edu/~dvb/preprints/diffforms. Using mathematics to model real world phenomena is useful." George Box.H. Garthwaite P.J. Used widely in grad school. http://hk. Statistical Inference .ox.R.edu/~mark/degroot/typos/typos. partly because people from other fields only learn this material in graduated schools. Lectures on Statistics (FREE! http://www. An Introduction to Mathematical Modelling (FREE! http://www.T.edu/~r-ash/Stat. Probability and Statistics .J.cmu. and Engelhardt M. sort of. The Nature of Mathematical Modeling Giordano F. Statistical Inference .ntnu.L. and Schervish M. and Jones B. Mann-Whitney U test. Wilcoxon signed rank test. A First Course in Mathematical Modeling Meerschaert M. you will learn here the Cramer-Rao bound.. Fowler A.html) Bain L.Detail explanation.45).ac.ac.uk/~fowler/courses/tech/technotes. and Berger R. Note a few texts say they are aimed at graduated level. McNemar's chi-square test.How to Become a Pure Mathematician Page 28 of 66 Morita S. some are useful.htm .net/lecnotes/model. Introduction to Probability and Mathematical Statistics Casella G.html).B. Kolmogorov-Smirnov two-samples test. basic bootstrap and robustness.pdf) Fowler A. Neyman-Pearson theory. Check out the typo page (http://www. Jolliffe I. stage 2 stuff) for them. stage 2 probability and statistics is the assumed knowledge for this text.math.ox. Geometry of Differential Forms Mathematical Modelling (optional): "All models are wrong.no/~ushakov/emner/ST2201/v07/files/sol. Spearman's R and Kendal's Tau).P.stat.A.com/mathphyweb/puremath. Bayesian inference. Unlike the rest recommended texts for this course.C. Techniques in Applied Mathematics (FREE! http://www. Weir M.maths. uniform minimum variance unbiased estimators. for example: "Probability Zero Does Not Mean Impossible" (p.pdf).uiuc.A standard text.. but not the main concern of pure mathematician. Can be served as a reference. and Fox W.Made a few good comments. Ash R.geocities.D.uk/~fowler/courses/o11intro/o11notes.pdf) Gershenfeld N.M.C. as a result the texts have to set up basic probability and statistics (you have done it.maths. Lectures Notes on Mathematical Modeling (FREE! http://javeeh.math.H. etc. Mathematical Modeling Veeh J. DeGroot M.17) and Statistical Swindles (p. introductory non-parametric (including the sign test. Solutions Manual available on web (http://www.pdf) Statistical Inference (optional): Build upon stage 2 statistics. but perhaps too concise as a textbook. Common misconceptions are mentioned. Wald-Wolfowitz runs test. Kruskal-Wallis analysis of ranks. html) Mood A.purdue. This topic is particularly useful for electrical and computer engineers.W. If you want to learn this with measure. A First Course in Stochastic Processes http://hk..C. and Taylor H.edu/mckean/HMC/).stat. Graybill F.edu/lowry/webtext.ac.Classical text modernised.The main feature of this book is that it contain lots of figures. Schaum's Outline of Probability. actuarial study. 11 pages of errata (http://www. Mukhopadhyay N.uk/~rrw1/stats/notes.G. and Davisson L. finance or things like that. queueing theory.ellerman.wmich. An Introduction to Probability and Statistical Inference Sprott D.M.org/Davids-Stuff/Maths/Rota-Baclawski-Prob-Theory-79. Probability and Statistical Inference Rice J.htm . Brownian motion and stationary processes. Statistics (FREE! http://www.R.M. Introduction to Statistical Signal Processing (FREE! http://ee. and Craig A. Introduction to the Theory of Statistics .As my lecturer Spiridon Penev have said.pdf) Young G.T."A colorful and rich introduction to probability theory and its applications" according to Karlin and Taylor.geocities. renewal theory.A.ndml/1175196978) Weber R. Lebanon G. An Introduction to Probability and Random Processes (FREE! http://www. Tutorial Notes (FREE! http://www. Introduction to Mathematical Statistics .A. but it "is limited in that it deals only with discrete probabilities. Probability and Random Processes Hsu H. A Course in Mathematical Statistics Roussas G.stat.html Roussas G. On the Principles of Statistical Inference (FREE! http://projecteuclid.berkeley. See also left material from stage 2 probability. not so user friendly for people from other fields.M.V. Markov chains.Read S1 S12. Random Variables.. Statistical Inference in Science Wald A. Quite professional. see stage 4 probability list.R. Lowry R. and Smith R. McKean J. this text is a bit outdated but it contains useful exercise problems. 1 .com/mathphyweb/puremath. Concepts and Applications of Inferential Statistics (FREE! http://faculty.-C. Essentials of Statistical Inference . Baclawski K.edu/~gray/sp. it's a pain for customers to correct such long list of typos.G. and Random Processes Karlin S. reliability theory. An Introduction to Probability Theory and Its Applications Vol. If you are serious about Statistics. Poisson process.html) Grimmett G.pdf) Feller W. Probability and Stochastic (Random) Processes (optional): Conditional expectation.D.stanford.edu/~lebanon/notes/) .. perhaps.L. and Stirzaker D.org/euclid.statslab.A bit advanced for a first course.cam.How to Become a Pure Mathematician Page 29 of 66 Hogg R. Check the at: http://stat-www.A.edu/~rice/Book3ed/index.. and Boes D. Mathematical Statistics and Data Analysis .A.vassar." Gray R. and Rota G. not an introduction to basic probability theory. in particular the section "an Introduction to R" is completely included free on web.mhhe.ehu. so I recommend books of this kind. and Venables B. R Tutorial (FREE! http://www. and Smith D.ac.pdf) Spector P.geocities. and Olson M.Note this is an introduction to probability and stochastic processes. analyses and examples from the book are available at http://cran. Dalgaard P. Lecture Notes on Probability Theory (FREE! http://people. Probability.org/pages/newhtm/r/tchtml/ and http://www.stat.r-project. data sets.et.ucla. S-PLUS uses the same language. A Handbook of Statistical Analyses Using R .biosino. you can't do statistics without computer.pdf) Spector P.J.edu/users/rice/Stat230A/SpectorIntroR.pdf) Yates R. This text could be quite hard to follow without knowing very basic probability theory. Introduction to Probability Models . no graph has been used when the author introduce Markov chains for the first time.eecs. Krause A.purdue.Many functions.pdf) Ripley B.S.ats. in practice. Random Variables and Stochastic Processes . so he includes the topic survival analysis in this introductory text.org/).D. R 丁國徽譯： http://www.html .pdf) Venables B.How to Become a Pure Mathematician Page 30 of 66 Krishnan V.edu/~lebanon/notes/) .M. Lecture Notes on Probability Theory and Random Processes (FREE! http://robotics. Mörters P. Introductory Statistics with R .Read R1 R5.pdf) 《R 導論》《R 文檔》(FREE! ， Venables B. Everitt B. The Basics of S and S-Plus Kuhnert P. Also note that. and Smith D.berkeley.M.biosino. An Introduction to R: Software for Statistical Modelling and Computing (FREE! http://www.es/~etptupaf/pub/R/Rlecturenotes.edu/users/rice/Stat230A/SpectorRtutorial. Ross S.stat. and Hothorn T. and Goodman D.com/mathphyweb/puremath.org/doc/manuals/R-intro.org/R/R-doc/) http://hk.ps) Papoulis A.M.berkeley. Walrand J. Probability and Stochastic Processes: An Friendly Introduction for Electrical and Computer Engineers Statistical Computing (optional): These days.stat. Probability and Random Processes Lebanon G.edu/~wlr/126notes. An Introduction to R (FREE! http://www.bath.uk/maspm/prob.r-project.bs.See their web (http://www.com/engcs/electrical/papoulis/) for resources.org/web/packages/HSAUR/index.berkeley. An Introduction to R (FREE! http://cran.htm . R is a freeware (http://www. Tutorial Notes (FREE! http://www.edu/stat/SPLUS/library/sguide.D. Introductory Guide to S-Plus (FREE! http://www.r-project.The author is a biostatistician. How to Become a Pure Mathematician Page 31 of 66 Venables W. See their web (http://www.edu/~calvinw/MthSc885/yakirsplus. Yakir's notes (http://www.N.ac. http://hk.com/mathphyweb/puremath. Modern Applied Statistics with S .pdf).geocities.htm .D.stats.clemson.ox. See also B. and Ripley B.uk/pub/MASS4/) for extra material.ces.More advanced topics are covered. staff. Wittgenstein L. further reading in Number Theory and algebraic geometry in Geometry.. Foundations of Mathematics (FREE! http://www. Principia Mathematica to *56 .edu/simpson/courses/math558/fom. let me know if you have a suggestion! Note that at this stage.G.Oxford Texts in Logic 3.uk/teaching/STDN. Moreover.First part of the landmark work. What Is Mathematics? An Elementary Approach to Ideas and Methods Mac Lane S.. (ed.psu. Remarks on the Foundations of Mathematics Logic. further reading should belong to stage 5.) Handbook of Mathematical Logic . or the basic graduated year in other places. What is Mathematical Logic? Barwise J.Interested reader may find his paper Logic and Mathematics (http://www. and Williams N.umich.math. Set Theory (FREE! http://www.H.org/bertrand-russell/the-principles-of-mathematics/) Simpson S.: Ash C.g. and Robbins H.Not for beginners.C. A few subfields may be highly related. Brickhill C. Church A.edu/simpson/foundations. Whitehead A.J. I'm far from being an expert. and Hodges W. and Russell B.J.com/mathphyweb/puremath.shef.psu.pdf) .N.Oxford Texts in Logic 2.html) worth reading.math.geocities. Proof and Disproof in Formal Logic: An Introduction for Programmers .0001. Foundations and Discrete Mathematics: Foundation: Courant R.001) Whitehead A. I shall outline the key topics and give short lists of references. the levels or difficulties of books within a certain topic may vary a lot.. Crossely J. Mathematical Logic .ac. and Russell B. Principia Mathematica (FREE! http://name. Set Theory.N.How to Become a Pure Mathematician Page 32 of 66 Stage 4 Equivalent to the (fourth) honours year in Australia. The Principles of Mathematics (FREE! http://fair-use.peter-dixon. e. you may scout around to see what makes you interest. etc. Chiswell I.G. mind you.edu/AAT3201. Stillwell J.PDF) http://hk.N. Introduction to Mathematical Logic Dixon P.umdl.htm . Bornat R. Mathematics: Form and Function Russell