Maths and academia

Below are lecture notes (or review papers, etc.) I have studied/read (partially or entirely) during the past years, and which I found particularly useful and/or well written.

"Classic" stuff many people learn in their bachelor/master courses:

• Introduction to topology by Marius Crainic

• First- and second-year comprehensive courses on mathematical analysis by Paolo Acquistapace [in italian]

• Measure theory and functional analysis by Paolo Acquistapace [in italian]

• Classical PDEs by Paolo Acquistapace [in italian]

• The three "classical introductions" to topological manifolds, smooth manifolds and Riemannian manifolds by John Lee 

• Vector and principal bundles, connections and G-structures by Marius Crainic

• Symplectic geometry by Ana Cannas da Silva

• Lie groups and representation theory by Erik van den Ban

• Analysis on Manifolds by Marius Crainic and Erik van den Ban

• Various lecture notes on Analytical/rational mechanics and mathematical physics (scroll down the page) by Sergio Benenti [partially in Italian]

Less classic/more advanced topics:

• Lie groupoids, Lie algebroids and integrability by Marius Crainic and Rui Fernandes

• Lie groupoids and algebroids by Eckhard Meinrenken

• Morita equivalence and differentiable stacks by Matias del Hoyo

• Poisson geometry by Eckhard Meinrenken

• Poisson geometry by Rui Fernandes and Ioan Mărcuț (now partly incorporated into this book)

• Tangent distributions (topological aspects) by Álvaro del Pino

• Dirac Geometry by Ioan Mărcuț

• Exterior differential systems by Robert Bryant

• Diffeological groupoids by Christian Blohmann

• Various notes/books on Algebraic topology, K-theory, etc. by Allen Hatcher

• Differential geometry and PDEs by Luca Vitagliano

• Relativistic Theories, Gravitational Theories and General Relativity (including a detailed coordinate approach to tensor calculus, fibre bundles, connections, etc.) by Lorenzo Fatibene - see also the video recordings of his course on Relativistic Models

Online seminars born during the pandemic (for a more comprehensive list, see https://researchseminars.org/):

• Friday Fish online seminars

• Junior/Senior Global Poisson events

• Global noncommutative geometry seminar

• Virtual symplectic seminar and symplectic zoominar

• Cibercoloquio Latinoamericano de Matemáticas [in Spanish and Portuguese]

Recordings of physical events (pre-pandemic):

• SwissMAP (Switzerland): events since 2018

• ESI (Vienna): events since 2017

• IMPA (Rio de Janeiro): events since 2015

• CIRM (Marseille): events since 2014

• IHES (Paris): events since 2013

• HCM (Bonn): events since 2013

• BIRS (Banff): events since 2005

Recordings of the recent Poisson events (when available):

• 2022 edition (Barcelona and Madrid): conference

• (2020 edition, scheduled in Salerno and Napoli: cancelled due to the pandemic)

• 2018 edition (Toronto): school

• 2016 edition (Genève and Zürich): school and conference

• 2014 edition (Urbana Champaign): school and conference

• 2012 edition (Utrecht): school and conference

• 2010 edition (Rio de Janeiro): apparently broken link?

Lists of free maths books:

• Online Mathematics textbooks

• Free (and legal) mathematics textbooks

• Free mathematics books

• Books online

• Textbooks in mathematics

• Differential geometry textbook recommendations and historically interesting works

List of lists of free maths books:

• Free, high quality mathematical writing online

• Math resources

• Mathbooks

• Books on differential geometry

I will not mention the well known online library which generously offers many books from Russia and elsewhere.

Miscellanea:

• Early Career: a project of the Notices of the American Mathematical Society, started in 2018, where senior academics provide information and advice to junior ones on a variety of topics

• How to learn math and physics by John Baez

• Advice for the young scientist by John Baez

• Advice to a young mathematician by Micheal Atiyah, Béla Bollobás, Alain Connes, Dusa McDuff and Peter Sarnak

• The "Three Things" Exercise for getting things out of talks by Ravi Vakil

• The most common errors in undergraduate Mathematics by Eric Schechter

• A PhD student or a Jedi Padawan? by Liubov Vetoshkina

Writing:

• Tips on mathematical writings by Tom Leinster

• Practical suggestion for mathematical writing by Bjorn Poonen

• Writing tips by Claire Debord [in French]

• On Writing by Terence Tao

• How to write the summary of a talk by Stefan Waldmann

• How to write mathematical papers by Bruce Bernd

Career:

• Career advice by Terence Tao

• On doing a PhD by Marcello Seri

• The illustrated guide to a PhD by Matt Might

• How to get a PhD in mathematics in a timely fashion (in the US system) by Sarah Billey 

• Advice by Lauren Williams, by Catherine Cannizzo and by Remy van Dobben de Bruyn on finding postdoc positions (mostly) in the US

• Cultivating an Online Presence for the Academic Job Market by Holly Krieger (Notices Amer. Math. Soc. 66:6 (2019), 853-854)

• Names/titles of academic positions through Europe: PhD/Postdocs and Professors by Informatics Europe

Refereeing:

• How to Referee a (Math) Paper by Álvaro Lozano-Robledo

• Writing, and Reading, Referee Reports by Arend Bayer (Notices Amer. Math. Soc. 66:3 (2019), 363-364)

• Writing for Mathematical Reviews by Kelly Jabbusch (Notices Amer. Math. Soc. 66:2 (2019), 197-198)

