Maths and academia
Lecture notes
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:
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
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
Morita equivalence and differentiable stacks by Matias del Hoyo
Poisson geometry by Rui Fernandes and Ioan Mărcuț (now partly incorporated into this book)
Tangent distributions (topological aspects) by Álvaro del Pino
Various notes/books on Algebraic topology, K-theory, etc. by Allen Hatcher
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
Videos of talks/lectures
Online seminars born during the pandemic (for a more comprehensive list, see https://researchseminars.org/):
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?
Books available online
Lists of free maths books:
List of lists of free maths books:
I will not mention the well known online library which generously offers many books from Russia and elsewhere.
Advice on mathematics and/or the academic world for young people
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
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
Writing:
Practical suggestion for mathematical writing by Bjorn Poonen
Writing tips by Claire Debord [in French]
Career:
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:
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)
Advice on mathematics and/or the academic world for everyone
Miscellanea:
How to balance research with everything else we do by David Zureick-Brown (Notices Amer. Math. Soc. 67:5 (2020), 659-669)
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 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]
Movies
(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) *
* = despite 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, etc.)
Other online resources
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:
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)
Real-world applications of mathematics, by arxiv subject area?
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)