B. Graphs and Homomorphisms . Graph Theory Hell P.rhul.no/~michal/und/i227/book/book.A. and Moerdijk I.ac.htm . Lectures in Logic and Set Theory.G. and Algorithms Bang-Jensen J. A Course on Mathematical Logic Suppes P.Discuss in their blog (http://blogs. and Zaring W. Graph Theory: Modeling. and Jech T. Set Theory Hedman S. Hrbacek K.B. Scheinerman E. A First Course in Logic: An Introduction to Model Theory.uk/books/dbook/) Bollobas B.pdf) Srivastava S.pdf) Graph Theory: Agnarsson G. and Greenlaw R. Applications. Mathematical Logic (FREE! http://www.geocities. Computability.cs. and Nesetril J. Modern Graph Theory Bondy J. and Ullman D.html) for errata.M. Fractional Graph Theory: A Rational Approach to the Theory of Graphs Tutte W.I.theory/) Harary F. Volume I: Mathematical Logic Tourlakis G.com/bondyandmurty/).See their web (http://www. Introduction to Logic (FREE! http://www. A Course in Mathematical Logic Rautenberg W.com/mathphyweb/puremath.springer.M. Graph Theory (FREE! http://www. Introduction to Graph Theory http://hk. Sheaves in Geometry and Logic: A First Introduction to Topos Theory Manin Y.T. Introduction to Set Theory Jech T.math.Oxford Texts in Logic 1. and Complexity .ca/~pavol/hombook.ii.ecp6.sfu.How to Become a Pure Mathematician Page 33 of 66 Hausdorff F. Set Theory Komjáth P.fr/pageperso/bondy/books/gtwa/gtwa. Lectures in Logic and Set Theory. Algorithms and Applications (FREE! http://www. Graph Theory (FREE! http://www.de/home/diestel/books/graph.R. A Concise Introduction to Mathematical Logic Simpson S.math. and Murty U. Problems and Theorems in Classical Set Theory Mac Lane S. Diestel R. and Gutin G. Volume 2: Set Theory Walicki M.html) .R.psu. Axiomatic Set Theory Takeuti G. and Totik V.uni-hamburg. Introduction to Axiomatic Set Theory Tourlakis G.S. Digraphs: Theory. Proof Theory.cs.H.edu/simpson/courses/math557/logic.jussieu.uib. Graph Theory As I Have Known It West D. Number Theory.ac. Introduction to Modern Cryptography (FREE! http://www-cse. Applied Combinatorics van Lint J. A First Course in Coding Theory Humphreys J.phy.ucsd.H.com/mathphyweb/puremath. Introduction to Combinatorics Bona M. and Bellare M. Hill R. and Williamson S.W. The Mathematics of Ciphers: Number Theory and RSA Cryptography Goldreich O. and Rogaway P.inference. and Prest M.M.html) Berlekamp E.C. How to Count Without Counting Ryser H. Combinatorial Problems and Exercises Niven I. Coding and Information Theory (not so pure): First stage introductory discrete mathematics and basic linear algebra are usual assumed knowledge. The Foundations of Cryptography (FREE! http://www.How to Become a Pure Mathematician Page 34 of 66 Combinatorics: Bender E. this is the origin.htm .cam.edu/~wilf/DownldGF.F.ucsd. and Wilson R.weizmann.875/fall01/index.il/~oded/foc-drafts.Y. Groups and Codes . Combinatorics of Permutations Kisacanin B.mit. Mathematical Problems and Proofs: Combinatorics. Information Theory. Johnson O.math. Inference.An introductory text.B. Cryptography and Cryptanalysis (FREE! http://courses. Information Theory and the Central Limit Theorem Koblitz N. Coding and Information Theory .M.R. you must learn Hamming error correcting codes. and give me a reason why http://hk. Ash R.wisdom. and Learning Algorithms (FREE! http://www.edu/users/mihir/cse207/classnotes. Numbers.edu/~ebender/CombText/) Berman G. Bellare M.ucsd.csail.edu/6.math.html) MacKay D.uk/mackay/itila/) .G.html) Hamming R.upenn.ac.Friendlier than Roman. Lecture Notes on Cryptography (FREE! http://www-cse.geocities. A Course in Combinatorics Wilf H.html) Goldwasser S. and White Constructive Combinatorics Tucker A.html) Cryptography. Generatingfunctionology (FREE! http://www. Information Theory . and Geometry Lovasz L. Algebraic Coding Theory Coutinho S.A. A Course in Number Theory and Cryptography Lysyanskaya A.If you are serious about this field (I mean you are not merely interested in cracking ciphers like those the Zodiac Killer used). Foundations of Combinatorics with Applications (FREE! http://www. Combinatorial Mathematics Stanton D.edu/~mihir/papers/gb.Take a look. Mathematics of Choice: Or. uwaterloo.pdf) Shilov G.htm . Cryptography: The Science of Secret Writing . Milman V. and Vanstone S. with Wolff M. Vaudenay S.com/mathphyweb/puremath.A. van Lint J. Algorithms.A. A Course in Functional Analysis Eidelman Y. Functional Analysis (FREE! http://www. Elementary Functional Analysis .ca/hac/) Mollin R. Error-Correcting Codes and Finite Fields Roman S. Introduction to the Theory of Error-Correcting Codes Pretzel O. Menezes A.cacr.ch/crypto/) for extra material.edu/phd_students/vilensky/Functional_Analysis. Topological Vector Spaces Serfaty S. suitable for non-mathematicians. Functional Analysis: Belton A.math.Algebra and probability required. MIT Press.html) Treves F. Coding and Information Theory ..J. basic abstract algebra required. Topological Vector Spaces.html).math.edu/~garrett/m/fun/) Kreyszig E. Functional Analysis (FREE! http://www. A Classical Introduction to Cryptography: Applications for Communications Security . Introductory Functional Analysis with Applications Riesz F.P.See the web (http://www. Salomaa A. Functional Analysis Schaefer H.C.mat.ucc.pdf) http://hk. II.H.at/~gerald/ftp/book-fa/index. Distributions and Kernels Ward T. Functional Analysis . Functional Analysis Notes (FREE! http://www..univie. Handbook of Applied Cryptography (FREE! http://www. Applied Cryptography: Protocols. Rudin W.com/book-applied. Analysis: Materials listed in stage 3.See author's site (http://www.geocities.Riesz is a key figure in this field.ac.Republication of Mathematical Analysis Vol.Elementary.ie/en/euclid/StaffProfiles/BeltonAlex/Teaching/) Conway J. and Sz. van Oorschot P. Functional Analysis Lecture Notes (FREE! http://www. Introduction to Coding Theory .umn. and Source Code in C .H. Teschl G.vaudenay.uea.E. Functional Analysis (FREE! http://www.R.How to Become a Pure Mathematician Page 35 of 66 you shouldn't download it if you are studying this subject.S. Cambridge.B.mth.-Nagy B.B.C. An Introduction to Cryptography Pless V.ac.nyu. and Tsolomitis A. Mass. Smith L.schneier.uk/~h720/teaching/functionalanalysis/materials/FAnotes. Functional Analysis: An Introduction Garrett P. may be studied here.More advanced text.math. Public-Key Cryptography Schneier B. which have not been covered there. ac. Measure Theory . Analysis of Functions of a Single Variable (FREE! http://spot.pdf) Schilling R. Measure Theory (FREE! http://www-maths. Measure Theory Cohn D. C*-algebras (FREE! http://www.html) Yosida K.analysis/analysis. A Primer of Lebesgue Integration http://hk.kcl. Complex Analysis . A Modern Theory of Integration Bear H.com/mathphyweb/puremath. Ash R.ac.html) Bartle R.Classic in complex analysis. Integrals and Martingales . Functional Analysis (topological vector space version) (FREE! http://www. Oxtoby J.ac.htm) Halmos P. Measure and Category: A Survey of the Analogies between Topological and Measure Spaces Sattinger D. Hjorth G.R.edu/~r-ash/CV.F. Functional Analysis (FREE! http://www.L.mth. Measure Theory & Integration (FREE! http://www. Measures.H.kcl.edu/~greg/measure. Measure Theory Fremlin D.html Hilbert Spaces: Halmos P. A Hilbert Space Problem Book Halmos P.mth.Another nice text by Halmos.B.html) Wilde I.html) Baggett L.G.edu/~baggett/.motapa.uk/~iwilde/notes/calg/index. An Introduction to Hilbert Space Real and Complex Analysis: Ahlfors L.math.colorado. hints and solutions at http://www. Functional Analysis Measure Theory: See also the probability section below.uk/~iwilde/notes/fa1/index.R.mth.uiuc. Introduction to Hilbert Space And the Theory of Spectral Multiplicity Young N.I. Additional material.kcl.ac.C.pdf) Elementary notes.ac.math.R.How to Become a Pure Mathematician Page 36 of 66 Wilde I. Complex Variables (FREE! http://www.de/measures_integrals_and_martingales/index.edu/~dhs/MeasTheory.F.H.essex. Bogachev V.S.math. and Novinger W.swan.uk/maths/staff/fremlin/mt.geocities.html) Liskevich V.uk/staff/vl/c98.htm .usu.F.ucla.A friendly introduction to these topics.L. Measure Theory (FREE! http://www.html) Wilde I.uk/~iwilde/notes/fa2/index.P. Measure Theory (FREE! http://www. Real Analysis: Measure Theory. Abstract Harmonic Analysis: Volume 2: Structure and Analysis for Compact Groups.geocities. and Ross K.B.ps) Dales H. Classical Complex Analysis Krantz S. Rudin W.Big Rudin! Scheidemann V. Real Analysis Markushevich A.edu/~shlomo/docs/Real_Variables. make this book beautiful. Eschmeier J. Fourier Series Hewitt E.K.B.V. Geometric Integration Theory Fourier and Harmonic Analysis: Axler S. and Hilbert Spaces Sternberg S.M. Analysis Now Royden H.E. Aiena P.. and Churchill R.pdf) Whitney H.J.M.G. and Epstein B. Group Representations Hewitt E. and Mcshane Lang S. as the title suggested.W.net/HFT. Laursen K.Fairly good text.G. Harmonic Analysis and Partial Differential Equations (FREE! http://www.axler. and Shakarchi R. and Swartz C.html) Brown J. Pedersen G. and Shakarchi R.How to Become a Pure Mathematician Page 37 of 66 Conway J.A.math.G. Multidimensional Real Analysis I.B.harvard.Plenty of diagrams. Introduction to Complex Analysis in Several Variables Stein E.. Visual Complex Analysis . and Kolk J. and Harmonic Analysis Hardy G.math.I. Partial Differential Equations and Complex Analysis Kurtz D.H. and Ramey W.C. Theories Of Integration: The Integrals Of Riemann. Integration Theory. Functions of One Complex Variable I.htm . An Introduction to Harmonic Analysis http://hk. Theory of Functions of Real Variable (FREE! http://www. Real Analysis: Modern Techniques and Their Applications Freitag E.W. and Kenig C.A.A.chalmers. Fourier Series and Boundary Value Problems Dahlberg B.E.se/Math/Research/GeometryAnalysis/Lecturenotes/HAPE. and Rogosinski W.A.