homeowmorphismWebpage of a math cat.
http://localhost:4000/
Fri, 22 Oct 2021 14:06:31 -0400Fri, 22 Oct 2021 14:06:31 -0400Jekyll v3.9.0Reading List for Aspiring Mathematicians<p><span class="newthought">My first college major was neuroscience.</span> I learned about the beauty of mathematics by chance conversations, and eventually decided I deserved to give myself a chance to explore it. When I decided to switch my major to Honours Mathematics, I felt behind my peers who chose math as their major because they had been good at it growing up. After three years in graduate school, I have made my own contributions to mathematics and been invited to various places to present my work to the mathematical community. Although I still struggle with mathematics (as all mathematicians do), I no longer feel the same sense of insecurity that plagued me as an undergrad.</p>
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<p>Here are a few of the readings I found useful, especially at the start of my mathematical journey. Also included are a few pieces of writing I found later on I wish I had the chance to read at a younger age. I hope this can be useful to the person who admires mathematicians but does not identify as a “math person”. I fully believe that, if you choose, you can become not only a “math person”, but a full-fledged mathematician.</p>
<h2 id="readings-that-helped-me-really-understand-math">Readings that helped me really understand math</h2>
<p><strong>Note:</strong> none of these were part of my undergraduate curriculum, but extra resources I found which helped me bridge the gap between what my program implicitly assumed I should already know and what I actually knew as an incoming student of mathematics.</p>
<p><a href="https://betterexplained.com/">Better Explained</a> - this is the website which allowed me to truly understand how math was about concepts, not memorisation. It was especially enjoyable for me to re-learn some key concepts in trigonometry and functions and finally understand what they were about.</p>
<p><a href="https://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/">Gilbert Strang’s Calculus Book</a> - I took Calculus I, II, and III without really understanding much. I had memorised how to do the problems and gotten disappointing grades in exchange. When I decided to major in math, I decided this wouldn’t do and took it upon myself to relearn the whole course sequence, focusing on the ideas rather than the computations.</p>
<p><a href="https://store.doverpublications.com/0486474178.html">A Book of Abstract Algebra by Charles Pinter</a> - this was my first exposure to pure math and formal proofs. I found it illuminating to really attempt the exercises before checking the solutions. My approaches were embarrassingly wrong at first! It felt bad, but really taking the time to understand why I had been so wrong and why the solutions were right opened up the world of pure mathematics to me. To this day, coming up with a proof is the greatest joy of my life.</p>
<h2 id="readings-that-nourished-my-soul">Readings that nourished my soul</h2>
<p><a href="https://www.maa.org/external_archive/devlin/LockhartsLament.pdf">A Mathematician’s Lament by Paul Lockart</a> - an essay dissecting of how mathematics is misrepresented in pre-college education.</p>
<p><a href="https://mathyawp.wordpress.com/2017/01/08/mathematics-for-human-flourishing/">Mathematics for Human Flourishing by Francis Su</a> - an essay discussing who is mathematics for and what can we gain by learning mathematics.</p>
<p><a href="https://arxiv.org/pdf/math/9404236.pdf">On Proof and Progress in Mathematics by William Thurston</a> - an essay by one of the greats of mathematics about what mathematical progress is.</p>
<p><a href="https://www.amazon.com/Peak-Secrets-New-Science-Expertise-ebook/dp/B011H56MKS">Peak by Anders Ericsson</a> - a book dispelling the myth of born geniuses in various fields, including mathematics.</p>
<h2 id="popular-readings-which-propagate-harmful-ideas">Popular readings which propagate harmful ideas</h2>
<p><a href="https://www.math.ualberta.ca/mss/misc/A%20Mathematician%27s%20Apology.pdf">A Mathematician’s Apology by GH Hardy</a></p>
<h2 id="further-readings">Further Readings</h2>
<p><a href="http://ee.usc.edu/stochastic-nets/teaching-resources/documents/Su-Mathematical-Microaggressions.pdf">A list of mathematical microaggressions by Francis Su</a> - math students may recognise some, if not all, of the things listed. I believe it’s important to shift the focus away from how students should inherently feel already confident and competent in their mathematical abilities<label for="" class="margin-toggle sidenote-number"></label><input type="checkbox" class="margin-toggle" /><span class="sidenote">They are students to <em>develop</em> their abilities, after all. Moreover, this kind of assumption disproportionally affects the marginalized, who may have already internalized messages that mathematics is not for them. I know I had. </span> and put some of the burden onto the instructors, who are after all an integrate part of a student’s development.<label for="" class="margin-toggle sidenote-number"></label><input type="checkbox" class="margin-toggle" /><span class="sidenote">Unfortunately, not all instructors are as mindful as Francis Su, and despite what is said in the article I have known some who would <em>purposefully</em> use these sorts of microaggressions to gatekeep students under the veil of plausible deniability; it may be useful to desconstruct what exactly they are doing that is harmful, so to at least learn what - and who - not to listen to. </span></p>
<p><a href="https://blogs.ams.org/inclusionexclusion/2021/05/24/i-want-to-quit-my-job-part-1/">A letter to students about why professors have unearned power by Piper H</a>. This essay explains the systems that allows professors to have unearned power, and why you shouldn’t let them decide whether you belong.</p>
<p><a href="http://www.theliberatedmathematician.com/math/">Piper H’s PhD thesis</a> - Piper H graduated from Princeton (widely seen as the top school for mathematics), where for their thesis they laid out her mathematical research in layman’s terms with great social commentary.</p>
<p><a href="https://www.showingupforracialjustice.org/white-supremacy-culture-characteristics.html">An apt description of most math environments</a></p>
<p><em>Last updated: 22-10-2021.</em></p>
Sat, 08 May 2021 20:21:00 -0400
http://localhost:4000/2021/05/08/Reading-List-For-Aspiring-Mathematicians
http://localhost:4000/2021/05/08/Reading-List-For-Aspiring-MathematiciansMetaMathematics*FavouritesNew Talk About Research Motivations Available<p><span class="newthought">A new talk</span> is available for viewing. I gave a version of this talk today at UIC, and this recording was the practice. The content is motivated by growing pains I was having while writing my thesis, which I will elaborate a bit below.</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/R_DyxY0JpA4" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen=""></iframe>
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<p>In this talk, I’ve focused on explaining the research problem I’ve been trying to tackle in my PhD at the best of my abilities.</p>
<p>Oftentimes when I go to seminars or read papers, I feel like I don’t really understand the problem the math is trying to tackle in the first place because I am distracted by the opacity of the math. I’ve been guilty obscuring the broader point of my research myself, partly perhaps to hide that I don’t really have a good sense of the research questions I am pursuing.</p>