• Journal Refereeing: Merge with the Collective Mind by Ken Ono and Robert Schneider (Notices Amer. Math. Soc. 67:2 (2020), 188-189)

Miscellanea:

• How to balance research with everything else we do by David Zureick-Brown (Notices Amer. Math. Soc. 67:5 (2020), 659-669)

• Math-life balance by Mura Yakerson

• The materiality of mathematics: Presenting mathematics at the blackboard by Christian Greiffenhagen (The British Journal of Sociology, 65:3 (2014), 502-528)

• Various links on women underrepresentation and (lack of) diversity in academia by Anna Pachol and Catherine Cannizzo (often from a US perspective)

Presenting:

• How to teach stuff by John Baez

• Talks are not the same as papers by Terence Tao

• What to do in talks by Jordan Ellenberg

• How to talk mathematics by Paul Halmos (Notices Am. Math. Soc. 21:3 (1974), 155-158)

• Giving a Talk by Bryna Kra (Notices Amer. Math. Soc. 69:1 (2022), 52-55)

• Good talk? by Stefan Waldmann [in German]

• How to give a good colloquium by John McCarthy

(well known and less known) Movies on mathematics, mathematicians and academia, in cronological order:

• Donald in Mathmagic Land (1959): short animated film about Donald Duck discovering the beauty of mathematics

• Galileo (1968): film on G. Galilei [in Italian]

• Cartesius (1974): miniseries on R. Descartes [in Italian]

• Galileo (1975): film on G. Galilei

• It's my turn (1980): film with a scene on the Snake Lemma

• N is a Number: A Portrait of Paul Erdős (1993): documentary on P. Erdős

• A Beautiful Mind (2001): film on J. Nash

• Travelling Salesman (2012): film on the P versus NP problem

• Die Vermessung der Welt (2012): film on the lives of C. Gauss and A. Humboldt [in German]

• The Imitation Game (2014): film on A. Turing

• Theory of Everything (2014): film on S. Hawking

• The Discrete Charm of Geometry (2015): documentary on the work of a group of mathematicians

• The Man who knew Infinity (2015): film on S. Ramanujan *

• Hidden Figures (2016): film on K. Johnson, M. Jackson and D. Vaughan

• Secrets of the Surface: the Mathematical Vision of Maryam Mirzakhani (2020): documentary on M. Mirzakhani

• Genius, season 1 (2017): miniseries on A. Einstein (and M. Marić) *

• Le Théorème de Marguerite (2023): film on the (private and professional) life of fictitional PhD student *

* = despite many fictionalised elements, the representation of academia is relatively more faithful than the usual idealistic/stereotyped one (e.g. it is depicted the struggle to find a position, to get one's paper published, to collaborate with colleagues, etc.)

Basics:

• Arxiv: database of pre-prints (see recent uploads in differential geometry and symplectic geometry)

• MathSciNet: database of published papers + reviews (accessible only via your institution) - see a written guide and various video tutorials

• ZbMATH: database of published papers + reviews (freely accessible)

• MR collaboration distance: collaboration graphs of mathematicians with at least one paper published (as application, it computes Erdős Number)

• Math Genealogy: genealogical trees (of advisor/PhD student) through the centuries

• MathJobs: collection of academic jobs offers for mathematicians in North America (only very few jobs in the rest of the world are posted there - as far as I know, there is unfortunately no other comparable and comprehensive website for academic jobs in Europe and other continents)

• Learn Latex in 30 minutes: to learn the basics using Overleaf (for further information, this Latex Guide by Anna Marie Bohmann is a good collection of other online resources)

Tools:

• Scirev: website to read other researchers' experiences with scientific journals (in particular, the review time)

• Detexify: website to find latex symbols

• tikzcd-editor: website to create commutative diagrams with tikzcd by Yichuan Shen

• quiver: website to to create commutative diagrams with tikzcd (with a few more features than the one above) by Varkor

Interesting Stackexchange/Mathoverflow discussions:

• Academia varies more than you think it does

• What does the typical workflow of a journal look like?

• Education or employment: What is a post-doc? What is a PhD student?

• Admission process for PhD in various countries (Germany, The Netherlands, USA, Sweden, France, Japan, UK, Italy, Russia, Australia)

• Evaluation of PhD applications in the US

• What's so great about blackboards?

• Real-world applications of mathematics, by arxiv subject area?

• Proofs without words

• Proofs accessible to high school level

• Your favorite surprising connections in mathematics

• How to deal with Imposter Syndrome

Funny:

• PhD movies (1 and 2): very US-centric and biased towards experimental sciences, but still enjoyable

• Mathgen: randomly generated maths papers

• Maths-themed Scavenger Hunt (the access code is "Request Mission") by Alfonso Garmendia and Leonid Ryvkin

• Six Degrees of Wikipedia: for learning graph theory by playing the homonymous game (see also Getting to Philosophy)

• Exceptional MathReview by Kimball Martin

• Postdoc Hunters Club by Ana Ros Camacho

• A Mathematical Conventions Survey by Thomas Lam

• A systematic ordering of foods in the Soup-Salad-Sandwich phase space by Madelyn Leembruggen and Caroline Martin 

• If I Reacted to Other People’s Careers the Way They React to Me Becoming a Mathematician by Jordy Greenblatt

• Why Do We Pay Pure Mathematicians?  by Ben Orlin