-S. Function Theory of Several Complex Variables Krantz S. II Folland G. and Busam R.S. Lebesgue. Bourdon P. Real and Complex Analysis . Introduction to Banach Algebras. II Duistermaat J.J. Real Analysis . Complex Analysis Stein E.W. Operators. Complex Analysis Hahn L.L. Theory of Functions of a Complex Variable Needham T. Henstock-Kurzweil. Principles of Fourier Analysis Katznelson Y. and Ross K. Abstract Harmonic Analysis: Volume 1: Structure of Topological Groups. See the web for extra information.com/mathphyweb/puremath. Integration. Harmonic Function Theory (FREE! http://www. and Willis G. Analysis on Locally Compact Abelian Groups Howell K. Abstract Algebra: The Basic Graduate Year (FREE! http://www.James Wilson has a solutions manual: http://www.math.07w) Varadhan S.M.au/~danielch/) contains lots of algebra stuffs.B.edu/~tao/247b.ca/~ilaba/wolff/) Algebra: Materials mentioned in stage 3.berkeley. '[t]his will teach you how to run if you know how to walk. Only a few of us dare to ignore it. Algebra Course Notes (FREE! http://www. Topics in Algebra Hungerford T. Smith R. II . Not recommended for a beginner. Notes by A. Jacobson N.html) Bourbaki N. Abstract Algebra Herstein I.pdf. Dummit and Foote. To quote an Amazon customer.to.html) Cohn P.math.math.Non-standard treatment. Lectures in Abstract Algebra I.com/mathphyweb/puremath. for example. Grillet P. A Course in Universal Algebra The Millennium Edition (FREE! http://www. can be studied at this stage.L/).N.P.uiuc.uga. but if you like.math.math. Abstract Algebra (FREE! http://www. Algebra .My first Springer GTM.ubc. Algebra .M.1. Artin.N.C.S. Lang S. but I'm not going to do so.ucla. Fourier Analysis: An Introduction Stein E.edu/~jwilson7/math/Hungerford. one should be comfortable with Lang's Algebra first. Fourier Analysis 1.edu/~tao/247a.R.htm .' Some readers think that if one wanna study algebraic number theory. Lectures on Harmonic Analysis (FREE! http://www.Over 400 pages.A.ucla. II. Vol 1.How to Become a Pure Mathematician Page 38 of 66 Stein E. Algebra. Harmonic Analysis (FREE! http://www.edu/~gbergman/. 2. and Shakarchi R.ca/~snburris/htdocs/ualg. Abstract Algebra with Applications Vol I.Classic. Burris S.W.html) Wolff T.Nicolas Bourbaki is a group of French mathematician aimed to write rigour mathematic which based on set theory. My lecturer Daniel Chan's web page (http://web. Bergman's Companion (http://math.edu/~r-ash/Algebra.edu/faculty/varadhan/harmonic. 3 Garrett P. Ignoring it may look cool.M.maths. See George M. III .unsw. Isaccs I. II http://hk. Blank Modern Higher Algebra (Galois Theory) Ash R. and Sankappanavar H. In general: Artin E.geocities. which have not been covered there.edu/~garrett/m/algebra/) .math.uoregon.uwaterloo.edu.06f and http://www. 2 (FREE! http://www. Algebra: A Graduate Course .M.nyu. Elements of Mathematics: Algebra I.edu/~roy/) Spindler K.A.math.umn. Harmonic Analysis Tao T. search around for topics.math.1. -P. SL2(R) Rossmann W. Workbook in Higher Algebra (FREE! http://www.More advanced work. An Elementary Introduction to Groups and Representations (FREE! http://xxx.maths. Abstract Algebra (FREE! http://www.math. The Linear Algebra .uiuc.J.A Beginning Graduate Student Ought to Know Greub W. Compact Lie Groups Serre J.gov/abs/math-ph/0005032) Hall B. Groups and Representations Baker A.R.How to Become a Pure Mathematician Page 39 of 66 Surowski D. and Fama C. Introduction to Lie Algebras Hall B.L. Lie Algebras Kurzweil H.C. Matrix Groups: An Introduction to Lie Group Theory Chevalley C.htm . Linear Algebra Matthews K. Algebra: Volume I.S. and Bell R.math. Advanced Modern Algebra . and Representations: An Elementary Introduction . Lie Algebras Lecture Notes (FREE! http://hk.B.J.tcd. The Theory of Finite Groups: An Introduction Milne J. An Introduction to the Theory of Groups Schmidt O.lanl.jmilne. Lie Groups: An Introduction through Linear Groups Rotman J. II . Linear Representations of Finite Groups Serre J.edu/~dbski/book. Trees .S. A First Graduate Course in Abstract Algebra Wilkins D.H.F.L.See author's web (http://www.geocities.html) Lang S.org/courses/MP274/) Roman S.pdf) Rotman J.See his web for information.edu/~rotman/) for errata.U. Theory of Lie Groups Erdmann K. perhaps.com/mathphyweb/puremath. Lie Algebras. Linear Algebra Notes (FREE! http://www. Skorobogatov A.J.N.R.J. Abstract Group Theory Sepanski M.ie/~dwilkins/Courses/311/) Advanced Linear Algebra: Golan J. and Stellmacher B.-P.C. A Course in Group Theory Humphreys J. Algebraic Groups and Arithmetic Groups (FREE! http://www.R. and Wildon M. Lie Groups. Advanced Linear Algebra Groups and Lie Algebras: Alperin J.org/math/CourseNotes/aag.numbertheory. Humphreys J. Linear Algebraic Groups Jacobson N.F. Wickless W. van der Waerden B.ksu.For historical interest. pdf) Hadlock C. Ghorpade S.lehigh.R.pdf) Ghorpade S.gla.jmilne. Galois Theory (FREE! http://www.ac. 李華介：簡介 Galois 理論 (FREE! http://math.geocities.pdf) Borceux F.Assumed knowledge is standard algebra.B.Another well written book on this subject. Lie Algebras (FREE! http://www.iitb. Galois Theory Escofier J.uk/~ajb/dvi-ps/Galois.maths.ic.math. i.math.math.ntnu. J.com/mathphyweb/puremath.R.-P. the author also gives historical remarks for the interesting life of Galois.M.edu.html) Morandi P.edu/~shlomo/docs/lie_algebras. The Classical Groups: Their Invariants and Representations Rings.ac.in/~srg/Lecnotes/isant. Fraleigh's one.T. Galois Theory . Field Theory Stewart I.PDF) Sternberg S. Galois Theories Edwards H. Swallow J.iitb. A First Course in Noncommutative Rings Lam T. Exploratory Galois Theory Weintraub S.pdf) Tauvel P.e. Fields and Galois Theory: Note several stage 3 algebra texts also introduce Galois Theory.org/euclid.R. Galois Theory . Field and Galois Theory Roman S.org/math/CourseNotes/math594f. For example. Fields and Galois Theory Lam T.S.See author's website (http://www. and Janelidze G. Fields and Galois Theory (FREE! http://www.ma. stage 3 algebra above.tw/~li/galois-html/) Modules and Representation Theory: Fulton W. Lie Groups.htm .M. Lie Algebras and Algebraic Groups Varadarajan V. Despite the nice treatment of the subject itself. and Their Representation Weyl H. Galois Theory . and Harris J.How to Become a Pure Mathematician Page 40 of 66 http://www.H. and Yu R. Lectures on Field Theory and Ramification Theory (FREE! http://www.N. Noncommutative Rings Howie J.S.ac.in/~srg/Lecnotes/galois. Artin E.Y.uk/~anskor/LIE.ac.edu/~shw2/) for errata. Field Theory and Its Classical Problems Herstein I. Galois Theory (FREE! http://projecteuclid.ndml/1175197041) Baker A. Representation Theory: A First Course http://hk. Exercises in Classical Ring Theory Milne J. Lie Algebras. Notes on Galois Theory (FREE! http://www.W.harvard.Y. Primary Decompositions of Modules (FREE! http://www.cs.ac.cts..pdf) Ghorpade S.R.edu/artsci/math/wells/pub/ttt.Solutions by Jinhyun: http://www.ac.html a scanned version for FREE!) .maths. and Eilenberg S.B.B.G. and Wells C. Lectures on Commutative Algebra (FREE! http://www.edu/~r-ash/ComAlg.de/acc/) Awodey S. Homological Algebra http://hk.case.pdf) Woit P.math..html) Commutative Algebra: Ash R. Introduction to Commutative Algebra (FREE! http://www.uk/personal/ibf/als3/leno.math.iitb.W Lectures on Modular Forms and Galois Representations (FREE! http://math.edu/~dac/iva.ac. Varieties.columbia. and O'Shea D. etc. and Verma J.geocities.iitb.E.C.html) Atiyah M.math.in/~srg/Lecnotes/commalg. and MacDonald I.edu/~woit/repthy.html) for typo.edu/~jinhyun/sol2/comm. Toposes.uiuc.math.html Cox D.in/~srg/Lecnotes/primary_decomp. Herrlic H.R.Y.R. Ideals. K.edu/scans/252/Atiyah-MacDonald/index.pdf) Humphreys J. and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra . Undergraduate Commutative Algebra Zariski O. and Strecker G. Little J.amherst.washington.purdue.edu. Eisenbud D. Triples and Theories (FREE! http://www.pdf) Ghorpade S.html) Cartan H.nott.ac. Commutative Algebra Homological Algebra and Category: Adámek J.uni-bremen.F.in/~srg/Lecnotes/AfsPuneLecNotes.B.tw/Mathematics/english/lecnotes/2006/Lectures on Modular Forms and Galois Representations.math.iitb. Commutative Algebra: with a View Toward Algebraic Geometry Fesenko I.pdf) Reid M.See their web (http://www.nthu. Abstract and Concrete Categories: The Joy of Cats (FREE! http://katmat. A Quick Review of Commutative Algebra (FREE! http://www.com/mathphyweb/puremath. A Course In Commutative Algebra (FREE! http://www. Lectures on Modules and Rings Li W.E.math.math.math.htm . Lie Groups and Representations (FREE! http://www.A. and Samuel P. Introduction to Lie Algebras and Representation Theory Lam T. Introduction to Commutative Algebra (http://modular. Category Theory Barr M.How to Become a Pure Mathematician Page 41 of 66 Ghorpade S. M. algebraic number theory employs algebra result to number theory.A.A.uk/personal/ibf/ha/ha.html) Freyd P. In general: Baker A.math.purdue. Algebraic Number Theory Archives (http://www. Unsolved Problems in Number Theory Hardy G. Introduction to Homological Algebra Weibel C.html Number Theory: There are several branch of Number Theory and we focus in algebraic and analytic here.tac.utwente.W. An Introductory to the Theory of Numbers .math. Conceptual Mathematics: A First Introduction to Categories Mac Lane S. Abelian Categories (FREE! http://www.html) .org/ntw/lecture_notes.H.maths.edu/Algebraic-Number-Theory/) provide papers for further study.See his web (http://www. Introduction to Modern Number Theory: Fundamental http://hk.H.math.cs. http://www.pdf) Fokkinga M.uiuc.mta. and Panchishkin A.math.nl/~fokkinga/mmf92b. An Introduction to Homological Algebra – Correction: http://www. and Pomerance C. Crandall R.M. and Stammbach U.This site (http://www.htm .ca/tac/reprints/articles/3/tr3abs.K.mta. Introduction to Homological Algebra (FREE! http://www.ca/tac/reprints/index. Number Theory: Volume II: Analytic and Modern Tools .tac.fr/~cohen/) for errata list. and Schapira P.geocities. and Rosen M.A must have for this field.rutgers.J.html) consists dozens of Reprints in Theory and Applications of Categories for further reading. A Classical Introduction to Modern Number Theory Manin Y. Prime Numbers: A Computational Perspective Guy R.nott.I. Categories for the Working Mathematician Mac Lane S. Categories and Sheaves Lawvere F. A Gentle Introduction to Category Theory: The Calculational Approach (FREE! http://wwwhome. It is recommended you