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Jiří Lebl
Open textbooks & study tools · Analysis · English
Author-written open analysis texts.
Use / preparation: Follow each author’s prerequisites and use the legally available edition. · Checked 2026-10-11
Sheldon Axler
Open textbooks & study tools · Analysis & linear algebra · English
Author information and legally available editions of selected books.
Use / preparation: Follow each author’s prerequisites and use the legally available edition. · Checked 2026-10-11
Evan Chen
Open textbooks & study tools · All mathematics · English
Broad personal exposition; not a replacement for a detailed subject textbook.
Use / preparation: Follow each author’s prerequisites and use the legally available edition. · Checked 2026-10-11
David Tong
Open textbooks & study tools · Mathematical physics · English
Author theoretical-physics notes with mathematically relevant topics.
Use / preparation: Follow each author’s prerequisites and use the legally available edition. · Checked 2026-10-11
ProofWiki contributors
Open textbooks & study tools · Proofs & foundations · English
Community definitions and proofs; check arguments and references independently.
Use / preparation: Follow each author’s prerequisites and use the legally available edition. · Checked 2026-10-11
Xinyi Yuan / 袁新意 · 学习代数几何的建议
Learning & research advice · Algebraic geometry · Chinese
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.
Use / preparation: For students choosing a first scheme-theoretic course; compare the suggested depth of reading with your existing algebra background. · Author advice · 2026-04-30; v2.2 · Difficulty ratings and textbook preferences are the author's judgments. · Checked 2026-10-16
Yongqi Liang / 梁永祺 · 我为什么选择了代数几何和数论
Learning & research advice · Algebra & number theory · Chinese
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.
Use / preparation: Read as one mathematician's route into a field, alongside the more systematic preparation advice on his teaching page. · Personal learning account · 2012-12-16 · An autobiographical account, not a current admissions guide; the older article uses a legacy web layout. · Checked 2026-10-16
Ravi Vakil
Learning & research advice · Algebraic geometry & research habits · English
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.
Use / preparation: For advanced undergraduates and beginning graduate students learning how research differs from completing a course. · Author advice · The opening supervision expectations describe this author's advising style, not a universal doctoral admission threshold or a current vacancy. · Checked 2026-10-16
Timothy Gowers
Learning & research advice · Analysis & proofs · English
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.
Use / preparation: For a first proof-based analysis course; try the proof moves on an exercise before returning to the explanation. · Worked learning advice · 2014-02-03 · Checked 2026-10-16
Terence Tao
Learning & research advice · Mathematical learning · English
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.
Use / preparation: For deciding what it means to revisit a course after your mathematical perspective has changed. · Author advice · Checked 2026-10-16
Michael Atiyah; Béla Bollobás; Alain Connes; Dusa McDuff; Peter Sarnak
Learning & research advice · Mathematical learning & research · English
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.
Use / preparation: Read section VIII.6; the PDF begins with the end of the preceding entry and continues beyond the advice essays. · Collected essays · 2008; The Princeton Companion to Mathematics, VIII.6 · Checked 2026-10-16
Wen-Wei Li / 李文威 · 数学写作漫谈
Learning & research advice · Mathematical writing · Chinese
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.
Use / preparation: For students writing reading reports, expository notes or a first thesis; useful to contributors preparing explanations for this site. · Author advice / edited talk · 2015-04-24; based on a 2014 talk · Checked 2026-10-16
J. S. Milne
Learning & research advice · Mathematical writing · English
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.
Use / preparation: Read the recommendations in reverse when editing your own mathematical exposition. · Satirical writing advice · Satire: the literal instructions are deliberately bad advice. · Checked 2026-10-16
Noah Snyder; discussion by named commenters
Learning & research advice · Mathematical learning & research · English
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.
Use / preparation: Read the comments as well as the post; the author explicitly frames the advice around his own graduate-school setting. · Personal research advice & discussion · 2009-05-13 · Historical personal advice; graduation, funding and employment conditions must not be inferred from this 2009 discussion. · Checked 2026-10-16
Gian-Carlo Rota
Learning & research advice · Mathematical communication & research · English
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.