<p>I’ve tried to do better here, and spent some time thinking more deeply about the motivations of my research, having important conversations with my supervisor, and doing some readings perhaps I should’ve done a long time ago. In the talk itself, I tried to elaborate on my new understanding of the problem enough to leave the direction of the research itself wide open to criticism, <a href="https://michaelnielsen.org/blog/principles-of-effective-research/">which I am now able to accept as a good thing</a>.<label for="" class="margin-toggle sidenote-number"></label><input type="checkbox" class="margin-toggle" /><span class="sidenote">For example, in the actual live talk, someone pointed out that I did not formalise the fact that we implicitly want our normal forms to evaluate with low complexity to positive group elements. We have only formalised that we want the normal forms to be recognisable with low complexity. I think that is an excellent point that I have overlooked. </span></p>
<p>If you are interested in my line of research and would like a gentle starting point, I would recommend this talk to you.</p>
<p>PS: I think the audio should be better in this recording than in the previous ones also.</p>
Mon, 19 Apr 2021 20:21:00 -0400
http://localhost:4000/2021/04/19/New-Talk-Research-Motivations-Available
http://localhost:4000/2021/04/19/New-Talk-Research-Motivations-AvailableUpdateMetaSecond Paper on arXiv<p><span class="newthought">My second paper</span> has been uploaded on <a href="https://arxiv.org/abs/2104.04475">arXiv</a>. It is titled “Regular left-orders on groups” and is joint with <a href="https://sites.google.com/site/yagoanpi/">Yago Antolín</a> and <a href="http://www.mat.usach.cl/index.php/2012-12-19-12-50-19/academicos/183-cristobal-rivas">Cristóbal Rivas</a>. In case of interest, I have given talk back in November at Queen’s University on some of the results. A version of the talk is available on <a href="https://www.youtube.com/watch?v=5VsovXUjsJI&feature=emb_title">YouTube</a>.</p>
Mon, 12 Apr 2021 16:21:00 -0400
http://localhost:4000/2021/04/12/Second-paper-on-the-arXiv
http://localhost:4000/2021/04/12/Second-paper-on-the-arXivUpdateLooking For My Next Job<p><span class="newthought">I am on the lookout for a job</span> in either academia or industry. I have uploaded my academic research statement in the “About” section in case of interest. Currently, I am in the thesis-writing stage of my PhD and would be looking for something starting in the Fall or Winter. I am open to working on problems outside of the scope of the research statement, especially in an industry context. I think serendipity can be highly beneficial in my current situation, and unfortunately the pandemic does not allow for much of this, hence me posting this on my website. Please do not hesitate to contact me if you have a lead!</p>
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<p>PS: The Lambda Calculus post’s align environment should be fixed.</p>
Sun, 28 Mar 2021 16:21:00 -0400
http://localhost:4000/2021/03/28/Looking-For-My-Next-Job
http://localhost:4000/2021/03/28/Looking-For-My-Next-JobUpdateMathJax Errors Fixed<p><span class="newthought">The broken math should have been fixed.</span> Recently, I have upgraded the website from using MathJax 2 to MathJax 3. In the process, I have accidentally broken some math environments, causing some lines of maths to disappear for some months. This seems to have affected mostly the more mathematics-heavy posts (which are incidentally in the favourite category of posts - whoops). I think it should be fixed now. Apologies for the inconvenience.</p>
Sun, 17 Jan 2021 19:21:00 -0500
http://localhost:4000/2021/01/17/MathJax-Errors-Fixed
http://localhost:4000/2021/01/17/MathJax-Errors-FixedUpdateTwo New Talks<p><span class="newthought">Two new seminar recordings </span> are up on YouTube. The first one is an introductory talk on left-orderable groups, with some motivation on why it can be interesting to study left-orders under the lens of formal languages and a quick survey of my research niche.</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/MUzO-nIHY84" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen=""></iframe>
<p>You can download the slides for the above talk <a href="https://www.dropbox.com/s/s3nrrbfaieoz5f8/short-slides.pdf?dl=0">here</a>.</p>
<p>The second is more advanced and is about my upcoming paper joint with my supervisor Y. Antolín and C. Rivas, which will be tentatively named ``Left-orders of low language complexity’’.</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/5VsovXUjsJI" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen=""></iframe>
<p>You can download the slides for the above talk <a href="https://www.dropbox.com/s/2060kk5dxvabc7b/slides.pdf?dl=0">here</a>.</p>
<p>The talks have a slight overlap at the beginning of about ten minutes.</p>
Thu, 03 Dec 2020 19:21:00 -0500
http://localhost:4000/2020/12/03/Two-New-Talks
http://localhost:4000/2020/12/03/Two-New-TalksUpdateDealing with Math Anxiety<p><span class="newthought">I still deal with math anxiety regularly</span> as a math PhD student. Here’s my advice on overcoming it.</p>
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<h2 id="step-1">Step 1</h2>
<p>Take a deep breath. Stop staring at those symbols for a bit. Make yourself tea.</p>
<h2 id="step-2">Step 2</h2>
<p>When you come back to it, ask yourself: what is the thing you want to understand? If writing notes, I like to write it down at the top of the page so I don’t get sidetracked by feelings of inadequacy or rabbit-hole into something.</p>
<h2 id="step-3">Step 3</h2>
<p>If the symbols are giving you anxiety, relax. This is normal. The notation is probably confusing you. Write down what every symbol means. You’ll probably have to backtrack and spend time looking them up. That’s well worth it and take however much time you need for this.</p>
<h2 id="step-4">Step 4</h2>
<p>Now that you know what everything means, what is this theorem or equation saying? Are there words left that are ambiguous in the statement? Go look them up. Sometimes it takes me re-reading something 5-6 times, out loud, to get it. Often, I have forgotten the meaning of one crucial word or condition or skipped over them completely and focusing my attention there makes everything makes sense. Math has heaps and heaps of definitions and a perfectly normal English word might mean something super specific! (And it’s totally ok to not remember them at this moment.)</p>
<h2 id="step-5">Step 5</h2>
<p>Sometimes (but not always) it’s a good idea to figure out why what you’re reading is important (or if it’s not). The text probably addressed it somewhere above or below the theorem or equation, but you might’ve been too distracted the first time to notice. Now is a good time to re-read.</p>
<h2 id="step-6">Step 6</h2>
<p>Now onto the proof! This is probably the most technical part. Take a deep breath here again, and start small. Looking at the whole thing might be overwhelming. If a proof is giving me too much anxiety (and let’s be real, this happens a lot), I like to write it down word for word, and as I copy it, fill in every step so that every small change is <em>completely</em> clear to me. I try not to lie to myself about my understanding, as a small lie of understanding can snowball into a big pile of math anxiety again.</p>
<blockquote>
<p><strong>Remember:</strong> This has to make sense somehow.</p>
</blockquote>
<p>As you do this, you might notice that you can’t fill in every step sometimes or get really stuck. This happens more frequently as the complexity of what you read grows. Take time to try to isolate what it is that you don’t understand or are stuck at, and try to formulate a question whose answer would help you understand.</p>