have a solid background of algebra before you begin to study algebraic number theory. Fermat's last theorem and the Poincaré conjecture have been solved.u-bordeaux1. A Concise Introduction to the Theory of Numbers Cohen H. As the title indicated.ac. A Course in Homological Algebra Kashiwara M. Hilton P.com/mathphyweb/puremath. this is probably the biggest challenge to date. Riemann hypothesis is one of the Millennium Prize Problems. analytic one applies analysis theory.J.numbertheory. Homology Rotman J. Ireland K.edu/~weibel/Hbook-corrections. Number Theory: Volume I: Tools and Diophantine Equations Cohen H.How to Become a Pure Mathematician Page 42 of 66 Fesenko I.html consists links to online number lecture notes.html Solution by Jinhyun: http://www. and Schanuel S. and Wright E.edu/~jinhyun/sol2/weibel. html) Mollin R.G.M.secamlocal. and Tall D. Cohn H. Notes on Algebraic Numbers (FREE! http://www. Algebraic Number Theory Milne J.edu/~chai/coursenotes.u-bordeaux1.jmilne. A Course In Algebraic Number Theory (FREE! http://www. The Elements of the Theory of Algebraic Numbers Stein W.ac.math. Introductory Algebraic Number Theory Ash R. Introduction to Algebraic Number Theory (FREE! http://www. Algebraic Number Theory Murty R.How to Become a Pure Mathematician Page 43 of 66 Problems.edu/papers/ant/) Stewart I.math. Algebraic Number Theory Pollard H.pdf) Chapman R.L. and Diamond H.washington.edu/~r-ash/ANT. Basic Number Theory Algebraic Number Theory: Achter J. The Theory of Algebraic Numbers Reid L.ex.ac.S.in/~srg/Lecnotes/KielNotes.utah.English translation of Cours d'arithmétique.A.math.com/mathphyweb/puremath.pdf) Ghorpade S. A Brief Guide to Algebraic Number Theory http://hk.pdf) Cohen H.org/math/CourseNotes/math676.html) Chai C. Lectures on Topics in Algebraic Number Theory (FREE! http://www.html) Lang S.uk/people/staff/rjchapma/notes/ant2.ex.math. Algebraic Number Theory and Introduction to Automorphic L-functions (FREE! http://www.uk/people/staff/rjchapma/notes/algn.R.uk/personal/ibf/aln/aln. P. Lectures on Elementary Number Theory Serre J. and Esmonde J. A Brief Introduction To Classical and Adelic Algebraic Number Theory (FREE! http://modular. Algebraic Number Theory (FREE! http://www.B.html) Chapman R.edu/~sather/notes. Weil A.pdf) Gross D. A Course in Computational Algebraic Number Theory .See his web (http://www.W.html) Alaca S.htm .iitb.S.edu/~chai/coursenotes. A Course in Arithmetic . Ideas and Theories Rademacher H. and Williams K.ac. Algebraic Number Theory (FREE! http://www.-L.geocities. F.secamlocal. Algebraic Number Theory Notes (FREE! http://www.upenn.nott. Advanced Number Theory Fesenko I.maths.fr/~cohen/) for errata list. Problems in Algebraic Number Theory Neukirch J. and Chai C.math. Algebraic Number Theory (FREE! http://www.uiuc.upenn.-P.ac.math.math. Algebraic Number Theory and Fermat's Last Theorem Swinnerton-Dyer H. leidenuniv.math. Martin G.html) Neukirch J. and Montgomery H.harvard. Analytic Number Theory (FREE! http://www.G.jmilne.geocities.edu/~elkies/M259.ac.mth. Abstract Analytic Number Theory .html) Knopfmacher J.785/calendar.J.html) Davenport H.98/index.math. Introduction to Analytic Number Theory (FREE! http://www.html) Gras G.V. and Tate J.edu. Algebraic Groups and Class Fields Serre J.org/math/CourseNotes/math776.K.ca/~desmondl/ANT Notes with Greg. Introduction to Analytic Number Theory Chen W. Combinatorial and Analytic Number Theory (FREE! http://www.uk/~h090/antmain.math.uea.S.pdf) http://hk.pdf) Hildebrand A. Problems in Analytic Number Theory Newman D. Introduction to Analytic Number Theory . Distribution of Prime Numbers (FREE! http://www. Local Fields and Their Extensions (FREE! http://www. Bateman P.R. Class Field Theory (FREE! http://www.How to Become a Pure Mathematician Page 44 of 66 Weyl H.html) Everest G.D.M. Analytic Number Theory (FREE! http://www.-P.edu/~kedlaya/18.nott.com/mathphyweb/puremath. and Diamond H. Class Field Theory: From Theory to Practice Milne J.nl/~tijdeman/cant. Algebraic Theory of Numbers Class Field Theory: Artin E.mit.W. and Vostokov S.-P.ac.sfu.Another nice book by Apostol. Analytic Number Theory: An Introductory Course Chandrasekharan K.S. Class Field Theory Fesenko I.maths.mq. Analytic Number Theory (FREE! http://www-math.htm . Local Fields Analytic Number Theory: Apostol T.He developed this from analytic number theory and applied it to other areas. A Primer of Analytic Number Theory: From Pythagoras to Riemann Tijdeman R.edu/~hildebr/ant/) Kedlaya K.T.au/~wchen/lndpnfolder/lndpn.math. Multiplicative Number Theory Elkies N. An Introduction to Analytic Number Theory (FREE! http://www.B. Analytic Number Theory Stopple J. Class Field Theory Serre J.pdf) Murty M.L.uiuc. and Leung D.J.maths.uk/personal/ibf/book/book. math. Sieves in Number Theory Husemöller D. p-adic Analysis.ac. and Eberlein W.V. like Riemann's paper. Rooney B.math.geocities. Mazur B.edu/scans/252/Knapp/index. Apostol T. PI and the AGM: A Study in Analytic Number Theory and Computational Complexity Cassels J.htm .B.S.uni-heidelberg.edu/edu/2007/simuw07/index. p-adic Numbers. An Introduction to the Theory of the Riemann Zeta-Function Rockmore D.com/mathphyweb/puremath.): See Franz Lemmermeyer's page (http://www. Elliptic Functions Baker A. and Stein W. and van Frankenhuijsen M.How to Become a Pure Mathematician Page 45 of 66 Riemann Zeta Function and the Hypothesis: Borwein P.washington..rzuser.pdf) Greaves G.math.html) for extra material. and Frohlich A. Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings . and Shurman J..uk/personal/ibf/lf/lf. and Zeta-Functions Lapidus M.F.See Stein's page (http://modular. Elliptic Curves (http://modular.maths.W.C. The Theory of the Riemann Zeta-Function Further readings (including Modular Forms.L.W.washington. Riemann's Zeta Function Koblitz N. Transcendental Number Theory Borwein P.M.html) for further study. Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers Titchmarsh E. and Borwein P. Fractal Geometry. A First Course in Modular Forms Fesenko I. The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike Edwards H.de/~hb3/LN/ellc-intro. What is Riemann's Hypothesis? (FREE! http://modular.washington.nott. Algebraic Number Theory: Proceedings of an Instructional Conference Organized by the London Mathematical Society (http://modular.edu/edu/2007/simuw07/notes/rh.M.pdf) . and Weirathmueller A. Lectures on Complete Discrete Valuation Fields (FREE! http://www.html a scanned version for FREE!) http://hk.math.J.washington. Patterson S.edu/scans/252/cassels-frohlich/ a scanned version for FREE!) Diamon F. Modular Functions and Dirichlet Series in Number Theory Armitage J. etc.M. Choi S. Elliptic Curves. Computational Excursions in Analysis and Number Theory Borwein J. Elliptic Curves Knapp A.More advanced work. Nikulin V. see if you are with him. The Foundations of Geometry (FREE! http://www.edu/scans/252/lang-intro_modform/ a scanned version for FREE!) Lang S.math. Geometry and the Imagination Hvidsten M. Modular Functions in Analytic Number Theory Koblitz N.H.htm . Handbook of Number Theory I.How to Become a Pure Mathematician Page 46 of 66 Knopp M. Hilbert D.edu/scans/252/Miyake/index. Introduction to Elliptic Curves and Modular Forms Lang S.com/mathphyweb/puremath. Mitrinovic D. Topology and Geometry Eves H. Rational Points on Elliptic Curves Geometry: Just like number theory.M.edu/scans/252/Shimura-Intro/index.math. Modular Forms (http://modular.msp. Introduction to Arithmetic Theory of Automorphic Functions (http://modular. Introduction to Geometry Hartshorne R. algebraic geometry applies algebra result to study geometry.H.uk/gtm/) for further study.E. and Cohn-Vossen S.ac. Introduction to Modular Forms (http://modular. Geometry with Geometry Explorer .S.S.H.) Number Theory III: Diophantine Geometry (http://modular.edu/~hvidsten/geom-text/) for extra material. Algebraic geometry is related to algebra and number theory. blah. as he says. Elementary Dirichlet Series and Modular Forms . The Arithmetic of Elliptic Curves Silverman J. blah.I. II Shimura G.. Problems in Geometry Bredon G. Pansu P.math. Advanced Topics in the Arithmetic of Elliptic Curves Silverman J.washington. In general: Berger M..washington. and Tate J. See Geometry & Topology Monographs (http://www.edu/scans/252/lang-nt3/ a scanned version for FREE!) Miyake T. and Shafarevich I.See the web (http://homepages.Has a supplementary nature to his Introduction to Arithmetic Theory of Automorphic Functions. Berry J. Elliptic Functions Lang S.washington.I.html a scanned version for FREE!) Sándor J. Abelian Varieties with Complex Multiplication and Modular Functions Shimura G.-P. and Saint-Raymond X.warwick.He wrote an excellent text on algebraic geometry.. Shimura G.V.html a scanned version for FREE!) Silverman J. Begin with this. (ed. blah.gutenberg.gac.R. Geometry: Euclid and Beyond .geocities.washington. to some extend.org/etext/17384) Hilbert D. and Crstici B.math. Survey of Geometry Coxeter H. Geometries and Groups http://hk. Algebraic Geometry Dolgachev I. Introduction to Algebraic Geometry (FREE! http://bearlair. Algebraic Geometry (FREE! http://www..E.-F. Algebraic Geometry Milne J.html) Reid M. Using Algebraic Geometry .html.uk/~anskor/AG.pdf) Drozd Y. and Roy M.A.pdf) Ewald G.Don't forget to check out the typological error at http://www.V. Debarre O. Coste M. and Harris J.berkeley.jmilne.org/math/CourseNotes/math631.geocities.com/mathphyweb/puremath. An Invitation to Algebraic Geometry Ueno K.edu/~idolga/topics1. Pollack R. and Roy M. Combinatorial Convexity and Algebraic Geometry Fulton W.purdue.ic.J.math.math. Real Algebraic Geometry Cox D. Introduction to Algebraic Geometry Lefschetz S.math. Richard Borcherds (http://math. Kahanpää L. Principles of Algebraic Geometry Harris J.lsa.edu/~jinhyun/sol2/hart.ac..B.V.cs.drozd. Undergraduate Algebraic Geometry Skorobogatov A.edu/~idolga/AG1. An Introduction to Algebraic Geometry Walker R..B.PDF) Smith K.org/Math/Solutions.harvard. Introduction to Algebraic Geometry (FREE! http://www.amherst.pdf) Dolgachev I.washington.