├── .$mathematics-roadmap.drawio.bkp ├── README.md ├── mathematics-roadmap-topics.md ├── mathematics-roadmap.drawio └── mathematics-roadmap.jpg /README.md: -------------------------------------------------------------------------------- 1 | # mathematics-roadmap 2 | A Comprehensive Roadmap for Mathematics _(in progress)_ 3 | 4 | Keywords: `mathematics roadmap`, `mathematics`, `roadmap`, `mathematics study plan`, `mathematics references`, `references`, `mathematics books`, `books` 5 | 6 | ## Audience 7 | This roadmap is primarily intended for students of Mathematics. This doesn't necessarily mean that students from other disciplines such as Physics and Computer Science won't benefit from it; however, looking at the roadmap could be overwhelming for them, but this is because Mathematics has many areas and the roadmap was intended to be comprehensive to include them. 8 | 9 | ## Roadmap Image 10 | The file `mathematics-roadmap.jpg` contains the image of the roadmap. 11 | ![roadmap](https://github.com/TalalAlrawajfeh/mathematics-roadmap/raw/master/mathematics-roadmap.jpg) 12 | 13 | ## Philosophy 14 | 15 | ### Problems with learning Mathematics 16 | There are several problems with the way Mathematics is presented and taught today which causes all the confusion and struggle that the students experience. In my opinion, the main problem is the way in which Mathematics is currently written in the textbooks. Mathematics is considered to be a deductive science, i.e., starts from first principles (called **Axioms** or **Postulates**) and a set of logical rules that are used to establish results (called **Theorems**) from these first principles; hence, it is typically written in that systematic order to reflect its underlying logical structure. I don't mean from this that it is a "bad" way to write Mathematics in, and I would even say that this is how mathematics should be written "rigorously". However, "rigorously" doesn't imply "pedagogically effective", that is, we don't "naturally" think within the bounds of the axiomatic method. This, also, doesn't imply that we will need to get entirely rid of writing axiomatically either, but to seek somewhere between logical rigor and effective pedagogy. The lack of motivation for the axioms and definitions and the precedence of abstractions to concrete examples (or instances) make students feel that the subject could only be fully understood by an elite few (geniuses). One can hardly find a textbook on any topic that includes its history, philosophy & motivation, and to also contain all the theorems and proofs that the other textbooks contain. 17 | 18 | ### Objective 19 | I don't intend here to offer solutions to the problems mentioned above; however, using the best _(pedagogically best)_ of available references, I wish to construct an effective and comprehensive roadmap for learning Mathematics which approximates my idea of good mathematical exposition. 20 | 21 | I emphasize the importance of the relation of other subjects to Mathematics. Of course, Philosophy lays down the conceptual framework that encompasses the entirety of human knowledge so it relates to any field or science not just Mathematics but Philosophy always had a special relationship with Mathematics and anyone who reads Philosophy can clearly see that. Philosophy impacts one's thought and makes him independent, aware, self-reflective, critical, rigorous, and always seeking for deep understanding. Although I started the roadmap with Philosophy because of my obvious bias, you can skip it but I highly recommend reading at least one book. Also, there are many other important subjects such as Computer Science and Physics which are strongly connected to Mathematics if not sometimes regarded as subsets of Mathematics. Throughout history, Mathematics was strongly influenced by these subjects, and in turn, Mathematics also influenced them. Many ideas in Mathematics have their origins in problems in subjects elsewhere so these subjects are extremely useful for motivating these ideas. 