<p>Then, just accept it. It’s ok. You won’t be able to fill in everything every time. Make a note of it, and maybe come back to it later with fresh eyes or accept you don’t know this step. This is also a moment where looking at what you wrote down wanted to understand in the first place helps. Is this tiny step you don’t get crucial for the idea you want to get? Is there someone who can help you figure it out? Book an appointment with them and don’t fuss about it until later.</p>
<h2 id="step-7">Step 7</h2>
<p>As you do this, you might get hit with waves of understanding. I like to sink into that feeling of happiness and do a little dance! This is one of the best feelings of doing math, so relish it!</p>
<h2 id="step-8">Step 8</h2>
<p>Wow, this took longer than you thought! Don’t worry, that’s normal. You’re not bad at this.</p>
<h2 id="extra-tip-thats-been-crucial-for-me">Extra tip that’s been crucial for me</h2>
<p>When given the choice, try to choose the resource that’s easiest for you to read. There’s a lot of resources floating around that aren’t written for a first-time understanding of a concept…or are just plain <em>terrible</em> because a lot of mathematicians aren’t great writers. Unfortunately, you’ll often be suggested those first because people who already know the subject don’t notice how confusing they are, as they’ve forgotten how it felt like the first time they’ve learned the subject<label for="or" class="margin-toggle sidenote-number"></label><input type="checkbox" id="or" class="margin-toggle" /><span class="sidenote">or they’ve toughed it out and now have survival bias, or they just think and get things in a very different way than you do, etc. Who cares! If you feel like what they’re suggesting isn’t right for you, keep looking and don’t feel bad. I don’t remember regretting doing this. </span>. I don’t like wasting my time on things that aren’t written with my situation in mind if I can avoid it. <label for="bang" class="margin-toggle sidenote-number"></label><input type="checkbox" id="bang" class="margin-toggle" /><span class="sidenote">Some say it’s good to bang your head against the wall. I say that hurts. </span></p>
<p>This is complicated if you have assigned notes or textbook for a class or something. In that case I like to follow those but independently google resources when I am confused by a certain topic - remember, not everything is written with uniform clarity and there is no shame trying to get clarification elsewhere!</p>
<p><em>This post was originally posted in Recurse Center’s internal forum, but made it to my blog at the suggestion of <a href="https://blog.wesleyac.com/">Wesley Aptekar-Cassels</a>.</em></p>
Sun, 06 Sep 2020 20:36:00 -0400
http://localhost:4000/2020/09/06/Math-Anxiety
http://localhost:4000/2020/09/06/Math-AnxietyMetaMathematicsHow to Make a YouTube Video Out of Your Beamer Seminar<p><span class="newthought">I posted my one-hour seminar</span> on YouTube. Here’s why I thought it was important for me to do so, and how you can make your own using iMovie.</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/pvBGdIH3zt8" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen=""></iframe>
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<ul id="markdown-toc">
<li><a href="#why-make-talks-available-online" id="markdown-toc-why-make-talks-available-online">Why make talks available online</a> <ul>
<li><a href="#why-slides" id="markdown-toc-why-slides">Why slides</a></li>
<li><a href="#how-time-consuming-is-this" id="markdown-toc-how-time-consuming-is-this">How time consuming is this?</a></li>
</ul>
</li>
<li><a href="#lets-make-a-video-in-two-minutes" id="markdown-toc-lets-make-a-video-in-two-minutes">Let’s make a video in two minutes!</a> <ul>
<li><a href="#step-1--take-screenshots" id="markdown-toc-step-1--take-screenshots">Step 1 : Take screenshots</a></li>
<li><a href="#step-2-getting-slide-into-imovie" id="markdown-toc-step-2-getting-slide-into-imovie">Step 2: Getting slide into iMovie</a></li>
<li><a href="#step-3-getting-the-audio" id="markdown-toc-step-3-getting-the-audio">Step 3: Getting the audio</a></li>
<li><a href="#step-4-sync-the-audio-and-the-slide" id="markdown-toc-step-4-sync-the-audio-and-the-slide">Step 4: Sync the audio and the slide</a></li>
<li><a href="#step-5-export" id="markdown-toc-step-5-export">Step 5: Export</a></li>
</ul>
</li>
</ul>
<h2 id="why-make-talks-available-online">Why make talks available online</h2>
<p>I think the people in the greatest need of intuition and a gentle introduction to a paper are students (undergrad, masters, PhDs). These are the same people who don’t have the power to fly someone over to give a seminar, but are tech saavy. Plus, not every person who would like to see an intuitive explanation of a paper can afford or want to fly someone over anyway. Seminars can save someone hours of work trying to decipher a paper.</p>
<p>I find the practice of flying just to give a talk carbon-wasteful and antiquated (but please invite me anyway - really, I can’t afford to not go at this point in my career and I’d love to visit your group too and talk math)!</p>
<p>Most people quit academia at some point. I find it very sad that most of this time-specific intuition gets lost (most of what we leave behind is formal writing which is less conducive to communicating intuition). And even if I stay in academia, what are the chances I’ll remember the details of my first talk as well as right now, the eve of me giving my seminar? I’d like to share this version of my intuition about my own paper, rather than the blurry version I’ll inevitably have a couple years down the road. And to be completely honest, preparing the slides of this talk was quite a struggle already as I’m giving this talk almost a year after the original paper submission on the arXiv.</p>
<h3 id="why-slides">Why slides</h3>
<p>Slides are digital so I can make them highly legible (unlike a lot of blackboard video talks I’ve seen, due to bad angles and lighting), and the audio can be a lot better since I’ll be talking directly into the microphone in an empty room.</p>
<h3 id="how-time-consuming-is-this">How time consuming is this?</h3>
<p>I’ve personally found this process totally ok time-wise. I record my talks in sections for practice before giving them anyway, as I’ve found re-listening to the audio enormously helpful to improve clarity, structure, and use of filler words. For this particular seminar, I practiced my talk for a solid day before giving it, once I had a first draft of my slides. I adjust the slides and what I’m going to say on the go during practice. The result is the YouTube video above.</p>
<p>Also, <em>it’s really super easy to make this video, I promise!</em></p>
<h2 id="lets-make-a-video-in-two-minutes">Let’s make a video in two minutes!</h2>
<p>I’m using iMovie 10.1.14 for these instructions.</p>
<h3 id="step-1--take-screenshots">Step 1 : Take screenshots</h3>
<p>Open your beamer slides and put them in fullscreen. On Skim, I use the <strong>Option+Command+P</strong> shortcut.</p>
<p>Take a screenshot of this fullscreen. I like to use the <strong>Command+Shift+3</strong> shortcut.</p>
<figure><figcaption>For example, this is a slide I would like to add to my talk.</figcaption><img src="/assets/img/2020/title-screenshot.png" /></figure>
<h3 id="step-2-getting-slide-into-imovie">Step 2: Getting slide into iMovie</h3>
<p>Open iMovie and click the <strong>+ Create New</strong> button and select <strong>Movie</strong>.</p>
<figure><figcaption></figcaption><img src="/assets/img/2020/create-new-movie.png" /></figure>