-F. Topics in Classical Algebraic Geometry (FREE! http://www.lsa.html). Kekäläinen P.math.html) and Chris (http://lomont.edu/scans/252/hartshorne/ a scanned version for FREE!) Solutions by William Stein (http://modular. The Geometry of Syzygies: A Second Course in Algebraic Geometry and http://hk. Introduction to Algebraic Geometry Lang S. Algebraic Curves Further Algebraic Geometry: Basu S.edu/~reb/courses/alggeom/).ma.umich. and O'Shea D. Jinhyun Park (http://www..edu/AG. Hassett B.N. Algorithms in Real Algebraic Geometry Bochnak J.edu/~dac/uag.umich. Little J. and Traves W. Algebraic Curves: An Introduction to Algebraic Geometry Griffiths P. Algebraic Geometry (http://modular. Higher-Dimensional Algebraic Geometry Eisenbud D.S.fas. Algebraic Geometry Lecture Notes (FREE! http://www.pdf).htm ..org/~yuriy/ageng.How to Become a Pure Mathematician Page 47 of 66 Algebraic Geometry: Bump D. Algebraic Geometry: A First Course Hartshorne R. geocities. Introduction to Differentiable Manifolds Lee J.edu/scans/252/lang-abvar/ a scanned version for FREE!) Lang S. 2 Kobayashi S.washington. Fundamentals of Differential Geometry Lang S. and Kneebone G.washington. Lie Groups.pdf) Ivey T. and Symmetric Spaces Hwang A.More advanced work. An Introduction to Noncommutative Differential Geometry and its Physical Applications .edu/scans/252/mumford-abvar/ a scanned version for FREE!) Reid M.H.edu/~ahwang/print/HDG.html) . 陳省身，陳維桓： 《微分幾何講義》 ，北京大學出版社 Helgason S.com/mathphyweb/puremath.M.math. Algebraic Geometry 1: From Algebraic Varieties to Schemes Ueno K. Chern S.D.math. Topological Methods in Algebraic Geometry Lang S. Hodge Theory and Complex Algebraic Geometry Vol 1. 2 Yang K.S.A. Introduction to Algebraic and Abelian Functions (http://modular.topology. Introduction to Topological Manifolds Madore K. and Landsberg J. and Lam K. Complex Algebraic Geometry Differential Geometry: Also consult mathematical physics section below.htm . Differential Geometry Reconstructed: A Unified Systematic Framework (FREE! http://www. Cartan for Beginners: Differential Geometry Via Moving Frames and Exterior Differential Systems Kennington A. Differential Geometry. Algebraic Projective Geometry Ueno K. Foundations of Differential Geometry Vol 1.math. Algebraic Geometry 3: Further Study of Schemes Voisin C.Basic topological concspts assumed.washington. Algebraic Geometry and Arithmetic Curves Mumford D. and Nomizu K.G. Lectures on Differential Geometry . Complex Manifolds and Hermitian Differential Geometry (FREE! http://cs. Chapters on Algebraic Surfaces (FREE! http://arxiv. Chen W. A chapter on Finsler geometry is added in the English translation.T.org/abs/alg-geom/9602006) Semple J. Kobayashi S.Comprehensive. Abelian Varieties (http://modular. Lectures on Vector Bundles Liu Q.edu/scans/252/lang-algab/ a scanned version for FREE!) Le Potier J.M.How to Become a Pure Mathematician Page 48 of 66 Commutative Algebra Hirzebruch F.holycross.S. Abelian Varieties (http://modular.org/tex/conc/dg. http://hk. Transformation Groups in Differential Geometry Lang S.U. Algebraic Geometry 2: Sheaves and Cohomology Ueno K. html) Jost J.djvu) Riemannian Geometry: Boothby W.pdf) O'Neill B. Topics in Differential Geometry (FREE! http://www. Chern.edu/~edgar/books/mtfg. The Fractal Geometry of Nature http://hk.htm .geocities.See author's page (http://www. SuperFractals . and Shen Z.F.R.M.univie. Fractal Geometry: Mathematical Foundations and Applications Mandelbrot B.. Riemann-Finsler Geometry do Carmo M.weizmann.W.pdf) Montiel S.com/). and Ros A.com/bookhtms/dg.gc. Topology.F. Differential and Riemannian Manifolds Moller J.tcd.B.pdf).dk/~moller/f05/genotes.edu/faculty/chavel/CORR-R. S.se/matematiklu/personal/sigma/Lecture-Notes.lu.ac.weizmann.ac.A. Riemannian Geometry: A Modern Introduction .W. Foundations of Differentiable Manifolds and Lie Groups (FREE! http://www.wisdom.matematik.maths.S.An epic. A Course in Riemannian Geometry (FREE! http://www. Read the first two volumes at least. 4.ie/~dwilkins/Courses/425/RiemGeom. Semi-Riemannian Geometry With Applications to Relativity Petersen P. A Comprehensive Introduction to Differential Geometry 1.com/mathphyweb/puremath.at/~michor/dgbook.T.il/~yakov/scanlib/Warner.superfractals. and Yau S. An Introduction to Riemannian Geometry (FREE! http://www.cuny.Check the correction and addition (http://math. Riemannian Geometry Gudmundsson S.il/~yakov/scanlib/Boothby.djvu) Chavel I. Content can be found at: http://www.wisdom. Riemannian Geometry Gallot S.P.ohio-state.mathpop. Riemannian Geometry (FREE! http://www. Riemannian Geometry and Geometric Analysis Lang S.html) for errata.How to Become a Pure Mathematician Page 49 of 66 Michor P. 3. Edgar G. and Lafontaine J. An Introduction to Differentiable Manifolds and Riemannian Geometry (FREE! http://www. Fractals Everywhere Barnsley M. Falconer K.ac. Lectures on Differential Geometry 丘成桐，孫理察： 《微分幾何講義》 ，高等教育出版社 Spivak M.math.math. 2.htm Warner F. 5 .mat. Hulin D.ku. Curves and Surfaces Schoen R.See the web (http://www.pdf) 伍鴻熙、沈純理、虞言林： 《黎曼幾何初步》 ，北京大學出版社 Fractals: Barnsley M.M. Riemannian Geometry Wilkins D. and Fractal Geometry . Measure. ucsd.A.Contains dozens of historical material.F. Convex and Discrete Geometry Huybrechts D.One quote by Ian Stewart is sufficient.com/mathphyweb/puremath. Lecture Notes on Fractal Geometry (FREE! http://people.A." Gay D. Lectures on Algebraic Topology Eilenberg S. Modern Geometry with Applications Matousek J. Algebraic Topology: A First Course Hatcher A.D. Surfaces and Knots Ranicki A. and Gardiner C. In general: Hocking J.W. Surface Topology .S.uk/~aar/knots/index. and Young G. and Tu L.warwick.L.springerlink. Knots Knotes (FREE! http://math.htm . Topology Kelley J. and Steenrod N. Explorations in Topology: Map Coloring.ps) Peitgen H.ac.msp. General Topology .geocities."What every young analyst should know" was the originally intended title. Lectures on Discrete Geometry Topology: See Geometry & Topology Monographs (http://www.com/content/978-0-387-21823-6/) Further readings: Audin M. Classical Topology and Combinatorial Group Theory Geometric Topology: Firby P.E.htm) . Chaos and Fractals: New Frontiers of Sciences (FREE! http://www. Morse-Kelley set theory was introduced in this book.-O. "This book is in the top ten for books conveying the dreadful secret that mathematics is fascinating. The Topology of Fibre Bundles Stillwell J.uk/maspm/fract.maths.G.ac.bath. Foundations of Algebraic Topology Fulton W. Differential Forms in Algebraic Topology Dold A.edu/~justin/Papers/knotes.How to Become a Pure Mathematician Page 50 of 66 Mörters P. and Saupe D.ac. Jürgens H. Roberts J.pdf) Algebraic Topology: Bott R..ed. Algebraic Topology (FREE! http://hk. Geometry Gruber P. History of Knot Theory (FREE! http://www. Steenrod N.uk/gtm/) for further study.M. Complex Geometry: An Introduction Jennings G. Algebraic Topology: Homology and Cohomology Wallace A. Differential Manifolds .ps. Differential Topology Kosinski A. R.htm . F.ku.cornell. Spivak and R.ch/svnbuild/math/at/0506-ws/at.edu/~hatcher/AT/ATpage. Elements of Homotopy Theory Wilkins D. Lefschetz S.G.edu/pub/matt/diff.Homology and Homotopy Vick J.rutgers.ethz.An appendix by John W.edu/pub/matt/diff.math. Notes taken by M.More advanced work.com/mathphyweb/puremath.H. Introduction to Differentiable Manifolds Milnor J.ie/~dwilkins/Courses/421/) K-theory: Hatcher A. http://hk. Morse Theory .Very beautiful book. Morgan on the work of Grigory Perelman is included in the 2007 Dover edition. Algebraic K-Theory and Its Applications Weibel C. An Introduction to Algebraic Topology Whitehead G. Differential Topology Hirsch M.binghamton.html) Differential Topology: Brin M.S. Algebraic Topology: An Introduction Mislin G.S. A Basic Course in Algebraic Topology Massey W. Algebraic Topology (FREE! http://www. Introduction to Differential Topology (FREE! ftp://ftp. Algebraic Topology .edu/~weibel/Kbook.dk/~moller/f04/algtop/AlgTopnotes.H.W.How to Become a Pure Mathematician Page 51 of 66 http://www. K-theory: An Introduction Rosenberg J.edu/~hatcher/VBKT/VBpage. Elements of Algebraic Topology Switzer R.R.W. Algebraic Topology Notes (FREE! http://www.cornell.math.R. Maunder C. Algebraic Topology Massey W.math. Wells. Vector Bundles and K-theory (FREE! http://www.tcd.ams.M. influential.vmp. Lang S.html) .Lefschetz developed new result in this monograph.pdf) Moller J.maths.pdf or ftp://ftp. The K-book: An Introduction to Algebraic K-theory (FREE! http://math.binghamton. and Pollack A.A.gz) Guillemin V.org/online_bks/coll27/) .geocities.math. Homology Theory: An Introduction to Algebraic Topology Wallace A. Algebraic Topology (FREE! http://shamir. Algebraic Topology (FREE! http://www.M.H.math.html) Munkres J.html) Karoubi M. Differential Topology: First Steps 張筑生： 《微分拓樸》 Further Calculus (optional): Ordinary Differential Equations: Aronld V.edu/~driver/231-02-03/lecture_notes.e. electrostatics. Partial Kuttler K.pdf) Driver B. and this is a textbook. equations with partial derivatives. Ordinary Differential Equations Ince E. understanding generalised solutions is fundamental. Partial Differential Equations .com/mathphyweb/puremath. e. Partial Differential Equations of Applied Mathematics (FREE! http://www. Entropy and Partial Differential Equations (FREE! http://math.As an introduction.As Evans emphasise.math.pdf) Markowich P. An Introduction to Partial Differential Equations Pivato M. Wallace A.berkeley.htm .geocities. Analytic Methods for Partial Differential Equations Evans L.edu/~evans/) for errata. Linear Partial Differential Equations and Fourier Theory (FREE! http://xaravve.ca/305/305. Otto S.I. Billingham J.berkeley. Lectures on Partial Differential Equations Bass R. PDE is not a branch of functional analysis.math. etc. Ordinary.math. Ordinary Differential Equations Tenenbaum M. Applied Partial Differential Equations: A Visual Approach Pinchover Y. Nonlinear.R. electrodynamics. and Rubinstein J.H.How to Become a Pure Mathematician Page 52 of 66 Milnor J.byu.ucsd.W. and Michel A.N.trentu. Blackledge J. and Pollard H..C. PDE From a Probability Point of View (FREE! http://www.H.uconn. Jost J. Partial Differential Equations King A..pdf) Evans L.C.htm) Evans G.edu/~klkuttle/547notesB.F. i. Check out his page (http://math.I. Ordinary Differential Equations Partial Differential Equation: Differential equation with several variables.L. Notes for Partial Differential Equations (FREE! http://www. and Yardley P.and. not a reference book. Analysis Tools with Applications and PDE Notes (FREE! http://math.pdf) Sattinger D. Ordinary Differential Equations Miller R.A.usu.g.edu/~evans/entropy. Particularly useful in physics.edu/~dhs/pde. Differential Equations: Linear. many interesting equations are non-linear.PDE.edu/~bass/pdeprob.C.pdf) http://hk. heat. Arnold V.K. Topology from the Differentiable Viewpoint . good theory is as useful as exact formulas. Quantum Physics of Atoms. Feynman Lectures On Physics . and Fomin S. Fowler M. 2 Showalter R. Modern Physics (FREE! http://galileo.phys.M. 2.E.edu/internet/physics/cbphysics/index. 