22 | 23 | ### Learning Tips 24 | Learning Mathematics is a tedious task that requires long periods of conscious effort and patience. I offer some tips which I consider to be of great importance when learning any subject within Mathematics (which could be applied elsewhere). 25 | 26 | * The main goal of learning is to understand the ideas and concepts at hand as "deeply" as possible. Understanding is a mental process we go through to see how a new idea is related to previous ideas and knowledge. By "deeply" we mean to grasp as much of the ideas and relations between them as possible. A good metaphor for this is picturing knowledge as a web of ideas where everything is somehow related to everything else, and the more dense the web is, the stronger it becomes. This means that there might be no "perfect" state of understanding, and otherwise it is an on-going process. You could learn a subject and think you understand it completely, then after learning other subjects, you come back to the first subject to observe that now you understand it deeper. Here we can use a famous quote from the mathematician John V. Neumann: "Young man, in mathematics you don't understand things. You just get used to them", which I think really means that getting "used to" some subject in Mathematics might be the first step in the journey of its understanding! Understanding is the journey itself and not the final destination. 27 | 28 | * Motivation for any new concept is a must. This includes historical development of the subject which is sometimes crucial to understanding, analogies, drawings, and many other methods. Thought is induced by problems, questions, and misconceptions; thus, knowing what questions were asked in the mind of the mathematician who developed the subject and the problems he confronted really helps guiding thought in the right direction of understanding. 29 | 30 | * Always question the way the subject is presented. This includes questioning everything from the way terms are defined, to the way theorems are proved, even questioning whether the subject deserves the time and effort mathematicians put to it. Sometimes, there could be many different ways to define something; however, a particular definition is chosen among others for some conveniences and goals, so learning about these conveniences and goals would motivate the use of that definition. Some other times, more than one definition are studied independently so one can easily see the consequences of different definitions. We could use a good quote here from the mathematician Paul Halmos: "Don't just read it; fight it! Ask your own question, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?". 31 | 32 | * Solve as many exercises as you can to challenge your understanding and problem-solving skills. Exercises can sometimes reveal weaknesses in your understanding. Unfortunately, there is no mathematical instruction manual for problem-solving, it is rather an essential skill that requires practice and develops over time. However, it could be greatly impacted by your level of understanding of the subject. The processes of learning and problem-solving are interrelated and no one of them is dispensable in the favor of the other. There are also general techniques that could be helpful in most cases which are found in some books on problem-solving (which are included in the roadmap). 