<p>In a new project, simply drag the screenshot you took into the timeline.</p>
<figure><figcaption></figcaption><img src="/assets/img/2020/drag-screenshot.png" /></figure>
<p>Now it might be that the background is in “Ken Burns” (the weird zoom mode by default). Change this by clicking the <strong>crop icon</strong> and then choosing <strong>fit</strong> as your option. In general, you can fix this default mode by going to the <strong>Preferences…</strong> menu.</p>
<figure><figcaption></figcaption><img src="/assets/img/2020/ken-burns-fit.png" /></figure>
<p>You will be able to extend the length of time this slides appears in. We will adjust this later. But first, let’s put an audio.</p>
<h3 id="step-3-getting-the-audio">Step 3: Getting the audio</h3>
<p>To get audio, click the <strong>mic button</strong>, and make sure the audio feed is connected to your <strong>built-in microphone</strong>. Record what you’d like to say about this slide.</p>
<figure><figcaption></figcaption><img src="/assets/img/2020/built-in-mic.png" /></figure>
<h3 id="step-4-sync-the-audio-and-the-slide">Step 4: Sync the audio and the slide</h3>
<p>Simply <strong>drag the audio file (in green)</strong> to time the audio input with the visual input.</p>
<figure><figcaption></figcaption><img src="/assets/img/2020/drag-audio.png" /></figure>
<p>Next, drag the <strong>visual file</strong> so that your slide ends at the same time as your audio.</p>
<figure><figcaption></figcaption><img src="/assets/img/2020/drag-audio2.png" /></figure>
<p>That’s pretty much it! Repeat step 2-4 until you have your talk in video form.</p>
<h3 id="step-5-export">Step 5: Export</h3>
<p>Choose <strong>share > file</strong>. This may take a while.</p>
<figure><figcaption></figcaption><img src="/assets/img/2020/export-vid.png" /></figure>
<p>They’ll ask for some settings for the quality of your exported video. I just chose these ones.</p>
<figure><figcaption></figcaption><img src="/assets/img/2020/export-settings.png" /></figure>
<p>Once you have the file, upload it on <a href="https://www.youtube.com/">YouTube</a>! If you have issues uploading the video because it is longer than 15 minutes, you can follow <a href="https://support.google.com/youtube/answer/71673">this tutorial</a> to verify your YouTube account.</p>
<p>By the way, here is the finished product from our little project!</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/O0zmf1niqUM" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen=""></iframe>
<p>Happy sharing!</p>
Mon, 02 Mar 2020 19:36:00 -0500
http://localhost:4000/2020/03/02/YouTube-Video-Slide-Talk
http://localhost:4000/2020/03/02/YouTube-Video-Slide-TalkMetaUpdateConference Notes Posted <p><span class="newthought">It’s been a while.</span> Lots has happened since I started graduate school<label for="" class="margin-toggle sidenote-number"></label><input type="checkbox" class="margin-toggle" /><span class="sidenote">which I may or may not elaborate on at some point </span>. Anyhow, lately I’ve been thinking about how to be useful to others via this blog again, while not overloading myself with the pressure to produce new content specifically for this blog. As a step into this direction, I’ve created a new section for this website where I post my notes from various courses/conference I’ve attended, and included some links to extra content. I hope this will be useful to some!</p>
Sun, 08 Dec 2019 14:36:00 -0500
http://localhost:4000/2019/12/08/Conference-Notes-Posted
http://localhost:4000/2019/12/08/Conference-Notes-PostedUpdateIntro to Mapping Class Groups, Teichmuller Spaces and Hyperbolic Surfaces. <p>I asked <a href="http://www.math.toronto.edu/~rafi/">Kasra Rafi</a> about the link between hyperbolic surfaces, Teichmüller spaces and Mapping Class Groups with the goal of getting an intuitive overview of these concepts. During the span of several office hours, he generously replied with a beautiful torus story linking the three, which I adapt and fill in below.</p>
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<ul id="markdown-toc">
<li><a href="#quick-recap-of-definitions" id="markdown-toc-quick-recap-of-definitions">Quick recap of definitions</a></li>
<li><a href="#a-humble-beginning-with-lattices" id="markdown-toc-a-humble-beginning-with-lattices">A humble beginning with lattices</a></li>
<li><a href="#torus-time" id="markdown-toc-torus-time">Torus time!</a></li>
<li><a href="#scale-equivalence-and-textsl_2mathbbr-action" id="markdown-toc-scale-equivalence-and-textsl_2mathbbr-action">Scale equivalence and $\text{SL}_2(\mathbb{R})$ action</a></li>
<li><a href="#rotation-and-identification-of-torus-with-point-in-upper-half-plane" id="markdown-toc-rotation-and-identification-of-torus-with-point-in-upper-half-plane">Rotation and identification of torus with point in upper half-plane</a></li>
<li><a href="#the-teichmüller-space-of-the-torus" id="markdown-toc-the-teichmüller-space-of-the-torus">The Teichmüller space of the torus</a></li>
<li><a href="#markings-on-a-torus" id="markdown-toc-markings-on-a-torus">Markings on a torus</a> <ul>
<li><a href="#topological-definition" id="markdown-toc-topological-definition">Topological definition</a></li>
<li><a href="#group-theoretic-connection" id="markdown-toc-group-theoretic-connection">Group theoretic connection</a></li>
<li><a href="#interesting-combinational-fact" id="markdown-toc-interesting-combinational-fact">Interesting combinational fact</a></li>
</ul>
</li>
<li><a href="#mapping-class-group-of-the-torus" id="markdown-toc-mapping-class-group-of-the-torus">Mapping Class Group of the Torus</a> <ul>
<li><a href="#group-theoretic" id="markdown-toc-group-theoretic">Group Theoretic</a></li>
<li><a href="#topological" id="markdown-toc-topological">Topological</a></li>
<li><a href="#combinatorial" id="markdown-toc-combinatorial">Combinatorial</a></li>
<li><a href="#note-on-moduli-spaces" id="markdown-toc-note-on-moduli-spaces">Note on Moduli Spaces</a></li>
</ul>
</li>
<li><a href="#putting-it-all-together" id="markdown-toc-putting-it-all-together">Putting it all together</a></li>
<li><a href="#sources-and-resources" id="markdown-toc-sources-and-resources">Sources and Resources</a></li>
<li><a href="#acknowledgements" id="markdown-toc-acknowledgements">Acknowledgements</a></li>
</ul>
<h2 id="quick-recap-of-definitions">Quick recap of definitions</h2>
<blockquote>
<p><strong>Hyperbolic surface:</strong> A topological surface endowed with a hyperbolic metric such that the space has constant negative curvature. Recall from differential geometry that for Riemann surface with metric $ds$ where $ds^2 = g_1 dx_1^2 + g_2 dx_2^2$, the curvature is given by $K = -\frac{1}{\sqrt{g_1 g_2}}\left(\frac{\partial}{\partial x_1}\left(\frac{1}{\sqrt{g_1}} \frac{\partial \sqrt{g_2}}{\partial x_1}\right) + \frac{\partial}{\partial x_2}\left(\frac{1}{\sqrt{g_2}} \frac{\partial \sqrt{g_1}}{\partial x_2}\right) \right).$<label for="source hyp" class="margin-toggle sidenote-number"></label><input type="checkbox" id="source hyp" class="margin-toggle" /><span class="sidenote">Adapted from <em>Hyperbolic Geometry</em> by Caroline Series and Wikipedia. </span></p>
</blockquote>
<blockquote>
<p><strong>Teichmüller space:</strong> Let S be a compact surface. The Teichmüller space $\tau(S)$ is the set of isotopy classes of complex structures on $S$. By a complex structure on $S$, we mean a fixed homeomorphism $f: S \to X$, where $X$ is a metrized surface with distinguished curves. Equivalently, $\tau(S)$ is a space that parametrizes complex structures on $S$ up to the action of homeomorphisms that are isotopic to the identity. We may record the complex structure $f: S \to X$ by the triple $(S,X,f)$. The homeomorphism is referred to as a <em>marking</em> which has named curves, and either $X$ or $(X,f)$ can be referred to as a <em>marked surface</em>.<label for="source tech" class="margin-toggle sidenote-number"></label><input type="checkbox" id="source tech" class="margin-toggle" /><span class="sidenote">Adapted from <em>A Primer on Mapping Class Groups</em> by Farb and Margalit and Wikipedia. </span></p>
