{"id":2258,"date":"2024-05-14T16:01:48","date_gmt":"2024-05-14T16:01:48","guid":{"rendered":"https:\/\/diversityinmath.ssmu.ca\/?page_id=2258"},"modified":"2024-05-14T18:35:07","modified_gmt":"2024-05-14T18:35:07","slug":"how-to-read-a-math-paper","status":"publish","type":"page","link":"https:\/\/diversityinmath.ssmu.ca\/fr\/resources\/how-to-read-a-math-paper\/","title":{"rendered":"How to Read a Math Paper"},"content":{"rendered":"\n<div class=\"wp-block-columns alignwide is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>How to start &#8211; outline of steps<\/summary>\n<ul class=\"wp-block-list\">\n<li>Read over the introduction, even if you don\u2019t understand a lot of it, and check the references being mentioned throughout.\u00a0<\/li>\n\n\n\n<li>Read over the introductions of a couple of those, to get a better understanding of the context of your article\/problem, see what else has been done in the area.\u00a0<\/li>\n\n\n\n<li>Once you feel like you know the general (not all the details yet) idea of what\u2019s being asked and how it\u2019s being approached, read through the article\u00a0<\/li>\n\n\n\n<li>The intent of reading it the first time is not to understand everything, and in fact you may not understand anything of some of the sections; it\u2019s to be able to fill in the gaps on re-reads, and with other resources<\/li>\n\n\n\n<li>What might help during these steps:\n<ul class=\"wp-block-list\">\n<li>Find survey articles (google \u201csurvey article [my topic]\u201d)<\/li>\n\n\n\n<li>Find PhD thesis or MSc thesis papers about your question or a related one (they might give more examples and details)<\/li>\n\n\n\n<li>Take notes on at least the introduction, and maybe the first section (at least for now, start at the beginning)<\/li>\n\n\n\n<li>Look for textbooks that explain these topics in more depth, especially if they have examples written out<\/li>\n\n\n\n<li>Look for seminars\/talks about the paper<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/details>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>If you\u2019re stuck:<\/summary>\n<ul class=\"wp-block-list\">\n<li>Sometimes details are deeply hidden or omitted for page constraints\n<ul class=\"wp-block-list\">\n<li>An extended version of the paper may be on arxiv<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Be aware that there are gonna be typos (footnote: \u201cHowever, when, as a graduate student, one encounters the task of reading a technical mathematical paper for the first time, it is often the case that one loses much of one\u2019s higher reading skills, reverting instead to a more formal and tedious line-by-line interpretation of the text. As a consequence, a single typo or undefined term in the paper can cause one\u2019s comprehension of the paper to grind to a complete halt, in much the same way that it would to a computer.\u201d from <a href=\"https:\/\/terrytao.wordpress.com\/advice-on-writing-papers\/on-compilation-errors-in-mathematical-reading-and-how-to-resolve-them\/\">Terry Tao\u2019s blog<\/a>)\n<ul class=\"wp-block-list\">\n<li>Compare with similar sections of the paper<\/li>\n\n\n\n<li>Recognize what the author is trying to do, and how the step fits into the overall argument, don\u2019t just formally read line by line<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Try and find a simpler paper, or a paper which uses the same argument to solve an easier problem (check the references)\n<ul class=\"wp-block-list\">\n<li>A paper will usually tell you when they\u2019re following another paper for a section of the proof<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Find something to try!\n<ul class=\"wp-block-list\">\n<li>Even if you\u2019re completely lost on how to justify a step, just try the first thing which comes to mind. It might not work, but understanding why it doesn\u2019t is still progress<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Find a way to turn large parts of the paper into one step\n<ul class=\"wp-block-list\">\n<li>A paper may be going through a very complicated construction or a long computation which can be summarized as simply \u201cgetting a bound\u201d or something similar.<\/li>\n\n\n\n<li>Look at what the step you\u2019re stuck on is trying to prove, and how it comes into play in the larger structure.\u00a0<\/li>\n\n\n\n<li>In this case, there is no need to spend time understanding \u201cunnecessary\u201d parts of the proof, unless you\u2019re incorporating the arguments into your own research<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/details>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>General advice:<\/summary>\n<ul class=\"wp-block-list\">\n<li>Check your understanding of the ideas (not just the technical details)\n<ul class=\"wp-block-list\">\n<li>Can you approximately reconstruct the proof after looking at it<\/li>\n\n\n\n<li>Explain the paper to someone else (or to a rubber duck)<\/li>\n\n\n\n<li>Simple outline: first step leads to second, etc, in a few sentences<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Don\u2019t be a passive reader!