33 | 34 | * Be **metacognitive** (from **Metacognition** which literally means "beyond cognition", i.e., "beyond knowledge"), that is, be aware of your own knowledge and thoughts and consciously think about how you think and acquire knowledge. Thought is not passive, but an active process that could reflect on itself. Metacognition and consciousness help us monitor and regulate our thought processes to increase our potential to learn. This gives us the ability to evaluate our own performance by utilizing past thought experience. 35 | 36 | * Teach what you have learned to someone else or at least imagine that you are explaining what you learned to someone in the best possible way (which is also known as the Feynman Technique). This forces you to elaborately rethink what you have learned which could help you discover any weaknesses in your understanding. 37 | 38 | ## Reading Tips 39 | How should one approach books? Should the reader go through every word from the first page to the last page? Should you solve every single problem? These questions are typical regarding book reading, and answering them is not a straightforward task. I will provide general guidelines, and accordingly the reader should find suitable answers for these questions. 40 | 41 | * What is the most fundamental purpose of reading? To learn, of course. So determining what you want to learn, determines what you should read. Not only what books to read, but also what chapters within a book to read. Sometimes it suffices to read the first chapter of the book, and in other times you have to go through all the chapters. However, one isn't always sure what to read and what to leave, and in that case only read the part you are sure you will need, then after going through other books you will eventually know whether you need to return to the book to read more. Moreover, reading books is not always a "linear" process, that is, sometimes going back and forth between multiple books is necessary. The reader should be critical to himself, and he has to assess precisely what he knows and understands and what he doesn't. 42 | 43 | * Sometimes, skimming (pass quickly through the text to note only the important points by looking for certain keywords) is possible; however, in some cases, you might arrive at a paragraph that you will need to read word by word. That is left to the assessment of your understanding. Patience is the key when dealing with books, so don't expect to go through a 100+ page book in one day and understand everything completely unless you have reasonable prior knowledge of the subject. 44 | 45 | * When reading about a new concept, try to predict what the writer will say before you read it. What (important) questions would you ask about this concept? how would you answer them? and what would you deduce from these answers? Before you read a proof of a theorem, try to prove it yourself first. If you could carry out the proof entirely on your own, then you will become more confident of your knowledge. If you become stuck, then when you read the proof you would embrace what you didn't know and you would hardly forget the proof afterwards. 46 | 47 | * It is possible to find repetitive exercises, in other words, you could go through several exercises that have the same idea which could be solved by the same method. In this case, solving one of them could suffice. Don't always count on your intuition, since one can think he has solved the exercise by just looking at it and at the end he finds out otherwise. Going through all the exercises of a chapter/section is up to you and your assessment of how good you did with the exercises you solved (and again, depending on the assessment of your understanding of the subject). 