</blockquote>
<blockquote>
<p><strong>Mapping class group:</strong> The group of orientation-preserving homeomorphisms of a surface up to isotopy, where the group structure is inherited from functional composition of homeomorphisms. In other words, $\text{MCG}(S) = \text{Homeo}^+(S)/\text{Homeo}_0(S)$.<label for="iso" class="margin-toggle sidenote-number"></label><input type="checkbox" id="iso" class="margin-toggle" /><span class="sidenote">where $\text{Homeo}_0(S)$ are homeomorphisms isotopic to the identity. </span> The group multiplication of the mapping class group of $S$ gives it a natural action on the Teichmüller space of $S$ by composition of homeomorphisms.<label for="source tech" class="margin-toggle sidenote-number"></label><input type="checkbox" id="source tech" class="margin-toggle" /><span class="sidenote">Adapted from <em>A Primer on Mapping Class Groups</em> by Farb and Margalit and Wikipedia. </span></p>
</blockquote>
<p>At first glance, these seem tricky and confusing – let’s see what they mean in the context of <em>one</em> example.</p>
<h2 id="a-humble-beginning-with-lattices">A humble beginning with lattices</h2>
<p>Our story starts with a simple problem. We can define a <em>lattice</em> in $\mathbb{R}^2$ to be an injective homomorphism from $\mathbb{Z}^2 \to \mathbb{R}^2$.</p>
<figure><figcaption>We will be calling the two vectors defining a lattice the *basis* of the lattice. In the picture above, the left (green) represents the linear map $\{(1,0), (0,1)\} \mapsto \{(1,0), (0,1)\}$ and the right represents the linear map $ \{(1,0), (0,1)\} \mapsto \{(1,0), (1,1)\}.$</figcaption><img src="/assets/img/2017/hyperbolic/lattice-map.jpeg" /></figure>
<p><strong>Key idea</strong>: <em>Up to what should these maps be equivalent? What is a good notion of equivalence to use?</em></p>
<p>One way to interpret this question is to define up to equivalence maps which have the same images. For example, the linear map ${(1,0), (0,1)} \mapsto {(1,0), (0,1)}$ and the linear map ${(1,0), (0,1)} \to {(1,0), (1,1)}$ have the same image, which is $\mathbb{Z}^2 \subset \mathbb{R}^2.$</p>
<figure><figcaption>Picture illustrating how the two maps have the same image as they span the same lattice points.</figcaption><img src="/assets/img/2017/hyperbolic/lattice-map2.jpg" /></figure>
<p>On the other hand, the linear map ${(1,0),(0,1)} \mapsto {(2,0), (1,2)}$ (in blue) does not have the same image as the other two (in brown). For example, the point $(1,1)$ (in pink) is not in the image of the linear map.</p>
<figure><figcaption>In blue: the parallelogram formed by vectors $\{(2,0), (1,2)\}.$ In brown: the tesselation formed by the previous lattice. In pink: points not covered by the blue tesselation that are covered in the brown tesselation.</figcaption><img src="/assets/img/2017/hyperbolic/lattice-othermap.jpg" /></figure>
<p>Already, we have a new notion of equivalence. Since we can choose to care about only the span of the image of the chosen basis elements for $\mathbb{Z}^2$, two maps could be equivalent under change of basis which preserves the span of the image. Thus, an equivalence of maps from $\mathbb{Z}^2 \to \mathbb{Z}^2 \subset \mathbb{R}^2$ must necessarily be area preserving (so that both base and height are $1$ in the domain as fractions are not allowed). IF we choose to preserve orientation,<label for="orientation" class="margin-toggle sidenote-number"></label><input type="checkbox" id="orientation" class="margin-toggle" /><span class="sidenote">which we will (spoiler alert!) because we will want to associate each lattice with a marked torus, and a torus is an oriented surface. </span> the group acting on these maps must be $\text{SL}_2(\mathbb{Z})$ the special linear group of two-dimensional integer matrices with determinant $1$.</p>
<figure><figcaption>Picture illustrating how $\text{SL}_2(\mathbb{Z})$ works on bases of $\mathbb{Z}^2$ the domain of the lattice maps. The map pictured here is the natural embedding (in blue) of $\mathbb{Z}^2$ into $\mathbb{R}^2,$ but it could be anything that is an injective linear map.</figcaption><img src="/assets/img/2017/hyperbolic/change-of-basis.jpeg" /></figure>
<p>$\text{SL}_2(\mathbb{Z})$ is also known as the <em>mapping class group of the torus</em>. What torus? How do we get there?</p>
<h2 id="torus-time">Torus time!</h2>
<p>Since each parallelogram is exactly the same in a given lattice, it doesn’t matter which one we pick as a representative!</p>
<figure><figcaption>Map each lattice to a single parallelogram representative. </figcaption><img src="/assets/img/2017/hyperbolic/rep-lattice.jpeg" /></figure>
<p>Under the assumption that all parallelograms in a given lattice is the same, we may identify the sides of the parallelograms accordingly. The resulting surface is a torus identified with a particular lattice $\mathbb{Z}^2 \to \mathbb{R}^2$.</p>
<p><strong>Key idea:</strong> <em>How do we transfer the equivalence relation between parallelograms onto its torus representative?</em></p>
<p>Unfortunately, genus 1 tori are all the same as topological spaces, which we write as $\mathbb{R}^2/\mathbb{Z}^2$. Since there is only one equivalence class for genus 1 tori under homeomorphisms, we will need finer notion of equivalence to get “interesting” equivalence classes”.</p>
<h2 id="scale-equivalence-and-textsl_2mathbbr-action">Scale equivalence and $\text{SL}_2(\mathbb{R})$ action</h2>
<p>Notice we have only talked about equivalence in the domain space $\mathbb{Z}^2$. In the image space, the completeness of $\mathbb{R}^2$ allows for richer equivalences.</p>
<p>Indeed, maybe instead of the image of maps being the distinguishing factor, we only care about the <em>shape</em> of the parallelograms in the lattice. That translates to equivalences up to <em>scaling and rotation</em>. Notice that both are features that we think of as invariant in a torus. These equivalences will lead us to the Teichmüller space of the torus.</p>
<p>Let’s start with scaling and see what natural group action arises from it. If we decide to mod the lattices by a scaling factor, we may consider only the set of lattices whose parallelograms have area $1$.<label for="stretch" class="margin-toggle sidenote-number"></label><input type="checkbox" id="stretch" class="margin-toggle" /><span class="sidenote">Imagine stretching your parallelogram freely until you get area $1$. </span></p>
<figure><figcaption>Scaling our parallelogram to have area $1$. </figcaption><img src="/assets/img/2017/hyperbolic/para-scale.jpg" /></figure>
<p>Then, the group acting on parallelograms with area $1$ is $\text{SL}_2(\mathbb{R})$, the special linear group of two-dimensional real matrices with determinant $1$.</p>
<figure><figcaption>Diagram showing where $\text{SL}_2(\mathbb{R})$ acts. </figcaption><img src="/assets/img/2017/hyperbolic/SL2R.jpg" /></figure>
<h2 id="rotation-and-identification-of-torus-with-point-in-upper-half-plane">Rotation and identification of torus with point in upper half-plane</h2>
<p>For a given set of two vectors in $\mathbb{R}^2$, we may rotate a parallelograms such that the second basis vector is always in the upper half-plane. This does not change the torus we are looking at and neither does the scaling.</p>
