As a member of an underrepresented group in mathematics, I’ve found that it is useful for everyone to have access to resources to help them on their journey of mathematical exploration (and learning, in general). Federico Ardila’s beautiful axioms for doing mathematics provide an inspiring framework in which we can all strive to do mathematics equitably and joyously:
Axiom 1. Mathematical talent is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
Axiom 2. Everyone can have joyful, meaningful, and empowering mathematical experiences.
Axiom 3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
Axiom 4. Every student deserves to be treated with dignity and respect.
Below I’ve included an amalgamation of resources I’ve come across over time that have helped me in some way, shape, or form. I hope it will do the same for all learners out there.
- For those who want to jump into typing math and collaborating with others online, Overleaf is a friendly introduction to the math typesetting language $\LaTeX$ that hosts everything right in your browser (no need to install any packages!). It also has templates for common document and presentation types, as well as helpful tutorials to help get started.
- If you want to install $\LaTeX$ on your machine and typeset math offline (which I recommend you try), I suggest using the text editor Sublime Text. It can be used for $\LaTeX$ as well as a host of other languages.
- Information about installing $\LaTeX$ can be found here.
- A quick way to know how to typeset a certain mathematical symbol in $\LaTeX$ is to use detexify to draw out the symbol. The program will then present you with typeset options from which to choose.
- A cheat sheet for quickly changing the appearance of your beamer slides.
- If you prefer to collaborate through handwriting (imagine standing at a blackboard with your peers), the Scribble app lets you share a whiteboard with anyone through the app and anyone can view a shared whiteboard through their browser. It is easy to use and the infinite white space provides the perfect landing ground for new ideas.
- For collaboration that can be succinctly done through some diagrams, the online tikzcd editor is great for this. You can draw any diagram, share the link and edit it with others in real-time. You can even extract the $\LaTeX$ code to use it later.
- For collaboration that needs some coding (read: Sage), CoCalc is an interactive environment that can help you get started coding math right away - again, without the need for installing any packages on your machine. Although it can be a little slow at times, it is great for mini computations and jotting down quick and short ideas.
- My personal notetaking outside of collaborations is a combination of pen-and-paper (and post-its!), GoodNotes, and Notability. The latter two are practically interchangeable and its utility highly depends on personal preference.
- For task management and all things organizational, I use a combination of Notion and Obsidian. More on this will be forthcoming in a blog post.
Blogs and expositions
- One of my all-time favourite math blogs is Tai-Danae Bradley’s blog Math3ma. It is a huge source of friendly expositions about important math topics, ranging from Galois theory all the way to enriched category theory. I highly recommend you check it out.
- As an aside, Bradley and coauthors have penned a book called Topology: A Categorical Approach that takes traditional point-set topology and infuses it with the language of category theory. An enjoyable read for those familiar with point-set topology and interested in delving into category theory and its applications.
- Terence Tao’s blog What’s new? has a wealth of resources for mathematicians at all stages of their journey. He also provides regular research updates that are highly accessible.
- Keith Conrad’s expository papers consist of rigorous, understandable, and complete expositions on important topics in algebra and analysis. It has more of an emphasis on algebra, with many papers on elementary and algebraic number theory.
- Emily Riehl’s expository writings are vital for anyone interested in category theory. Her category theory book, Category Theory in Context is a wonderful introduction to the subject, with an emphasis on drawing examples from all areas of mathematics.
- Scott Balchin gave a series of lectures called Model Categories by Example and his detailed notes are helpful for those looking for some intuition behind the definitions and examples of model categories.
- The nLab has a plethora of expositions, although I mostly use it for introductions and descriptions of topics in category theory.
- A heap of advice on research and writing from MIT is helpful for all students and aspiring researchers.
- A graduate student survival guide from the University of Maryland is filled with detailed advice and descriptions of graduate school, useful for both prospective and current students alike.
- Last but definitely not least, the arXiV: for all mathematical pre-prints, past, and current research.