\n<ul class=\"wp-block-list\">\n<li>Ask yourself why the definitions are set up the way they are<\/li>\n\n\n\n<li>As you read statements of theorems, think of how you would approach proving it or disproving it before reading ahead in the paper<\/li>\n\n\n\n<li>Try to anticipate what results follow from theorems<\/li>\n\n\n\n<li>A math paper is a polished version of a longer process of trials and errors &#8211; recreate this process for yourself!<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Prioritise which parts to read and skip\n<ul class=\"wp-block-list\">\n<li>Don&#8217;t read everything in detail if the paper or a specific proof is very long<\/li>\n\n\n\n<li>Some parts of papers are there for completeness, but might not give a lot of interesting insights, skim or skip those<\/li>\n\n\n\n<li>Ask your supervisor for advice and insight into which parts are interesting general tools, which examples are important in your field, what is background reading, which parts you can skip<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Visualize abstract\/general proofs\n<ul class=\"wp-block-list\">\n<li>Start with the proof\/explanations of a special case<\/li>\n\n\n\n<li>Have a leading example, which you refer to as you read through the paper and check how the general results apply in practice &#8211; this will help you develop intuition for the problem<\/li>\n\n\n\n<li>(for graph theory) Draw a graph or example illustrating the important ideas or proofs of the paper<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Once you have a good understanding, think beyond the paper\n<ul class=\"wp-block-list\">\n<li>Check why all the assumptions are necessary &#8211; what happens when you drop one of them?<\/li>\n\n\n\n<li>Can the results be generalised further? If not, what are insightful counter-examples?<\/li>\n\n\n\n<li>Variations &#8211; How would the problem and results change if you slightly change the setting?<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/details>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns alignwide is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>If it\u2019s your first paper\u2026<\/summary>\n<ul class=\"wp-block-list\">\n<li>Don\u2019t be discouraged if you need to read up on a lot of background. If you don\u2019t understand everything on your first read &#8211;\u00a0 don\u2019t hesitate to pause and go back to fill gaps<\/li>\n\n\n\n<li>It gets faster! As you will get comfortable with standard arguments<\/li>\n\n\n\n<li>Papers are written with a specific target audience in mind &#8211; look for a first paper that is suitable for your background<\/li>\n<\/ul>\n<\/details>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>Further links on how to read a math paper:<\/summary>\n<p>Now you know how to read a math paper, you\u2019re all set to go try your own things and let us know any helpful advice we could add to this page!<\/p>\n\n\n\n<p>Not sure how to even find which paper to read? Take a look at <a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/publications-search\">MathSciNet<\/a>, which you can access for free through the McGill VPN or WiFi on campus<\/p>\n\n\n\n<p>Thank you to Agn\u00e8s, Bart, Maddie and Sam for organizing and giving this workshop. Thank you to Prof. Sophie Spirkl for her insightful advice.<\/p>\n\n\n\n<p>Further links on how to read a math paper:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/docs.google.com\/presentation\/d\/1m-ex3UNvkjZrR_ciOH0KA9Bs8J48BQIKkyuyO9oHVTs\/edit#slide=id.p1\" data-type=\"link\" data-id=\"https:\/\/docs.google.com\/presentation\/d\/1m-ex3UNvkjZrR_ciOH0KA9Bs8J48BQIKkyuyO9oHVTs\/edit#slide=id.p1\">Slides from our How to Read a Math Paper Workshop<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/dl.acm.org\/doi\/10.1145\/1273445.1273458\">How to Read a Paper<\/a>, by S Keshav &#8211; a paper about reading papers<\/li>\n\n\n\n<li><a href=\"https:\/\/web.stonehill.edu\/compsci\/history_math\/math-read.htm\">How to Read Mathematics<\/a> by Shai Simonson and Fernando Gouvea &#8211; advice presented in the most poetic and interactive way<\/li>\n\n\n\n<li><a href=\"https:\/\/www.ams.org\/notices\/202005\/rnoti-p660.pdf\">How to Read a Research Paper<\/a> by Matt Baker, AMS Notices &#8211; on the three different speeds at which to read a math paper, based on what you hope to get out of it<\/li>\n<\/ul>\n<\/details>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":2,"featured_media":0,"parent":65,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-2258","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/diversityinmath.ssmu.ca\/fr\/wp-json\/wp\/v2\/pages\/2258","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/diversityinmath.ssmu.ca\/fr\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/diversityinmath.ssmu.ca\/fr\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/diversityinmath.ssmu.ca\/fr\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/diversityinmath.ssmu.ca\/fr\/wp-json\/wp\/v2\/comments?post=2258"}],"version-history":[{"count":21,"href":"https:\/\/diversityinmath.ssmu.ca\/fr\/wp-json\/wp\/v2\/pages\/2258\/revisions"}],"predecessor-version":[{"id":2288,"href":"https:\/\/diversityinmath.ssmu.ca\/fr\/wp-json\/wp\/v2\/pages\/2258\/revisions\/2288"}],"up":[{"embeddable":true,"href":"https:\/\/diversityinmath.ssmu.ca\/fr\/wp-json\/wp\/v2\/pages\/65"}],"wp:attachment":[{"href":"https:\/\/diversityinmath.ssmu.ca\/fr\/wp-json\/wp\/v2\/media?parent=2258"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}