<figure><figcaption>We may represent our two-dimensional space as the complex plane. Then, then since the vector $v$ can always be rotated to be on the upper half-plane, we may uniquely identify our torus with the rotated coordinates of our vector $v = \tau = a + bi,$ assuming the torus has area $1.$</figcaption><img src="/assets/img/2017/hyperbolic/para-rotate.jpg" /></figure>
<p>The group action of $\text{SL}_2(\mathbb{R})$ on the image of the lattice becomes $\text{SL}_2(\mathbb{R})/O(2)$ where $O(2)$ is the group of two-dimensional rotation.</p>
<p>Due to our equivalence of parallelograms being up to rotation and scale, we can fix our first basis vector to be of length $1$ lying on the real-axis. We may then uniquely identify each representative of our equivalence class by the coordinates of the second basis vector, which we will refer to as the parameter $\tau = a + bi$ in the complex upper half-plane.</p>
<blockquote>
<p><strong>Possible point of confusion:</strong> <em>Are irrational coordinates allowed?</em></p>
</blockquote>
<blockquote>
<p>Irrational coordinates are allowed by definition. The domain of the $\text{SL}_2(\mathbb{Z})$ action should not be confused with that of $\text{SL}_2(\mathbb{R})$.</p>
<figure><figcaption></figcaption><img src="/assets/img/2017/hyperbolic/irrational-coords.jpeg" /></figure>
</blockquote>
<h2 id="the-teichmüller-space-of-the-torus">The Teichmüller space of the torus</h2>
<p><strong>The space of possible coordinates of $\tau$ which uniquely identifies a torus under our new equivalence is exactly the Teichmüller space of the torus. Thus $\tau(T^2) = \mathbb{H}$, the complex upper half-plane.</strong></p>
<blockquote>
<p><strong>Note:</strong> From the identification of the complex upper half-plane to this (Teichmüller) space of tori, we inherit the hyperbolic metric $ds = \lvert{dz}\rvert/\Im z$, which gives a useful notion of distance between tori in the Teichmüller space (for reasons not explored here).</p>
</blockquote>
<p>But how do we make sense of that? In the definition of the Teichmüller space, there was mention of distinguished curves and markings, and of mapping class group action…</p>
<p><em>This concludes the first part of this post (yay! We did it!). In the second part, we will explore markings of the torus, the mapping class group and tie all the previous concepts we explored together. If you were new to the material, this is a good break point.</em></p>
<h2 id="markings-on-a-torus">Markings on a torus</h2>
<p>Let’s start with an example.</p>
<p>As seen previously, the maps which embed the bases ${(1,0), (0,1)}$ (in green) and ${(1,0), (1,1)}$ (in pink) have the same image, so one could consider them the same as maps. However, if we ‘mark’ the resulting torus by the curves they embed, they become different as marked tori.</p>
<figure><figcaption>Shortest curves in blue on the green torus, which both correspond with the generators of the fundamental group (the edges of the square).</figcaption><img src="/assets/img/2017/hyperbolic/marking1.jpg" /></figure>
<figure><figcaption>Shortest curves in blue on the pink torus, where only one correspond with the generators of the fundamental group (the edges of the pink parallelogram).</figcaption><img src="/assets/img/2017/hyperbolic/shortest-curves2.jpeg" /></figure>
<p>Indeed, the green basis maps to the shortest curves on the torus, whereas the pink basis maps the second vector to a curve that is not the shortest.</p>
<blockquote>
<p>A torus is fundamentally a product of two circles $S^1 \times S^1$. If you have a parallelogram with identified edges, you have infinitely many circles to choose from. The standard two circles (in blue) are supposed to represent the shortest two loops on any torus. As the blue curves illustrate, the shortest curves are not necessarily always the curves you pick to generate your torus (ex: pink torus).</p>
</blockquote>
<p>Here are different interpretations of markings.</p>
<h3 id="topological-definition">Topological definition</h3>
<p>Let $\gamma_1, \gamma_2$ be two circles, and let $S$ be the genus $1$ surfaces generated by $S = S^1 \times S^1 = \gamma_1 \times \gamma_2$.</p>
<blockquote>
<p>In our examples, such an $S$ would be a fixed parallelogram $P$ in $\mathbb{Z}^2$ with the sides identified - let’s take $P$ to be a square with sides $1$ for simplicity. The horizontal sides of $P$ would correspond to $\gamma_1$ and the vertical sides to $\gamma_2$.</p>
</blockquote>
<p>A <em>marking</em> of the torus is a orientation-preserving homeomorphism $f$ from $S$ to a <a href="https://en.wikipedia.org/wiki/Riemann_surface">Riemann surface</a> $X$ where $f$ is <em>fixed</em> by once-intersecting simple closed curve $(\lambda_1, \lambda_2)$ on $X = \lambda_1 \times \lambda_2$ such that $f(\gamma_1) = \lambda_1, f(\gamma_2) = \lambda_2$.</p>
<figure><figcaption></figcaption><img src="/assets/img/2017/hyperbolic/markings.jpeg" /></figure>
<blockquote>
<p>In the pink torus example, $X$ is the pink torus and $(\lambda_1, \lambda_2)$ are the pink curves, and they are the (non-standard) generators the fundamental group $\pi_1(X)$. The blue curves represent the <em>standard</em> generators of $\pi_1(X)$. The map sending the parallelogram $P$ to the pink torus is the homeomorphism map $f$ from $S$ to $X$. The green torus example is similar, except that the green curves coincide with the blue curves because they are the standard generators of the fundamental group.</p>
</blockquote>
<p>We say that $X$ and $Y$ are <em>equivalent as markings</em> if there is an isometry $\varphi$ between the homeomorphisms $f : S \to X$, $g : S \to Y$ such that $\varphi$ is isotopic (or homotopic) to the identity.</p>
<blockquote>
<p>An example of such a homeomorphism isotopic to the identity is “introducing a wiggle” to a curve. Indeed, two curves as isotopic if they bound a disk. The proof relies on the “Alexander trick” on a disk, which can be generalized to prove isotopies of hyperbolic surfaces. (More in <em>A Primer of Mapping Class Groups</em> section 2.2.)</p>
</blockquote>
<figure><figcaption></figcaption><img src="/assets/img/2017/hyperbolic/iso.jpeg" /></figure>
<blockquote>
<p>A non-example of a homeomorphism homotopic to the identity is a <a href="https://en.wikipedia.org/wiki/Dehn_twist">Dehn twist</a>, which consists of cutting, rotating and gluing back.</p>
</blockquote>
<figure><figcaption></figcaption><img src="/assets/img/2017/hyperbolic/non-iso.jpeg" /></figure>
<blockquote>
<p>The markings of green and pink tori differ because the pink marking has a Dehn twist. (Scroll up!)</p>
</blockquote>
<p>Schwartz’s <a href="https://www.math.brown.edu/~res/Papers/surfacebook.pdf">Mostly Surfaces</a> Chapter 20 offers good coverage of this point of view.</p>
<h3 id="group-theoretic-connection">Group theoretic connection</h3>
<p>We may consider different <em>choices</em> of <em>curves</em> representing the generators of the fundamental group as different markings of the torus.<label for="note" class="margin-toggle sidenote-number"></label><input type="checkbox" id="note" class="margin-toggle" /><span class="sidenote"><strong>Note:</strong> emphasis on choice of curves, not the generators themselves. </span></p>
<blockquote>
<p>Let $(X_1, \lambda_1, \lambda_2)$ be the green torus with green distinguished curves and $(X_2, \Lambda_1, \Lambda_2)$ the pink one with pink curves. Then $\pi_1(X_1) = \pi_1(X_2) = \langle a, b \mid ab = ba \rangle$. For the green torus, we have picked $\lambda_1 = a, \lambda_2 = b$ (the distinguished curves are the standard elements of the fundamental group as we expect). For the pink torus, we pick the pink curves to represent$\Lambda_1 = ba = a + b$, $\Lambda_2 = b$.</p>
