The complete advice collection. For a focused view, choose study advice or application experiences.
Study advice · Application experiences · Reading a mathematics text
学习代数几何的建议
A concrete route into Hartshorne-level algebraic geometry: prerequisite algebra, comparisons of Hartshorne, Vakil and Görtz–Wedhorn, exercise habits and topics that can wait. The author explains where to supplement a compressed presentation rather than treating a book list as a curriculum.
For students choosing a first scheme-theoretic course; compare the suggested depth of reading with your existing algebra background.
我为什么选择了代数几何和数论
A personal account of entering algebraic geometry and number theory, introduced on the author's teaching page as a record of his studies and reading. The accompanying teaching page also discusses exploring modern mathematics and choosing a research direction and advisor.
Read as one mathematician's route into a field, alongside the more systematic preparation advice on his teaching page.
Advice on learning from seminars, working out small examples, talking with other students and developing mathematical questions. The seminar section explains how to benefit from a talk before you can follow all of its technical details.
For advanced undergraduates and beginning graduate students learning how research differs from completing a course.
Shows how elementary analysis proofs can be built from smaller moves: expanding definitions, extracting useful hypotheses and rewriting the goal. A worked sequence argument makes the gap between recognizing a proof and finding one more tangible.
For a first proof-based analysis course; try the proof moves on an exercise before returning to the explanation.
A short reflection on continuing to learn familiar material, including basic ideas within one's own specialty. It encourages revisiting solutions and connections rather than treating a solved problem as exhausted.
For deciding what it means to revisit a course after your mathematical perspective has changed.
Five perspectives on mathematical taste, problems and theory, reading, setbacks and research life. The contributions need not agree: read the differences as evidence that mathematical development does not follow one universal recipe.
Read section VIII.6; the PDF begins with the end of the preceding entry and continues beyond the advice essays.
数学写作漫谈
Discusses mathematical writing as communicating ideas, not merely correcting English or formatting a paper. It treats readership, organization, exposition, examples, typography and talks, with practical comparisons of different writing styles.
For students writing reading reports, expository notes or a first thesis; useful to contributors preparing explanations for this site.
An intentionally sarcastic catalogue of bad writing practices, such as concealing definitions, omitting motivation and giving untraceable references. The joke works by making the reader's difficulties explicit.
Read the recommendations in reverse when editing your own mathematical exposition.
Discusses readable sources, mathematical narratives, taste, peer groups and finding an advisor. The comments include substantive disagreement about when to graduate and how to weigh publications, making this useful as a conversation rather than a checklist.
Read the comments as well as the post; the author explicitly frames the advice around his own graduate-school setting.
Reflections on giving lectures, communicating with an audience and habits within mathematical research. Rota's distinctive voice makes this a useful counterpoint to more procedural guides.
For students starting to give mathematical talks and participate in a research community.
如何读好数学书？｜展卷
Explains how precise definitions, simple cases and the decisive idea of a proof turn reading into understanding. A request to negate an epsilon–delta definition illustrates the difference between repeating a definition and being able to reason with it.
For readers who recognize textbook arguments but cannot yet adapt them to a new example; an authorized excerpt from the chapter 读书妙法.
【论教】数学分析教学体会浅谈
Discusses independent reading, purposeful exercises and reflection in a first university analysis course. Comparisons of Cauchy–Schwarz proofs and the nested-interval proof of the intermediate value theorem show why an illuminating argument need not simply be the shortest one.
Read the student questions and the author's replies as well as the initial essay, especially when deciding how to practise analysis.
教师代表关启安教授在数学科学学院2021年开学典礼上的致辞
Connects repeated textbook reading with active questions: are a theorem's hypotheses needed, can its proof be simplified, and can the conclusion be strengthened? The address also discusses explaining arguments and learning through seminars.
A short starting point for moving beyond knowing a theorem's statement toward asking your own mathematical questions.
USTC基础数学修课指南（第三版第一稿）
A 193-page student-written guide combining prerequisite planning, textbook discussion and explanations of different subjects. Selected sections discuss metric-space examples in topology and counting methods such as double counting, generating functions and the probabilistic method.
Use the contents to select a subject rather than treating the entire guide as a required linear reading plan.
如何學習數學
Organizes learning around a topic's motivation, its actual content and its connections to other ideas. Deriving the quotient rule through different routes illustrates how connections reduce dependence on memorized formulas.
For revising calculus or approaching a new subject; compare the explanation of why a method is needed with the method itself.