Use / preparation: For students starting to give mathematical talks and participate in a research community. · Personal professional reflections · 1997 · Linked from Yongqi Liang's teaching page; this copy is hosted there, not on the original author's site. · Checked 2026-10-16
Zajj Daugherty
Application advice & experiences · US mathematics PhD applications · English
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.
Use / preparation: Start planning with the preparation and letters sections, then make a separate checklist for each target program. · Faculty application advice · Testing schedules, bridge programs and funding eligibility linked in an undated guide require current program-specific checks. · Checked 2026-10-16
UC San Diego Department of Mathematics
Application advice & experiences · US mathematics PhD applications · English
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.
Use / preparation: Read the personal-statement questions as prompts for evidence, not as a template to copy. · Department guidance · The general testing discussion is not a statement of every program's current requirements. · Checked 2026-10-16
AcademicOverAnalysis and r/math commenters
Application advice & experiences · US mathematics PhD applications · English
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.
Use / preparation: Useful for identifying questions to ask faculty who know your mathematical work. · Forum advice & discussion · 2023 · The poster's professional identity was not independently verified; individual anecdotes are not admission thresholds or guarantees of funding. · Checked 2026-10-16
Qi Ma / 马骐 · 马马其关于法国留学的经验与信息分享
Application advice & experiences · French mathematics masters · Chinese
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 / preparation: Use the experience to build questions about your own M1/M2 route, scholarship process and doctoral plans. · Applicant / student experience · Some broad claims about institutions, application platforms and admission chances are the writer's interpretations, not verified current rules; the document's own date is not stated. · Checked 2026-10-16
Daniele Turchetti; Yongqi Liang; Ilaria Lovato; Ariyan Javanpeykar; Djordjo Milovic
Application advice & experiences · European mathematics masters · English
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.
Use / preparation: For understanding the academic and personal experience of a mobile European master's program. · Signed alumni accounts · Selected alumni testimonials are not a representative outcomes survey, current course catalogue or funding offer. · Checked 2026-10-16
University of Strasbourg, UFR de mathématique et d'informatique · Conseils pour candidater
Application advice & experiences · French masters / application writing · French
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.
Use / preparation: English reading guide: focus on explaining your academic path and reasons clearly; the examples are not a ready-made motivation letter. · Department application guidance · Checked 2026-10-16
Texas_Granger and r/etudiants commenters · Choix de master 2 (mathématiques fondamentales)
Application advice & experiences · French mathematics masters · French
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.
Use / preparation: English reading guide: compare the reasoning behind the choices rather than taking commenters' rankings as facts. · Forum experience & discussion · 2024 · Pseudonymous accounts and subjective institutional judgments, not verified comparative outcomes. · Checked 2026-10-16
yozora184 · 東大数理(修士) 院試体験記(2026年度入学)
Application advice & experiences · Japanese mathematics masters / geometry · Japanese
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.
Use / preparation: English reading guide: distinguish basic-paper practice, specialist-paper strategy and oral explanation; this is a domestic undergraduate's experience. · Applicant exam account · 2025 examination; 2026 entry · The entry year is 2026, but the exam described was held in 2025; it does not establish the next exam format or international eligibility. · Checked 2026-10-16
すい / sui · 東大/東工大 数学科 院試体験記
Application advice & experiences · Japanese mathematics masters / algebra · Japanese
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.
Use / preparation: English reading guide: look at how practice changed over the months and how the writer revised weak areas alongside the chosen specialty. · Applicant exam account · The historical Tokyo Tech name is retained from the original; exam weighting and pass-threshold impressions are not official rules. · Checked 2026-10-16
とらかめ / torakame · 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
Application advice & experiences · Japanese mathematics masters / mathematical statistics · Japanese
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.
Use / preparation: English reading guide: read for preparation gaps and choices under time pressure, not for a numerical prediction of admission. · Applicant exam account / setback · 2026-08-29 · Osaka Metropolitan's result was still pending when the article was written; do not turn the account into a final outcomes comparison. · Checked 2026-10-16