48 | 49 | * Learning how and when to take notes is not easy. You don't want to waste your time copying the entire book. Most modern books have nice ways to display important information such as definitions and theorems, so it's a waste of time to write these down since you can always return to them quickly. What you should do is take notes of how you understood a difficult concept (that took you a relatively long time to understand) or anything that you would like to keep for yourself which is not included in the book, or to rewrite something in the book with your own words. Notes are subjective and they should be a backup memory that extends your own memory. 50 | 51 | * Read critically. Books are written by people and they are not perfect. Don't take everything for granted. Think for yourself, and always ask yourself how would you write whatever you are reading. If you found out a better way to explain a concept, then write it down and keep it as a note. 52 | 53 | ### How to use the Roadmap 54 | 55 | The roadmap consists of topics, each represented as a labeled group of rectangles. The arrow connecting two groups (say from A to B) represents a dependency (B depends on A). Sometimes the dependency of either of the topics on the other is vague or they are interrelated so a two-sided arrow is used. Colors indicate whether the topics are essential (you can't skip them), optional but recommended (they are not essential but very beneficial), or optional (reading them or skipping them is up to you so they are just regarded as additional information). The legend indicates these colors. 56 | 57 | Each topic has multiple books. This is since any two books in a single topic are either mostly similar but differ in a few aspects (e.g. how they explain some concepts or how the subjects are ordered) or they complement one another (so one book has subjects not discussed in the other). I recommend that the reader should look at the preface and/or introduction and the table of contents to see whether the book satisfies his/her needs and to be able to compare the differences between the books easily. 58 | 59 | The file [mathematics-roadmap-topics.md](mathematics-roadmap-topics.md) contains all the topics of the roadmap with the books in text format if it suits you better, which can also help if you want to arrange your own roadmap. 60 | 61 | ## Software 62 | The software used to create these diagrams is [draw.io](https://www.draw.io/). Just open the file `mathematics-roadmap.html` and you can start editing. 63 | -------------------------------------------------------------------------------- /mathematics-roadmap-topics.md: -------------------------------------------------------------------------------- 1 | ### **Introduction to Philosophy (Optional but Recommended):** 2 | 3 | Think: A Compelling Introduction to Philosophy 4 | Simon Blackburn 5 | 6 | Philosophy: A Text with Readings 7 | Manuel Velasquez 8 | 9 | Thinking It Through: An Introduction to Contemporary Philosophy 10 | Kwame Appiah 11 | *** 12 | ### **Introduction to Logic (Optional but Recommended):** 13 | 14 | The Power of Logic 15 | Frances Howard-Snyder, Daniel Howard-Snyder, Ryan Wasserman 16 | 17 | A Concise Introduction to Logic 18 | Patrick J. Hurely 19 | *** 20 | ### **Precalculus :** 21 | 22 | Precalculus: Mathematics for Calculus 23 | James Stewart, Lothar Redlin, Saleem Watson 24 | 25 | Precalculus 26 | Michael Sullivan 27 | *** 28 | ### **Calculus :** 29 | 30 | Calculus: Early Transcendentals 31 | James Stewart 32 | 33 | Infinite Powers: How Calculus Reveals the Secrets of the Universe 34 | Steven H. Strogatz (Optional but Recommended). 