This page exists to record a collection of advice and resources that I often find useful to share with my students, other people I've mentored, and newcomers to research or academia.
Writing mathematics is a very different skill from ordinary English writing. One may be a fluent or even native speaker of English and still not know how to write mathematical prose well in English. The best way to learn this is by practice, reading well-written mathematical books and papers, but there are some useful resources as well:
An academic CV is very different from a non-academic CV. The best resource I know for how to prepare an academic CV is on Karen Kelsky's blog The Professor Is In: check out Dr Karen's Rules of the Academic CV.
Any young mathematician (or computer scientist) who is strongly motivated to become a researcher should consider applying to the Heidelberg Laureate Forum. Every year in September, 200 young mathematicians and computer scientists (doing their bachelor's, master's, PhDs, or up to 5 years after their PhDs) come to Heidelberg to spend a week meeting with Fields Medallists, Abel Prizewinners, Turing Awardwinners, and basically all the superstars of maths and computer science. It's by invitation only (you can apply, but no guarantee you'll be chosen), with all expenses paid (accommodation, food, and also travel if you can prove that you can't get travel funding elsewhere).
Many students are scared of real analysis as a course. It may become easier after reading Timothy Gowers's "Why easy analysis problems are easy" - ignore the very basic typesetting (A and E instead of ∀ and ∃); this is a really useful way to understand theorems and proofs in real analysis.
Some online repositories of recorded videos of research presentations, relevant to my fields of interest:
Fractional Calculus Seminars (organised by Pavan Pranjivan Mehta and myself). Videos on this Youtube channel comprise a series of webinars about topics related to fractional calculus and its applications; more details here.
Operator Theory and Harmonic Analysis (organised by Alexey Karapetyants and Vladislav Kravchenko). Videos include an online webinar series and recordings of in-person talks at various conferences, workshops, and events; more details here.
Fractional Differential Equations (organised by Vassili Kolokoltsov, Jozsef Lörinczi, Eulalia Nualart, Michael Roeckner, Laura Sacerdote, as a four-month research programme at the Isaac Newton Institute, Cambridge). Both the weekly seminars of the programme, and the more numerous talks given in each of three week-long workshops, are recorded and can be viewed online.
Ravil Vakil's webpage "For potential Ph.D. students" is a treasure trove of useful advice and resources - some of it specific to him or to algebraic geometry, but much of it applicable to any budding young mathematician.