</blockquote>
<p>If we go back to thinking of the marked tori as parallelograms in $\mathbb{R}^2$ and then are markings as different choices of bases of $\mathbb{R}^2$ which we identify with arrows.</p>
<h3 id="interesting-combinational-fact">Interesting combinational fact</h3>
<p>Let $X$ be the torus identified with the distinguished curves $\alpha, \beta$ of rational slope $p/q, r/s$. In general, slopes need not be rational as the Teichmüller space admits irrational coordinates. But let’s stick to rational slopes for the sake of the interesting result that will follow.</p>
<p>We may re-shape the torus such that the distinguished curves lie at the center of the square-looking torus.</p>
<figure><figcaption>In this example, $\text{slope}(\alpha) = 2/1, \text{slope}(\beta) = 2/1$.</figcaption><img src="/assets/img/2017/hyperbolic/combinatorial-1.jpeg" /></figure>
<p>Then, we may lift torus to its universal cover $\mathbb{R}^2$ along with the distinguished curves, such that a unit square is a copy of the square torus, and the distinguished curves become segments from $(0,0)$ to $(q,p)$ and $(r,s)$ respectively.</p>
<figure><figcaption>In this example, $(q,p) = (1,2), (r,s) = (2, 1).$</figcaption><img src="/assets/img/2017/hyperbolic/combinatorial-2.jpeg" /></figure>
<p>We are interested in knowing how many lattice points there are on the on the curves $\alpha, \beta$, as these correspond to the number of intersection points $i(\alpha, \beta)$. The interior lattice points biject to the intersection points via redrawing the distinguished slopes (see below)</p>
<blockquote>
<p>In our example, the number of intersection point is clearly $3$ by inspection.</p>
</blockquote>
<figure><figcaption>Added green dots illustrate the bijection. Note: The bijection took me a while to get. </figcaption><img src="/assets/img/2017/hyperbolic/combinatorial-3.jpeg" /></figure>
<p>Using <a href="https://en.wikipedia.org/wiki/Pick%27s_theorem">Pick’s theorem</a>, the area of a parellogram is given by $A = j + \frac{b}{2} - 1$ where $j$ is the number of interior points and $b$ is the number of points on the boundary. Some interior points, will biject (from our previous correspondance) to points on the boundary. Since we don’t want to double count for degenerate cases, we say that there are $4$ points on the boundary which are the $4$ corners of the parallelogram – the rest will be counted by $j$.</p>
<figure><figcaption>Example of double counting of points on the boundary. The lift of $\beta$ has a lattice point on its at (1,0), which is already counted by the green interior point.</figcaption><img src="/assets/img/2017/hyperbolic/combinatorial-4.jpeg" /></figure>
<p>Thus, the number of intersection points in the interior of the lattice is given by
\(\begin{align*}
& j = A - \frac{b}{2} + 1 \\
& j = \det(P) - \frac{4}{2} + 1 \\
& j = \left|rq - ps\right| - 1
\end{align*}\)
Now, $j + 1 = i(\alpha, \beta)$, since the number of intersections is one plus the number of interior points by the argument above. Thus
\(i(\alpha, \beta) = \left| rq-ps \right|\)</p>
<p>This number is also known as the <em>algebraic intersection</em> of the torus.<label for="algebraic" class="margin-toggle sidenote-number"></label><input type="checkbox" id="algebraic" class="margin-toggle" /><span class="sidenote">algebraic as opposed to geometric. Algebraic intersection numbers are oriented whereas geometric intersection numbers are not. </span></p>
<p><strong>Punchline</strong> A torus has intersection number $1$ if and only if it’s determinant is $1$!</p>
<h2 id="mapping-class-group-of-the-torus">Mapping Class Group of the Torus</h2>
<p>The Mapping class group of the torus is $\text{SL}_2(\mathbb{Z})$ . It is well-known that $\text{SL}_2(\mathbb{Z})$ are generated by the elements<label for="well-known" class="margin-toggle sidenote-number"></label><input type="checkbox" id="well-known" class="margin-toggle" /><span class="sidenote">Not that <em>I</em> knew lol. Prof. Rafi subsequently remarked that the operations which do not change the determinant are adding one row with one another, and switching two rows, which is exactly what these two matrices represent. <em>Now we know :)!</em> </span>
\(A =
\begin{pmatrix}
1 & 1 \\ 0 & 1
\end{pmatrix},
\quad
B =
\begin{pmatrix}
0 & 1 \\ -1 & 0
\end{pmatrix}.\)</p>
<p>This is the change of basis group for $\mathbb{Z}^2$ that we have hinted at at the beginning. It has a natural action on the Teichmüller space, as we will see in the next section.</p>
<p>For now, we are going to offer a few interpretations of the action of the mapping class group as relating to the torus itself.</p>
<blockquote>
<p><strong>Note: Computing the mapping class group.</strong> Roughly speaking, the mapping class group can be computed using a variant of the Alexander method on the annulus. We are not going to do this here. Farb and Margalit do this in <em>Primer to Mapping Class Groups</em> section 2.2. There will be more references on this at the end.</p>
</blockquote>
<h3 id="group-theoretic">Group Theoretic</h3>
<p>Each element of the mapping class group takes the standard basis to a to unique basis of $\mathbb{Z}^2$. Equivalently, the mapping class group acts faithfully on the isomorphism maps between $\pi_1(S) \to \pi_1(X)$.</p>
<h3 id="topological">Topological</h3>
<p>The action of the map is given by natural the composition of the homeomorphism $h$ on the homeomorphisms given by the markings.</p>
<p>$A$ corresponds to making a Dehn twist on the second generator,</p>
<figure><figcaption></figcaption><img src="/assets/img/2017/hyperbolic/topogen1.jpeg" /></figure>
<p>whereas the $B$ interchanges the markings and reverses the direction of one of them.</p>
<figure><figcaption></figcaption><img src="/assets/img/2017/hyperbolic/topogen2.jpeg" /></figure>
<p>At first, these maps can seem odd and difficult to picture as homeomorphisms compared to our vanilla interpretations of homeomorphism (at least in the homotop sense which we are used to when we picture a cup changing into a donut), and made me doubt they were actual homeomorphisms<label for="homeomorphisms" class="margin-toggle sidenote-number"></label><input type="checkbox" id="homeomorphisms" class="margin-toggle" /><span class="sidenote">so continuous and bijective </span> but then we realize that’s because the mapping class group is <em>exactly</em> the homeomorphisms which are <em>not</em> isotopic to the identity!</p>
<p>Then, I remembered the <a href="/2016/07/03/Torus-vs-Its-Embedding-in-Three-Space">torus vs its embedding in three-space</a>. Viewing the tori as what they actually are, which is the cross-section of two circles, the continuity and bijectivity of the maps are clearer.</p>
<figure><figcaption>This map is continuous since every small variation in its image is induced by a small variation in its domain. It is bijective since it has an inverse, which is marking an arc in the opposite direction on the second coordinate. Note that this picture is misleading since the domain is over integers, not continuous circles. Thus, there are no arcs that don't form a loop such as the purple one (that is drawn for intuition).</figcaption><img src="/assets/img/2017/hyperbolic/topogen12.jpeg" /></figure>
<figure><figcaption></figcaption><img src="/assets/img/2017/hyperbolic/topogen22.jpeg" /></figure>
<h3 id="combinatorial">Combinatorial</h3>
<p>(<em>Nice connection with Farey graph</em>)</p>
<p>Perhaps the most beautiful connection,</p>
<p>$ \mid rq - ps \mid = 1 \iff \frac{p}{q}, \frac{r}{s}$ form an edge on the Farey graph. $\text{SL}_2(\mathbb{Z})$ acts on the graph in the following way.