35 | 36 | Thomas' Calculus 37 | George B. Thomas, Jr. 38 | *** 39 | ### **Problem Solving (Optional but Recommended):** 40 | 41 | How to Solve It 42 | G. Polya 43 | 44 | Solving Mathematical Problems 45 | Terence Tao 46 | 47 | How to Solve Mathematical Problems 48 | Wayne Wickelgren 49 | 50 | Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving, George Polya 51 | *** 52 | ### **Introduction to Physics (Optional):** 53 | Physics for Scientists and Engineers with Modern Physics 54 | Raymond A. Serway, John W. Jewett Jr. 55 | 56 | An Introduction to Physical Science 57 | James T. Shipman, Jerry D. Wilson, Charles A. Higgens Jr. 58 | 59 | Physics for Scientists and Engineers: A Strategic Approach with Modern Physics 60 | Randall D. Knight 61 | *** 62 | ### **Introduction to Linear Algebra :** 63 | 64 | Elementary Linear Algebra 65 | Howard Anton, Chris Rorres 66 | 67 | Introduction to Linear Algebra 68 | Gilbert Strang 69 | 70 | Linear Algebra and Its Applications 71 | David C. Lay, Steven R. Lay, Judi J. McDonald 72 | *** 73 | ### **Introduction to Differential Equations :** 74 | 75 | Elementary Differential Equations and Boundary Value Problems 76 | William E. Boyce, Richard C. DiPrima 77 | 78 | Differential Equations 79 | Dennis Zill, Warren Wright 80 | 81 | Fundamentals of Differential Equations 82 | David Snider, Edward B. Saff, and R. Kent Nagle 83 | *** 84 | ### **Introduction to Partial Differential Equations :** 85 | 86 | Partial Differential Equations for Scientists and Engineers 87 | Stanley J. Farlow 88 | 89 | Linear Partial Differential Equations 90 | Tyn Myint-U & Lokenath Debanath 91 | 92 | Partial Differential Equations An Introduction 93 | Walter A. Strauss 94 | 95 | Partial Differential Equations 96 | Mark A. Pinsky 97 | *** 98 | ### **Naive Set Theory, Mathematical Reasoning, Proofs:** 99 | 100 | How to Prove It 101 | Daniel J. Velleman 102 | 103 | Introduction to Mathematical Proofs 104 | Charles E. Roberts 105 | 106 | Book Of Proof 107 | Richard Hammak 108 | 109 | Proofs and Fundamentals 110 | Ethan D. Bloch 111 | *** 112 | ### **Discrete Mathematics :** 113 | 114 | Discrete Mathematics with Applications 115 | Susanna S. Epp 116 | 117 | Discrete Mathematics and Its Applications 118 | Kenneth H. Rosen 119 | 120 | Mathematics: A Discrete Introduction 121 | Edward R. Scheinerman 122 | *** 123 | ### **Introduction to Axiomatic Set Theory :** 124 | 125 | Classic Set Theory 126 | Derek Goldrei 127 | 128 | Introduction to Set Theory 129 | Karel Hrbacek, Thomas Jech 130 | 131 | Elements of Set Theory 132 | Herbert Enderton 133 | 134 | Foundations of Set Theory 135 | A.A. Fraenkel. Y. Bar-Hillel, A. Levy 136 | 137 | A First Course in Mathematical Logic and Set Theory 138 | Michael L. O'Leary 139 | *** 140 | ### **Introduction to Mathematical Logic and Model Theory :** 141 | 142 | Mathematical Logic 143 | Ian Chiswell, Wilfrid Hodges 144 | 145 | A Mathematical Introduction to Logic 146 | Herbert Enderton 147 | 148 | A First Course in Logic 149 | Shawn Hedman 150 | 151 | Propositional and Predicate Calculus 152 | Derek Godrei 153 | 154 | A Friendly Introduction to Mathematical Logic 155 | Christopher C. Leary 156 | 157 | Introduction to Mathematical Logic 158 | Elliott Mendelson 159 | *** 160 | ### **Introduction to the Theory of Computation :** 161 | 162 | Introduction to the Theory of Computation 163 | Michael Sipser 164 | 165 | Computability Theory 166 | Herbert Enderton 167 | 168 | Introduction to Languages and the Theory of Computation 169 | John C. Martin 170 | *** 171 | ### **Introduction to Lattice Theory :** 172 | 173 | Lattices and Ordered Sets 174 | Steven Roman 175 | 176 | Introduction to Lattices and Order 177 | B.A. Davey 178 | *** 179 | ### **Universal Algebra :** 180 | 181 | Universal Algebra 182 | P.M. Cohn 183 | 184 | An Invitation to General Algebra and Universal Constructions 185 | George M. Bergman 186 | 187 | Universal Algebra 188 | G. Gratzer 189 | 190 | A Course in Universal Algebra 191 | Stanley Burris, H.P. Sankappanavar 192 | 193 | Universal Algebra 194 | Clifford Bergman 195 | 196 | Post-Modern Algebra 197 | Jonathan D. H. Smith, Anna B. Romanowska 198 | *** 199 | ### **Introduction to Combinatorics :** 200 | 201 | Combinatorial Reasoning: An Introduction to the Art of Counting 202 | Duane DeTemple, William