不確定自己寫的證明對不對，可以如何學習
Treats proof writing as learning precise communication through practice, feedback and revision. The author compares revising a corrected proof with rewriting a marked composition, rather than checking only the final grade.
For a first proof course or reading group; a correct proof can differ from a model solution, but still needs its logical steps checked.
要学概率论？（作者主页链接名）
Outlines routes from undergraduate probability and real analysis toward measure-theoretic probability, stochastic processes and stochastic analysis. Separate sequences for continuing undergraduates and incoming graduate students make the preparation differences concrete.
Use the course sequence to locate gaps before advanced probability; this is a roadmap, not a sustained essay or a prerequisite for every probability course.
青年数学家刘思齐教授：从高中到大学的转变
Discusses taking responsibility for university study, comparing textbooks and writing and revising one's own mathematical notes. A useful suggestion is to prepare notes as though explaining the subject in a small textbook, then record where the lecturer's account differs.
Relevant to beginning undergraduates and independent learners organizing their first university-level courses.
证明需要全部记住吗？（导读标题）
Reflects on an early ambition to memorize every theorem's proof, then focuses on reusable ideas instead. Sequence-limit examples show how pairing terms, adding and subtracting an expression, and the triangle inequality recur in analysis.
For students who understand an argument while reading but forget its details later; extract the proof's goal and reusable moves.
A substantial guide to advanced coursework, research opportunities, letters, personal statements and funding. It also discusses master's and bridge routes for students who need more preparation before applying to a doctorate.
Start planning with the preparation and letters sections, then make a separate checklist for each target program.
The personal-statement section asks applicants to describe mathematical preparation, contributions and independent study with substance. It gives concrete advice on relevance, explaining setbacks briefly and obtaining honest feedback on a draft.
Read the personal-statement questions as prompts for evidence, not as a template to copy.
A long discussion of research experience, recommendation letters, earlier weak grades and returning to study after employment. Read replies as well as the opening post to see where experiences and opinions differ.
Useful for identifying questions to ask faculty who know your mathematical work.
2022申请季：纯数学与数学物理（导读标题）
A first-person diary covering preparation, recommendation letters, interviews, rejections and alternatives across US and European programs. The account follows interests in mathematical physics toward a reported decision to study algebraic geometry at Warwick.
Read the timeline and changes of plan rather than treating the reported grades or individual results as an admissions bar.
马马其关于法国留学的经验与信息分享
A signed account by a student who reports studying a funded M1 in Toulouse in 2023–24. It describes master's applications, separate scholarship applications and the later search for doctoral supervision and funding.
Use the experience to build questions about your own M1/M2 route, scholarship process and doctoral plans.
Several alumni describe studying across countries, choosing courses, writing a master's thesis and finding their place in mathematical research. The accounts illustrate different experiences within a joint mathematics program rather than a single admissions recipe.
For understanding the academic and personal experience of a mobile European master's program.
Conseils pour candidater
A department's concrete advice for presenting a CV and motivation letter, with good and bad examples. One example shows how to explain a change of university rather than leaving the reader to guess why it happened.
English reading guide: focus on explaining your academic path and reasons clearly; the examples are not a ready-made motivation letter.
Choix de master 2 (mathématiques fondamentales)
A student weighs mathematical interests, institutional reputation, moving costs and doctoral ambitions after receiving M2 offers. Replies discuss course content, potential supervision and everyday experience, sometimes disagreeing.
English reading guide: compare the reasoning behind the choices rather than taking commenters' rankings as facts.
東大数理(修士) 院試体験記(2026年度入学)
A writer identifying as a Tokyo mathematics undergraduate describes application questions, mathematics and English preparation, past-paper seminars and oral-exam preparation. The specialized discussion is particularly relevant to a geometry-oriented candidate.
English reading guide: distinguish basic-paper practice, specialist-paper strategy and oral explanation; this is a domestic undergraduate's experience.
東大/東工大 数学科 院試体験記
An algebra/number-theory-oriented writer compares written and oral exams at two institutions and describes a staged past-paper study group. The account includes differences between the basic and specialist papers rather than only listing a final result.
English reading guide: look at how practice changed over the months and how the writer revised weak areas alongside the chosen specialty.
東大数理の院試に落ちました――数学科B4の大学院入試体験記
A writer identifying as a Hiroshima mathematics undergraduate recounts preparation and exam-day experiences at Tokyo and Osaka Metropolitan, with interests in mathematical statistics. Including an unsuccessful Tokyo attempt offers a different perspective from success-only accounts.
English reading guide: read for preparation gaps and choices under time pressure, not for a numerical prediction of admission.