3. Calculus of Variations Jost J. Classical Mechanics http://hk. Calculus-Based Physics (FREE! http://www.anselm. A PDE Primer (FREE! http://www. Partial Differential Equations: An Introduction .geocities. Molecules. and Li-Jost X. Dolgachev I.html) Schnick J. Introductory level: Crowell B. Calculus of Variations van Brunt B. Light and Matters (FREE! http://www.com/mathphyweb/puremath.The author's principle: "Motivate with physics but then do mathematics.P." Symes W. and Resnick R.Including Newtonian Physics. etc.edu/~show/docs/pde.virginia. Optics and the Modern Revolution in Physics. Solids.V.math. Partial Differential Equations of Mathematical Physics (FREE! http://www.A very nice quantum physics text. these two subjects are highly related.umich. and Particles . Nuclei.html) Sneddon I.How to Become a Pure Mathematician Page 53 of 66 Sauvigny F. Calculus of Variations Mathematical Physics (optional): Advanced studies of modern physics rely heavily on mathematics.edu/~idolga/lecturenotes.Modern classics.caam.edu/classes/252/) Goldstein H.edu/~caam436/caam436notes.html) Relatively advanced level: Eisberg R.N.M. Partial Differential Equation Vol 1.math.lsa. Feynman R. Electricity and Magnetism. In fact.rice.lightandmatter.com/area1. throughout history.A. Vibrations and Waves. Introduction To The Calculus Of Variations Gelfand I.html) .oregonstate.V.htm .pdf) 郇中丹、黃海洋： 《偏微分方程》 ，高等教育出版社 Calculus of Variations: Dacorogna B. Conservation Laws. The Calculus of Variations Weinstock R. Elements of Partial Differential Equations Strauss W. Introduction to Physics 1.W. Fundamentals of Modern Physics Eisberg R. 4 and literature (FREE! http://www.W. Nakahara M.html) Fitzpatrick R. with Problems and Solutions (FREE! http://www.The bible of many engineers. Modern Differential Geometry for Physicists Jeffreys H. classical electromagnetism. and Freeman I.Stellar atmospheres.edu/~phys16/Textbook/) Rzchowski M. introduction to plasma physics.com/intro. Courant R. electricity and magnetism.com/mathphyweb/puremath. Mathematical methods in physics: Arnold V.html) Heinbockel. Mathematical Tools for Physics (FREE! http://www. Physics Topics (FREE! http://astrowww. and Jeffreys B. Mathematical Physics: A Modern Introduction to Its Foundations Isham C. Courses Taught at UT Austin (FREE! http://farside.uk/lectures/public. Green's function and group theory. Introduction to Groups. Fourier Series. celestial mechanics. Advanced Engineering Mathematics . Mathematical Physics Byron F.ca/~tatum/index.html) Heisenberg W.L.M. Hawking S. Hilbert space. Multivariable http://hk.odu. heat and thermodynamics. classical mechanics.wisc. Invariants & Particles (FREE! http://www. Methods of Mathematical Physics Koks D. Methods of Mathematical Physics Vol 1. Topology and Geometry for Physicists Nearing J.html) .physicsforfree.fas.utexas.physics. J. etc.ODE. Group Theory and Its Application to Physical Problems Hassani S.edu/~jhh/counter2.How to Become a Pure Mathematician Page 54 of 66 Firk W.physics.html) . e.g. and Hilbert D.ph. Ideas of Modern Physics (FREE! http://uw. and Kern K.Quantum mechanics. Theoretical Physics Morin D.edu/~rzchowski/phy107/lectures. geometric optics. Physical Principles of the Quantum Theory Joos G. Motion Mountain (FREE! http://motionmountain.K.geocities.dse.math.J.edu/~nearing/mathmethods/) . analytical classical dynamics.htm . Introductory Classical Mechanics. and Sen S.I.phys.courses.nl/index.miami. Mathematics of Classical and Quantum Physics . Geometry.W. Public Lectures (FREE! http://www.harvard. Differential Geometry and Lie Groups for Physicists Hamermesh M. 2 Fecko M.org. calculus of variations.uvic. Explorations in Mathematical Physics: The Concepts Behind an Elegant Language Kreyszig E.hawking.B.This text covers a wide range of mathematics topics that are often used in physics. Mathematical Methods of Classical Mechanics Boas M. Topology and Physics Nash C.htm) Schiller C.H Introduction to Tensor Calculus and Continuum Mechanics (FREE! http://www. thermodynamics and statistical mechanics.html) Tatum J. planetary photometry notes.W.edu/teaching. and Fuller R. Mathematical Methods in the Physical Sciences Butkov E. Fourier Analysis.etsu.edu/math/gardner/5310/notes.edu/Stefan_Waner/diff_geom/tc. Introduction to Differential Geometry and General Relativity (FREE! http://people.mat.K. Classical Electrodynamics and Theory of Relativity (FREE! http://uk. Tensors.uu. A Course in Modern Mathematical Physics: Groups.pdf) Waner S.htm) Lugo G.Stanford Little Lectures.phys.at/~gerald/ftp/book-ode/index. Differential Geometry (and Relativity) (FREE! http://www. Advanced Quantum Mechanics Sakurai J.W.A very nice quantum mechanics text. Introductory Quantum Mechanics .J Modern Quantum Mechanics http://hk.univie.htm) .arxiv.Aimed to built up mathematical background to study general theory of relativity. Sakurai J.W. Differential Geometry in Physics (FREE! http://people. Introduction to General Relativity (FREE! http://www.nl/~thooft/lectures/genrel. Numerical Analysis.uncw. and Wu H.geocities.M. Matter . etc.J. Theory of Relativity . Szekeres P. Relativity . Space.For general reader.ac.edu/~carroll/notes/ or http://arxiv. Pauli W. PDE. Mathematical Methods for Physics Relativity: Carroll S.org/abs/gr-qc/9712019) Einstein A.Published at the young age of relativity by a 21 years old young man.edu/lugo/COURSES/DiffGeom/dg1. Quantum Mechanics: Griffiths D.html) Weyl H.More advanced work. Frauendiener J. Ordinary Differential Equations and Dynamical Systems (FREE! http://www. Analytical and Numerical Approaches to Mathematical Relativity . Sachs R. Vector Calculus. still stand as a good exposition today. The Meaning of Relativity . Introduction to Quantum Mechanics Liboff R.J. Complex Variable.-H. Einstein A.html) Wyld H. Gardner R.uchicago.How to Become a Pure Mathematician Page 55 of 66 Calculus.com/mathphyweb/puremath. Giulini D. Calculus of Variations.hofstra. and Perlick V. Hilbert Space and Differential Geometry Teschl G.B. General Relativity for Mathematicians Sharipov R. Time. Lecture Notes on General Relativity (FREE! http://pancake.htm .org/abs/physics/0311011) 't Hooft G.J.Weyl describes a failed attempt of gauge invariance in this book. Introduction to Chaos (FREE! http://www.com/mathphyweb/puremath. http://www.lanl.org/abs/hep-th/0107094) Siegel W. Vattay G.html) Smith L.pdf) Forshaw J.phys. Graduate Course in String Theory Zwiebach B.html) von Neumann J. Tanner G.pdf..phys.uu.cmp.gov/abs/hep-th/9411028) Siopsis G. Lectures String Theory (FREE! http://www. A First Course in String Theory Chaos: Cvitanović P. String Theory (FREE! http://aesop. Whelan N..uk/u/forshaw/NorthWest/QFT.ias. 2006.mat.ac.html) Uranga A.gov/abs/hep-th/9912205) .nl/~bdewit/qft06. The Quantum Theory of Fields. 2005 (FREE! http://www.phys.How to Become a Pure Mathematician Page 56 of 66 Schulten K. Notes on Quantum Mechanics (FREE! http://www.. Mainieri R.at/~gerald/ftp/book-schroe/index.nl/~thooft/lectures/string. Mathematical Foundations of Quantum Mechanics Quantum Field Theory: de Wit B. http://www.phys. 2: Modern Applications Weinberg S. Mathematical Methods in Quantum Mechanics (FREE! http://www.uiuc.hep. Artuso R.html) Shankar R.edu/~mcc/Chaos_Course/Outline. Vol.lanl.nl/~bdewit/qft05.uu.htm . and Wirzba A. What is String Theory? (FREE! http://xxx.html) IAS Quantum Field Theory (FREE! http://www.edu/Services/Class/PHYS481/lecture.edu/QFT/) Siegel W.Over 800 pages of comprehensive text. Introduction to String Field Theory (FREE! http://arxiv. Principles of Quantum Mechanics Teschl G. The Quantum Theory of Fields.man. 3: Supersymmetry String Theory: Polchinski J. The Quantum Theory of Fields.pdf.org/) Cross M.uu.ks..uu. Fields (FREE! http://xxx. Vol. Chaos: A Very Short Introduction http://hk. Vol.caltech.ac.nl/~bdewit/qft07. Chaos: Classical and Quantum (FREE! http://chaosbook. Weinberg S. 1: Foundations Weinberg S.univie.phys.math..edu/strings/) 't Hooft G. Lecture Notes Quantum Field Theory 2007.M. Quantum Field Theory (FREE! http://www.utk.geocities. html) Gut A. He draws nice comparison between set-theoretic interpretation and interpretation in probability theory.edu/~bass/prob.html) .pdf) Billingsley P. Probability and Stochastic Processes with Applications (FREE! http://abel. Foundations of the Theory of Probability (FREE! http://www. you are probably ready for the followings.com/Kolmogorov/index. Probability Theory: The Logic of Science (FREE! http://omega.geocities.The material is based on a three-semester course of lectures. Functional for Analysis for Probability and Stochastic Processes: An Introduction Capinski M.wustl. Probability Theory I. 2 Gray R. Analysis and Probability: Wavelets. Probaility built upon Measure Theory: Ash R. Probability . Introduction to Probability Theory and Stochastic Processes (FREE! http://statmath. Check out Probability Theory As Extended Logic (http://bayes.N. Advanced Probability Theory (FREE! http://hk.stanford.E. II .wu-wien. Knill O. II.B. Strasser H. If you have done some baby measure theory in stage 3.edu/~gray/arp. and Kopp P.M.mathematik. Probability & Measure Theory Athreya K. and Doleans-Dade C.math.albany. and Lahiri S.math.A.html) Jorgensen P.htm . Stochastic Processes. Signals.com/mathphyweb/puremath.N.Nice historical and bibliographical notes are included at the back. Probability and Measure Bobrowski A.stanford.pdf) Dudley R. studying probability without measure is like studying physics without calculus. Random Processes.E. Shiryaev A.How to Become a Pure Mathematician Page 57 of 66 Probability (optional): These days.pdf) Tsirelson B.at/~strasser/STATS 25-01-2006. Measure Theory and Probability Theory Bass R. Foundations of Modern Probability .edu/) for collected probability papers.T.For historical interest. given by the author in the Moscow State University.ac.T. I. Fractals Kallenberg O. Probability Theory (FREE! http://www. Integral and Probability Dembo A.math.136-137. Statistics.M.edu:8008/JaynesBook. Probability: A Graduate Course Jaynes E. An Introduction to Probability Theory and Its Applications Vol.uconn.pdf) Loève M.edu/~adembo/stat-310a/lnotes.F.Fairly comprehensive. Probability.N.edu/~knill/teaching/math144_1994/probability. Measure. Kolmogorov A.B. and Ergodic Properties (FREE! http://www-ee.harvard. Real Analysis and Probability Feller W. Probability Theory (FREE! http://www-stat. see p. under the title Probability. M.ps) Seppalainen T. Brownian Motion (FREE! http://www. Stochastic Analysis Mörters P. An Introduction to Stochastic Differential Equations (FREE! http://math.edu/~adembo/math-136/nnotes.math. Probability with Martingales Stochastic Processes: Brzezniak Z.L. Stochastic Processes (FREE! http://www.berkeley.cmu.bath. Stochastic Processes (FREE! http://www-stat.cs.com/mathphyweb/puremath. Brownian Motion and Stochastic Calculus Klebaner F. Mörters P.pdf) .edu/faculty/varadhan/processes.nl/~rmeester/onderwijs/stochastic_processes/sp_new. Introduction to Stochastic Calculus with Applications Kurtz T.edu/~rdlyons/pdf/StochProc.edu/users/peres/bmbook.S.edu/~kurtz/735/main735.uk/~iwilde/notes/mip/index. Stochastic Calculus (FREE! http://www. and Williams R.html) Williams D. Stochastic Processes (FREE! http://www.org.pdf) Jacod J.L.iu.F.kcl.ac.M.greenend.How to Become a Pure Mathematician Page 58 of 66 http://www.stat. Basics of Stochastic Analysis (FREE! http://hk. Stochastic Processes . and Shiryaev A.J.Amazon user Yan Zhang comments that this is a must-have if you wanna go to Wall Street. and Zastawniak T. Basic Stochastic Processes Doob J.stanford.pdf) Varadhan S. An Introduction to Stochastic Processes and Their Application van Zanten H.C. and Peres Y. Lecture Notes on Stochastic Analysis (FREE! http://people. Measures. Lectures on Stochastic Analysis (FREE! http://www.html) Stochastic Analysis: Bain A. A Second Course in Stochastic Processes .Evans wrote a popular book on PDE.N.edu/~cshalizi/754/) Todorovic P.vu. Dembo A.math. and Taylor H. Stochastic Processes . An Introduction to Stochastic Processes in Continuous Time (FREE! http://www.geocities.tau.course.wisc.ac.uk/maspm/stoa. Integration and Probability (FREE! http://www.Considered to be classics in the field.mth.pdf). Limit Theorems for Stochastic Processes Karlin S.See also Russell Lyons' course notes (http://mypage.chiark.berkeley.E. Shalizi S.uk/~alanb/) Chung K.C.pdf) Ross S.math.htm .html) Wilde I.pdf) Malliavin P. An Introduction to Stochastic Integration Evans L.R.il/~tsirel/Courses/AdvProb03/syllabus.ac. Karatzas I. and Shreve S.nyu.edu/~evans/SDE.G.stat. ac. Classical Methods of Statistics Lehmann E. and Kokoska S.math.F. Statistical Design for Research Ramsay J. and Romano J.edu/~larry/all-of-statistics/index.edu/~seppalai/sa-book/etusivu. and Silverman B. All of Statistics: A Concise Course in Statistical Inference .quantlet.pdf.J. and Smith H.O. The Little Handbook of Statistical Practice (FREE! http://www. Härdle W.I. Applied Functional Data Analysis Ramsay J. and Silverman B.E. Elements of Large-Sample Theory Lehmann E. Permutation.L.stat. Applied Linear Regression Set 1.W.The author aimed to write a short course in this area. CRC Standard Probability and Statistics Tables and Formulae Statistical Models and Regression: DeCoster J.kcl. Testing Statistical Hypotheses .ncsu. Check the errata list (http://www. Set 2 (FREE! http://www.edu/faculty/varadhan/fall06.R. why don't use just get a handbook? Wasserman states in page 25 that if the reader is unfamiliar with "inf" then just treat it as the minimum.math.L.B.stat.nyu. I can't accept this for if "inf" can be treated as minimum there is no point to introduce "sup" and "inf".W.Classical work on hypotheses testing. rather than a textbook that explains everything in detail.com/mathphyweb/puremath.edu/~gdallal/LHSP.htm .html or http://www.tufts. Parametric.S.. and Bootstrap Tests of Hypotheses Kardaun O. (ed. Functional Data Analysis Rao C.F. And if you want a handbook.quantlet.html) Varadhan S.pdf) Good P.uk/~iwilde/notes/sa/) Statistics (optional): In general: Balakrishnan N.stat-help. Preparation for Statistical Research (FREE! http://www4. But if it's a handbook. Zwillinger D. A Primer on Statistical Distributions Dallal G.cmu.How to Become a Pure Mathematician Page 59 of 66 http://www.HTM) Davidian M.O. Kish L.mth.P.E.stat-help.com/cohen 2006-09-26. and Mori Y. Stochastic Analysis (FREE! http://www.html) Wilde I.com/mdstat/scripts/csa/pdf/csa.pdf) Draper N.edu/~davidian/st810a/index.html) Gentle J. http://www.W.geocities.com/mdstat/scripts/csa/html/index.com/linreg. it's kind of incomplete. Stochastic Analysis (FREE! http://www.R. Applied Regression Analysis http://hk. It's kind of encyclopedic.html).wisc.) Handbook of Computational Statistics (FREE! http://www. Linear Statistical Inference and Its Application Wasserman L. and Nevzorov V. html or http://www.How to Become a Pure Mathematician Page 60 of 66 Fox J.umn.K. Sperlich S. and Sincich T. Applied Regression Analysis..com/mdstat/scripts/mva/htmlbook/mvahtml.A. Theory of Multivariate Statistics Härdle W.cz/~hlavka/sms/html/index.J.Jr. Neter J.cuni.R.edu/Nachtsheim/). Well explanation.quantlet. and Applications Weisberg S.R.Data set and solution on web.umn. Multilevel Statistical Models (FREE! http://www. A User's Guide to Principal Components http://hk. for 4th ed. Correlated Data Analysis: Modeling.quantlet. Multivariate Statistics: Exercises and Solutions (FREE! http://www.com/mdstat/scripts/anr/pdf/anrpdf.K.mff. Linear Models: Least Squares and Alternatives Rao C.ats. Jammalamadaka Linear Models: An Integrated Approach . Song P.. and Stephan P. Linear Models and Generalizations: Least Squares and Alternatives Schroeder L.E. Linear Regression Analysis Sengupta S.mff.E.quantlet. (http://apps. and Werwatz A. and Simar L. Lindsey J.J.edu/stat/examples/msm_goldstein/) Härdle W. for 5th ed.. Applied Multivariate Statistical Analysis (FREE! http://www.L. Regression Modeling Strategies Jiang J.edu/Nachtsheim/5th/).karlin. errata etc. Linear Models. Applied Linear Statistical Models . Shalabh and Heumann C. and Brenner D.com/mdstat/scripts/mva/pdf/mvapdf.quantlet.karlin.. and Related Methods Goldstein H.cz/~hlavka/sms/pdf/smspdf. data set.com/mdstat/scripts/anr/html/anrhtml.com/mdstat/scripts/spm/html/spmhtml. Toutenburg H.com/mdstat/scripts/spm/spm.E. Applying Generalized Linear Models Mendenhall W. Nachtsheim C.htm .pdf) Harrell F.R. and Hlavka Z. and Toutenburg H.ucla. Understanding Regression Analysis: An Introductory Guide Seber G.csom.html or http://www. with reasonably amount of illustration. Linear and Generalized Linear Mixed Models and Their Applications Kutner M.. Solution. and Li W..-K. Applied Linear Regression Multivariate Analysis: Bilodeau M.New edition of the old title Applied Linear Regression Models.pdf) Härdle W. and S.csom.html or http://www.geocities.com/mathphyweb/puremath.cuni. Applied Nonparametric Regression (FREE! http://www. Nonparametric and Semiparametric Models (FREE! http://www.html or http://www. and Lee A. Sjoquist D. A Second Course in Statistics: Regression Analysis Rao C. Analytics.quantlet.pdf) Jackson J. (http://apps..pdf) Härdle W.X. Müller M.quantlet.L. and Wichern D. solutions.edu/~milesd/multivar.stat.Check out Gelman's website (http://www.edu/psych/cw/statmanual/index.T.com/tabachnick/stats/index.See author's website for this book (http://www.edu/~aa/cda/cda. http://hk.A.umn.edu/~brad/data.html) / 2nd (http://www. and Smith A.ufl.uk/aphome/multivariate.G.M..htm) Stockburger D. Carroll D. and McCarthy B.K.biostat.Data set can be found at: http://www. Tharp I.Check out their page for the 4th edition here (http://www. Applied Multivariate Statistical Analysis Kachigan S.Correction to 1st (http://www.com/mathphyweb/puremath. An Introduction to Bayesian Analysis: Theory and Methods Koch K.B. Bayesian Data Analysis . Introduction to Bayesian Statistics Carlin B.P.cats.pdf) edition. Categorical Data Analysis .stat.edu/~gelman/book/) for extra information. Manual for Univariate and Multivariate Statistics (FREE! http://www. Carlin J.. Delampady M. solution and errata etc.W. including data sets.html Congdon P. Multivariate Bayesian Statistics: Models for Source Separation and Signal Unmixing Categorical Data Analysis: Agresti A.uwsp.com/ab_tabachnick_multistats_5/). and Green P. Bayesian Statistical Modelling Gelman A. and Bibby Multivariate Analysis Miles D.B.M.edu/~aa/intro-cda/corrections/corrections. that provides datasets used for examples.geocities. and Louis T.. Bayesian Theory Bolstad W.ufl.stat.edu/~aa/intro-cda/corrections.B.html).F.S. Agresti A.ohiou. Models.C.. Selected Multivariate Statistical Techniques (FREE! http://homepages. and Fidell L. BIOS870 Multivariate Statistics (FREE! http://oak.html) and the 5th here (http://wps.ufl. Bayes and Empirical Bayes Methods for Data Analysis . and Samanta T. Kent J.html) Wendorf C.. and Applications (FREE! http://www. An Introduction to Categorical Data Analysis . Stern H.ablongman.A.M.ablongman. Multivariate Statistical Analysis: A Conceptual Introduction Lattin J.V.edu/MultiBook/mlt00.ac.htm) Tabachnick B.stat.htm .gold.