Webb 203 | 204 | Introductory Combinatorics 205 | Richard A. Brualdi 206 | 207 | Combinatorics: An Introduction 208 | Theodore G. Faticoni 209 | 210 | How to Count: An Introduction to Combinatorics and Its Applications 211 | Robert A. Beeler 212 | *** 213 | ### **Introduction to Probability Theory :** 214 | 215 | introduction to Probability 216 | Dimitri Berstekas, John N. Tsitsiklis 217 | 218 | A First Course in Probability Theory 219 | Sheldon Ross 220 | 221 | A Natural Introduction to Probability 222 | R. Meester 223 | 224 | Introduction to Probability 225 | Joseph K. Blitzstein, Jessica Hwang 226 | *** 227 | ### **Introduction to Game Theory :** 228 | 229 | Game Theory 230 | Steven Tadelis 231 | 232 | Strategy 233 | Joel Watson 234 | 235 | Game Theory: A Critical Introduction 236 | Shaun P. Hargreaves Heap, Yanis Varoufakis 237 | 238 | Game Theory Evolving: A Problem-Centered Introduction to Modeling Strategic Interaction 239 | Herbert Gintis 240 | *** 241 | ### **Introduction to Mathematical Statistics :** 242 | 243 | Introduction to Probability and Mathematical Statistics 244 | Lee J. Bain, Max Engelhardt 245 | 246 | An Introduction to Mathematical Statistics and Its Applications 247 | Richard Larsen, Morris Marx 248 | 249 | Mathematical Statistics with Applications 250 | Dennis D. Wackerly, William Mendenhall, Richard L. Scheaffer 251 | 252 | Introduction to Mathematical Statistics 253 | Robert V. Hogg, Joseph W. McKean, Allen T. Craig 254 | 255 | A Modern Introduction to Probability and Statistics: Understanding Why and How 256 | F.M. Dekking, C. Kraaikamp, H.P. Lopuhaä, L.E. Meester 257 | 258 | Modern Mathematical Statistics with Applications 259 | Jay L. Devore, Kenneth N. Berk 260 | *** 261 | ### **Advanced Probability Theory :** 262 | 263 | An Introduction to Probability and Statistics 264 | V. K. Rohatgi, A. K. Md. E. Saleh 265 | 266 | A First Look at Rigorous Probability Theory 267 | Jeffrey S. Rosenthal 268 | 269 | A User's Guide to Measure Theoretic Probability 270 | David Pollard 271 | 272 | An Introduction to Probability Theory and Its Applications 273 | William Feller 274 | 275 | Probability and Measure 276 | Patrick Billingsley 277 | 278 | Probability 279 | A. N. Shiryayev 280 | *** 281 | ### **Advanced Mathematical Statistics :** 282 | 283 | Statistical Inference 284 | George Casella, Roger L. Berger 285 | 286 | Mathematical Statistics: Basic Ideas and Selected Topics 287 | Kjell A. Doksum, Peter J. Bickel 288 | 289 | Theory of Statistics 290 | Mark J. Schervish 291 | 292 | Robust Statistics 293 | Frank  R. H., Elvezio M. R., Peter J. R., Werner A. S. 294 | 295 | All of Statistics 296 | Larry Wasserman 297 | 298 | All of Nonparametric Statistics 299 | Larry Wasserman 300 | *** 301 | ### **Introduction to Real Analysis:** 302 | 303 | Introduction to Real Analysis 304 | Robert G. Bartle, Donald R. Sherbert 305 | 306 | Introduction to Real Analysis 307 | William F. Trench 308 | 309 | How to Think About Analysis 310 | Lara Alcock 311 | 312 | From Calculus to Analysis 313 | Steen Pedersen 314 | 315 | Writing Proofs in Analysis 316 | Jonathan M. Kane 317 | *** 318 | ### **Introduction to Numerical Analysis :** 319 | 320 | Numerical Analysis 321 | Timothy Sauer 322 | 323 | Numerical Analysis 324 | Richard Burden Jr. Douglas Faires 325 | 326 | Numerical Methods That Usually Work 327 | Forman S. Acton 328 | 329 | An Introduction to Numerical Methods and Analysis 330 | James F. Epperson 331 | *** 332 | ### **Introduction to General Topology :** 333 | 334 | General Topology 335 | Stephen Willard 336 | 337 | First Concepts of Topology 338 | W. G. Chinn & N. E. Steenrod 339 | 340 | Topological Spaces 341 | Gerard Buskes, Arnoud van Rooij 342 | 343 | Topology Without Tears 344 | Sidney A. Morris 345 | 346 | Introduction to Metric and Topological Spaces 347 | Wilson A. Sutherland 348 | 349 | Topology 350 | James Munkres 351 | *** 352 | ### **Introduction to Mathematical Analysis :** 353 | 354 | Principles of Mathematical Analysis 355 | Walter Rudin 356 | 357 | The Elements of Real Analysis 358 | Robert G. Bartle 359 | 360 | Mathematical Analysis 361 | Tom Apostol 362 | 363 | Methods of Real Analysis 364 | Richard R. Goldberg 365 | 366 | Advanced Calculus of Several Variables 367 | C. H. Edwards Jr. 368 | 369 | Calculus and Analysis in Euclidean Space 370 | Jerry Shurman 371 | *** 372 | ### **Introduction to Optimization Theory :** 373 | 374 | A First Course in Optimization Theory 375 | Rangarajan K. Sundaram 376 | 377 | Introduction to Linear Optimization 378 | Dimitris Bertsimas. John N. Tsitsiklis 379 | 380 | Applied Optimization 381 | Ross Baldick 382 | 383 | Practical Optimization 384 | Philip E. Gill, Walter Murray, Margaret H. Wright 385 | 386 | An Introduction to Optimization 387 | Edwin K.P. Chong, Stanislaw H. Zak 388 | *** 389 | ### **Convex Optimization :** 390 | 391 | Convex Optimization Theory 392 | Dimitri P. Bertsekas 393 | 394 | Convex Optimization 395 | Stephen Boyd, Lieven Vandenberghe 396 | 397 | Convex Analysis and Nonlinear Optimization 398 | Jonathan M. Borwein, Adrian S. Lewis 399 | 400 | Foundations of Optimization 401 | Osman Güler 402 | *** 403 | ### **Introduction to Differential Geometry :** 404 | 405 | A Differential Approach to Geometry 406 | Francis Borceux 407 | 408 | Differential Geometry of Curves and Surfaces 409 | Kristopher Tapp 410 | 411 | Differential Geometery of Curves and Surfaces 412 | Manfredo P. Do Carmo 413 | 414 | Elementary Differential Geometry 415 | Barrett O'Neill 416 | 417 | Elementary Differential Geometry 418 | Andrew Pressley 419 | *** 420 | ### **Advanced Differential Geometry :** 421 | 422 | Manifolds, Tensors and Forms 423 | Paul Renteln 424 | 425 | An Introduction to Differentiable Manifolds and Riemannian Geometry 426 | William M. Boothby 427 | 428 | Introduction to Smooth Manifolds 429 | John M. Lee 430 | 431 | A Comprehensive Introduction to Differential Geometry 432 | Michael Spivak 433 | 434 | First Steps in Differential Geometry 435 | Andrew Mclnerney 436 | *** 437 | ### **Introduction to Functional Analysis :** 438 | 439 | Introductory Functional Analysis with Applications 440 | Erwin Kreyszig 441 | 442 | Principles of Functional Analysis 443 | Martin Schechter 444 | *** 445 | ### **Advanced Mathematical Analysis:** 446 | 447 | Real Analysis 448 | Elias M. Stein, Rami Shakarchi 449 | 450 | Real and Complex Analysis 451 | Walter Rudin 452 | 453 | Real Analysis 454 | Gerald B. Folland 455 | 456 | Real Analysis 457 | H.L. Royden, P.M. Fitzpatrick 458 | *** 459 | ### **Introduction to Complex Analysis :** 460 | 461 | Complex Analysis 462 | John M. Howie 463 | 464 | Complex Variables 465 | Mark J. Ablowitz 466 | 467 | Visual Complex Analysis 468 | Tristan Needham 469 | 470 | Complex Variables and Applications 471 | James Ward Brown, Ruel Churchill 472 | 473 | Complex Analysis with Applications 474 | Dennis G. Zill 475 | *** 476 | ### **Introduction to Number Theory :** 477 | 478 | Elementary Number Theory 479 | David Burton 480 | 481 | Elementary Number Theory with Applications 482 | Thomas Koshy 483 | 484 | Elementary Number Theory 485 | Kenneth H. Rosen 486 | *** 487 | ### **Introduction to Abstract Algebra :** 488 | 489 | Contemporary Abstract Algebra 490 | Joseph A. Gallian 491 | 492 | Algebra 493 | Michael Artin 494 | 495 | Algebra Abstract and Concrete 496 | Frederick Goodman 497 | 498 | A First Course in Abstract Algebra 499 | Joseph J. Rotman 500 | *** 501 | ### **Advanced Linear Algebra :** 502 | 503 | Linear Algebra Done Right 504 | Sheldon Axler 505 | 506 | Linear Algebra 507 | Serge Lang 508 | 509 | Finite-Dimensional Vector Spaces 510 | Paul R. Halmos 511 | 512 | Linear Algebra Done Wrong 513 | Sergei Treil 514 | 515 | Linear Algebra 516 | Georgi E. Shilov & Richard A. Silverman 517 | *** 518 | ### **Advanced Abstract Algebra :** 519 | 520 | Algebra 521 | Thomas Hungerford 522 | 523 | Abstract Algebra 524 | David S. Dummit, Richard M. Foote 525 | 526 | Advanced Modern Algebra 527 | Joseph J. Rotman 528 | 529 | Algebra: A Graduate Course 530 | I. Martin Isaacs 531 | 532 | Algebra 533 | Paul M. Cohn 534 | 535 | A Course in Algebra  536 | E.B. Vinberg 537 | 538 | Algebra 539 | Serge Lang 540 | 541 | Algebra: Chapter 0 542 | Paolo Aluffi 543 | 544 | Introduction to Commutative Algebra 545 | M. F. Atiyah & I. G. MacDonald 546 | *** 547 | ### **Algebraic Number Theory :** 548 | 549 | Number Fields 550 | Daniel A. Marcus 551 | 552 | An Introduction to the Theory of Numbers 553 | Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery 554 | *** 555 | ### **Motivation for Category Theory :** 556 | 557 | Conceptual Mathematics 558 | F. William Lawvere, Stephen H. Schanuel 559 | 560 | Sets for Mathematics 561 | F.W. Lawvere, R. Rosebrugh 562 | 563 | Arrows Structures and Functors: The Categorical Imperative 564 | Michael A. Arbib 565 | *** 566 | ### **Introduction to Category Theory :** 567 | 568 | Category Theory 569 | Steve Awodey 570 | 571 | Category Theory in Context 572 | Emily Riehl 573 | 574 | Basic Category Theory 575 | Tom Leinster 576 | 577 | Abstract and Concrete Categories The Joy of Cats 578 | Jiri Adamek 579 | 580 | Category Theory: A Gentle Introduction 581 | Peter Smith 582 | *** 583 | ### **Algebraic Topology Motivation :** 584 | 585 | Invitation to Combinatorial Topology 586 | Maurice Frechet and Ky Fan 587 | 588 | Basic Concepts of Algebraic Topology 589 | Fred H. Croom 590 | 591 | Basic Topology 592 | M.A. Armstrong 593 | 594 | Topology 595 | John G. Hocking, Gail S. Young 596 | 597 | Topology 598 | Klaus Janich 599 | *** 600 | ### **Introduction to Algebraic Topology :** 601 | 602 | Algebraic Topology: An Introduction 603 | William Massey 604 | 605 | Algebraic Topology 606 | Allen Hatcher 607 | 608 | A first course in Algebraic Topology 609 | C. Kosniowski 610 | 611 | Elements of Algebraic Topology 612 | James Munkres 613 | *** 614 | ### **Algebraic Geometry Motivation :** 615 | 616 | Ideals Varieties and Algorithms 617 | David A. Cox, John Little, Donal O'Shea 618 | 619 | Introduction to Algebraic Geometry 620 | Brendan Hassett 621 | *** 622 | ### **Introduction to Algebraic Geometry :** 623 | 624 | Basic Algebraic Geometry 625 | Igor R. Shafarevich 626 | 627 | A Royal Road to Algebraic Geometry 628 | Audune Holme 629 | 630 | Algebraic Geometry: A First Course 631 | Joe Harris 632 | 633 | Elementary Algebraic Geometry 634 | Keith Kendig 635 | 636 | Introduction to Algebraic Geometry 637 | Justin R. Smith 638 | *** 639 | ### **Euclidean and Non-Euclidean Geometry :** 640 | 641 | Foundations of Euclidean and Non-Euclidean Geometry 642 | Ellery B. Golos 643 | 644 | Foundations of Geometry 645 | Gerard A. Venema 646 | 647 | Euclidean and Non-Euclidean Geometry 648 | Marvin Jay Greenberg 649 | 650 | The Four Pillars of Geometry 651 | John Stillwell 652 | 653 | Modern Geometries 654 | John R. Smart 655 | 656 | A Modern View of Geometry 657 | Leonard M. Blumenthal 658 | 659 | Classical Geometry 660 | I. E. L., J. E. L., A. C. F. L., G. W. T. 661 | 662 | Euclidean Geometry 663 | M. Solomonovich 664 | 665 | Introduction To Non-Euclidean Geometry 666 | Harold E. Wolfe 667 | 668 | Geometry 669 | D. A. Brannan, M. F. Esplen, J. J. Gray 670 | 671 | Modern Geometry with Applications 672 | George A. Jennings 673 | 674 | The Foundations of Geoemtry and the Non-Euclidean Plane 675 | George E. Martin 676 | 677 | Projective Geometry 678 | H. S. M. 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