-R. Multivariate Statistics: Concepts. Introduction to Bayesian Statistics Rowe D.S.W.psychstat.columbia. Ghosh J.B. Using Multivariate Statistics .A. and Rubin D. Analyzing Multivariate Data Mardia K.missouristate.How to Become a Pure Mathematician Page 61 of 66 Johnson R.html) Bayesian: Bernardo J. Time Series: Theory and Methods Chair of Statistics.How to Become a Pure Mathematician Page 62 of 66 etc. Simulation Rummukainen K.J. Statistical Analysis of Categorical Data Powers D.A. and Handscomb D. The Analysis of Time Series: An Introduction Simulation and the Monte Carlo Method: Hammersley J. and Kroese D.A. Monte Carlo Simulation (FREE! http://www. Simulation and the Monte Carlo Method Ross S.See their web (http://www-stat.edu/~rwilliam/xsoc73994/index. Categorical Data Analysis with Graphics (FREE! http://www.com/mathphyweb/puremath. Time Series Analysis: Forecasting & Control Brockwell P.A. Monte Carlo Simulation Methods (FREE! http://cc. and Xie Y.C.php?id=book) Chatfield C.H.html) Data Mining: Hastie T.cs. and Davis R.nd.fsu.edu/faculty/goodman/teaching/Monte_Carlo/Sokal. errata. Statistical Methods for Categorical Data Analysis Williams R.uni-wuerzburg. Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms (FREE! http://www.Y..edu/~tibs/ElemStatLearn/) for data.ps) Biostatistics (optional): When mathematics is applied to biology or medical science. and Friedman J. Monte Carlo Methods . and Davis R. Friendly M.edu/~mascagni/Hammersley-Handscomb.htm . University of Würzburg A Firset Course on Time Series Analysis: Examples with SAS (FREE! http://statistik.J.math.(FREE! http://www.nyu.pdf) Haugh M.columbia. The Elements of Statistical Learning: Data Mining. and Prediction .stanford.edu/~mh2078/MCS04.ca/SCS/Courses/grcat/) Lloyd C.fi/~tf/tiedostot/pub/montecarlo/) Sokal A. Tibshirani R.M.de/timeseries/index.oulu. Myatt G.mathematik.html) Rubinstein R.M.D. For http://hk. etc.J. Introduction to Time Series and Forecasting Brockwell P. Jenkins G.geocities. Inference. Making Sense of Data: A Practical Guide to Exploratory Data Analysis and Data Mining Time Series: Box G.P.M. Categorical Data Analysis (FREE! http://www. and Reinsel G.yorku.math. a subject with significant importance. exercises.ubc.edu/courses/st572-larget/Spring2007) Moye L.do?keyword=k16579&pageid=icb.See the online companion (http://www. Larget B. http://www.biostat-edu. Statistical Methods for Bioscience I. van Belle G. Biostatistics I.medstatsaag. Concepts of Epidemiology: An Integrated Introduction to the Ideas. In general: Brant R. Glidden D.ca/~rollin/teach/643.S.html) Broemeling L. and Streiner D. Biostatistical Methods in Epidemiology http://hk. and Hernandez M. etc..edu/faculty/gerstman/eks/index.A.More advanced.See their web (http://www.info/). and Fisher L.html.wisc. Principles and Methods of Epidemiology Gerstman B.B.R.stat.stat.A.stat.E. Bayesian Biostatistics and Diagnostic Medicine Feinstein A. Regression Methods in Biostatistics: Linear.How to Become a Pure Mathematician Page 63 of 66 more free online material. Statistical Reasoning in Medicine: The Intuitive P-Value Primer Norman G. Theories. Biostatistics: A Methodology for the Health Sciences . Biostatistics: The Bare Essentials Paciorek C.C.edu/courses/st572-larget/. and McCulloch C.f02/index.wisc. at: http://www. Biostatistical Analysis Statistical Methods in Epidemiology: Bhopal R.sjsu.html.V.stat.jhsph. and Repeated Measures Models Zar J..L.com/mathphyweb/puremath.H. Logistic.page99868) Petrie A. collection of research papers. Statistical Methods for Biostatistics and Related Fields .ca/~rollin/teach/643w04/index.edu/courses/st571-larget/stat571.biostat-text. Newman S.wisc. Glantz S. Survival. Shiboski S. 2007 (FREE! http://www. and Vieu P.C. see also John Hopkins Bloomberg School of Public Health's biostatistics section (http://ocw. Epidemiology Kept Simple: An Introduction to Traditional and Modern Epidemiology .Amazon reader Kalantar-zadeh comments it as a historical masterpiece in the field. http://www. Primer of Biostatistics Härdle W. Biostatistics: A Guide to Design.Review questions. Analysis and Discovery . Vittinghoff E. Lee E. Bayesian Methods in Biostatistics (FREE! http://isites.harvard. http://www.D.D.stat. Medical Statistics at a Galance . II.P.edu/Topics.htm .edu/icb/icb.See their site (http://www.R.htm Jewell N.com/).cfm?topic_id=33).. II (FREE! http://www. PowerPoints.ubc. Statistics for Epidemiology . and Sabin C.geocities.. Principles of Medical Statistics Forthofer R. Mori Y.com/).N. M.edu/homepgs/klein/book.ac.edu/~dkleinb/surv2.asp?cat_no=C415X Longitudinal Data Analysis: Davidian M. BSTT513 Longitudinal Data Analysis (FREE! http://www. Liang K.-Y.stats. Applied Longitudinal Analysis (FREE! http://biosun1. Epidemiology: Study Design and Data Analysis . Longitudinal Data Analysis Survival Analysis: Hosmer D.harvard.H. Chen M..M. and Moeschberger M.biostat.Considered to be classics in the field.M.How to Become a Pure Mathematician Page 64 of 66 Rothman K.W.htm#data Laws N.J.D.L.geocities. and Ware J. and Klein M.edu/classes/bstt/bstt513/index. Survival Analysis – See Klein's page (http://www. Kleinbaum D.P. Biostatistics and Epidemiology: A Primer for Health and Biomedical Professionals Woodward M...crcpress. and Gibbons R.ncsu. and Ware J.htm .Data files at: http://www. Data and errata page at: http://faculty.edu/~dzhang2/st745/) http://hk..ncsu.mcw. proofs are omitted. Analysis of Multivariate Survival Data Ibrahim J.html) Hedeker D.ox. Survival Analysis: A Self-Learning Text . CRC's Instruction and Download at: http://www. Survival Analysis MT (FREE! http://www..G.stat.W.Lecture slides used at the Harvard School of Public Health for BIO226.Mark insists that this book is written for both researcher and statistician.html) .-H. and Lemeshow S.html) for data sets and errors. Bayesian Survival Analysis Klein J. Fitzmaurice G.com/mathphyweb/puremath.edu/people/davidian/st732/) Diggle P.stat.G. Applied Longitudinal Data Analysis (FREE! http://www. Analysis of Survival Data (FREE! http://www4.uic. Coull B. Applied Survival Analysis: Regression Modeling of Time to Event Data Hougaard P.R Applied Longitudinal Data Analysis for Epidemiology: A Practical Guide Wassertheil-Smoller S. and Zeger S. ST 732. Epidemiology: An Introduction Twisk J.washington. and Sinha D.edu/~fitzmaur/ala/lectures.edu/heagerty/Books/AnalysisLongitudinal/ Fitzmaurice G.uk/~laws/survival/) Tsiatis A. Applied Longitudinal Analysis Hedeker D. Analysis of Longitudinal Data . Laird N.com/e_products/downloads/download.sph. Heagerty P.emory. and Zhang D. org/uk/series/sSeries. My list ends here. Yosida.cambridge.htm#grad The Electronic Library of Mathematics http://www.-Nagy. e.com/west/home/math?SGWID=4-10042-69-173621337-0 • Springer: Grundlehren der mathematischen Wissenschaften http://www.65/mathdoc/ • Stein W. abstract algebra or topology courses.ams. and possibly more than that.math. you may refer to them according to your interest. Believe it or not. Now you probably know what's going on and you should have an idea what you can read.org/cgi-bin/bookstore/bookpromo/gsmseries • Springer: Graduate Text in Mathematics GTM http://www. This is (partly) because the entry requirement of mathematics degree is usually not so high (but don't expect it's easy) and folks fall into such program accidentally and they don't have to pick those abstract courses.org/online_bks/online_subject. Harthsorne. Riesz and Sz.php Scanned text (mostly Chinese.htm • • .html • • • • MIT's graduate courses list: http://ocw. choices decrease as stages increase.html FreeScience http://freescience.e.cambridge.edu/OcwWeb/Mathematics/index.org/uk/series/sSeries.ams. Baby Rudin and Big Rudin.de/ELibM. For more advanced FREE! materials.'s Modular Abelian Varieties page http://modular. check out: American Mathematical Society Books Online: http://www.asp?code=LMSN http://hk. Halmos.com/mathphyweb/puremath.washington.edu/scans/252/ (I've listed a few of them in the above list) List of books: American Mathematical Society: Graduate Studies in Mathematics http://www.38. Several books in the list above give suggestions for further reading.emis.geocities. a. you should be ready for graduated schools.springer.com/west/home/math?SGWID=4-10042-69-173621309-0 • Springer: Universitext http://www.cambridge.asp?code=LMST • Cambridge University Press: London Mathematical Society Lecture Notes Series http://www.126. I've seen mathematics graduates haven't taken any analysis. Moreover.org/uk/series/sSeries. You've completed an undergraduate mathematics program (who cares about certificate?).How to Become a Pure Mathematician Page 65 of 66 Stage 5 By completing stage 4. etc) http://202.g.asp?code=CSAM • Cambridge University Press: London Mathematical Society Student Texts http://www.mit.info/Mathematics.springer.springer.com/west/home/math?SGWID=4-10042-69-173621316-0 • Cambridge University Press: Cambridge Studies in Advanced Mathematics http://www. harvard.cam.html#past After several extra stages. given in the 'Professional Issues and Ethics in Mathematics' course): Modelling of environment (related areas are global warming.html Want challenging questions? See: Mathematical Tripos Examination Papers http://www.htm .math. to inspire (or corrupt?) the next generation • • • • http://hk.princeton.geocities.edu/graduate/index.uk/teaching/pastpapers/ Harvard Mathematics Qualifying Exam http://www. if you feel like you are ready to read journals.org/ • • • • • After all. what can you do with your mathematical knowledge? There are a few options (ideas by James Franklin. etc) Optimization for resources planning Statistical research on effectiveness of drugs and medical procedures.com/by-subject-science-and-mathematics-mathematics.edu/ arXiv. if it turned out that you're not going to further your study in pure mathematics.com/mathphyweb/puremath.org http://arxiv. try: Annals of Mathematics http://annals. better diagnosis Teaching.How to Become a Pure Mathematician Page 66 of 66 • Dover books in mathematics http://store.ac.org/archive/math Project Euclid http://projecteuclid. weather prediction.maths.doverpublications.