\(\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} p & q \\ r & s \end{pmatrix} = \begin{pmatrix} p + r & q + s \\ r & s \end{pmatrix} \\
\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} p & q \\ r & s \end{pmatrix}= \begin{pmatrix} r & s \\ -p & -q \end{pmatrix}\)</p>
<p>which we interpret as
\(\left(\frac{p}{q}, \frac{r}{s}\right) \to \left(\frac{p+r}{q+s}, \frac{r}{s}\right) \text{ and } \left(\frac{p}{q}, \frac{r}{s}\right) \to \left(\frac{r}{s}, \frac{-p}{-q}\right)\) respectively.</p>
<p>The first generators sends the edge $(1/0, 0/1) \mapsto (1/1, 0/1) \mapsto (1/2, 0/1) \mapsto (1/3, 0/1)$ and so on. The second generator inverts the direction of the edge.</p>
<figure><figcaption>Picture illustrating the action of the MCG on the Farey graph. The orientation of the arrow indicates which vertex stays fixed at each step.</figcaption><img src="/assets/img/2017/hyperbolic/farey-gen1.png" /></figure>
<p>If we applied the second generator we would start with $(0/1, -1/0) \mapsto (-1/1, -1/0) \mapsto (-2/1, -1/0) \mapsto (-3/1, -1/0)$ and so on.</p>
<figure><figcaption>Picture illustrating the action of the MCG on the Farey graph. The orientation of the arrow indicates which vertex stays fixed at each step.</figcaption><img src="/assets/img/2017/hyperbolic/farey-gen2.png" /></figure>
<p>This is beautifully illustrated on the Farey graph, where $1/0 = -1/0 = \infty$.</p>
<p>$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}^{-1}$ sends $(1/0, 0/1) \mapsto (-1/1, 0/1)$ $\mapsto (-1/2, 0/1) \mapsto (-1/3, 0/1).$</p>
<figure><figcaption>Picture illustrating the action of the MCG on the Farey graph. The orientation of the arrow indicates which vertex stays fixed at each step.</figcaption><img src="/assets/img/2017/hyperbolic/farey-gen3.png" /></figure>
<p>The action of $SL_2(\mathbb{Z})$ is transitive on the Farey graph since we can associate each edge of the Farey graph to a pair of columns of $SL_2(\mathbb{Z})$ which acts transitively on itself.</p>
<blockquote>
<p><strong>Possible point of confusion:</strong> <em>Are irrational slopes not markings since they are elements of the Teichmuller space? Shouldn’t the mapping class group act transitively on those too?</em></p>
<p>I was confused about this for a while. Yes, the irrational slopes are also markings since they have representative in $\mathbb{H}$, which is the space of markings. They do not correspond to an element of the mapping class group in the sense there is no bijection between an element of the MCG and an irrational basis in the edges of the Farey graph like there is for rational bases (by sending $p/q$ to the element mapping $0 \mapsto p/q$.)</p>
<p>Indeed, the mapping class group has a natural <em>action</em> on the Teichmüller space (say by adding a Dehn twist to a marking or inverting the markings), but its definition is first and foremost a topological definition relating to classes of orientation-preserving homeomorphisms. <em>It’s very easy to confuse the mapping class group with the group permuting the markings, but it is not.</em> The group acting transitively the markings for the torus is $\text{PSL}_2(\mathbb{R})$,while the mapping class group is $\text{SL}_2(\mathbb{Z})$, which does have a natural $\text{PSL}_2(\mathbb{Z})$ action on the Teichmüller space, as we will see in the last section.</p>
</blockquote>
<h3 id="note-on-moduli-spaces">Note on Moduli Spaces</h3>
<p>You may be wondering what’s left of $\mathbb{H}$** once we mod by the action of $\text{SL}_2(\mathbb{R})$. That is precisely what the moduli space of the torus is.<strong>The moduli space of the torus is given by the Teichmüller space modulo the action of the mapping class group</strong>.</p>
<blockquote>
<p>The tori with basis ${(1,0), (0,1}$ and ${(1,0),(1,1)}$ we saw at the beginning have different $\tau$ parameters ($i$ and $i+1$ respectively) and thus are distinct as points in the Teichmüller space.
However, they represent the same point in the <em>moduli space</em> of the torus because they differ by a Dehn twist.</p>
</blockquote>
<p>To come back to the previous possible point of confusion, if the MCG acted transitively on the Teichmüller, then the moduli space would be trivial. Instead, it is represented by the fundamental domain of $\text{SL}_2(\mathbb{Z})$ over the $\mathbb{H}$.</p>
<figure><figcaption>In grey: the fundamental domain representing the moduli space of the torus. In blue: planar version of the Farey graph bouding free regular sets. Read more in source: [wikipedia](https://en.wikipedia.org/wiki/Fundamental_domain).</figcaption><img src="/assets/img/2017/hyperbolic/modspacetor.png" /></figure>
<h2 id="putting-it-all-together">Putting it all together</h2>
<p>Here comes the last punchline. Recall from complex analysis that $\text{PSL}_2(\mathbb{R})$ acts isometrically transitively on the upper-half plane $\mathbb{H}$<label for="recall" class="margin-toggle sidenote-number"></label><input type="checkbox" id="recall" class="margin-toggle" /><span class="sidenote">If you can’t recall this, that’s totally fine - it’s a fact of complex analysis that should be available in any textbook :) </span>, which can be thought of as the Teichmüller space of the torus. An element of $\text{PSL}_2(\mathbb{R})$ is of the form
\(\begin{pmatrix} a & b \\ c & d \end{pmatrix}, a, b, c, d \in \mathbb{R} \qquad ac - bd = 1.\)</p>
<p>and acts by Mobius transformation on an element $\tau \in \mathbb{H}$
\(\begin{pmatrix} a & b \\ c & d \end{pmatrix} \tau = \frac{a\tau + b}{c\tau + d}.\)</p>
<p>The elements of $\text{SL}_2(\mathbb{Z})$ are of the same form as those of $\text{PSL}_2(\mathbb{Z})$, but we picked its action to be linear over the bases of $\mathbb{Z}^2$ (change of basis action).</p>
<p><strong>Thus, there is a map from the mapping class group $\text{SL}_2(\mathbb{Z})$ of the torus to the Mobius transformation $\text{PSL}_2(\mathbb{R})$ of the upper-half plane, and this group turns out to be the group of isometries in the hyperbolic plane $\mathbb{H}$.</strong></p>
<p>Recall the generators for $\text{SL}_2(\mathbb{Z})$ and $\text{PSL}_2(\mathbb{Z}) = \text{SL}_2(\mathbb{Z})/\langle \pm 1 \rangle$<label for="pslsl" class="margin-toggle sidenote-number"></label><input type="checkbox" id="pslsl" class="margin-toggle" /><span class="sidenote">Meaning $A, -A \in \text{SL}_2(\mathbb{Z})$ represent the same element in $\text{PSL}_2(\mathbb{Z})$. This is because the minus sign cancel on the top and bottom of the fraction (write it out if you do not see it)! </span>:
\(A =
\begin{pmatrix}
1 & 1 \\ 0 & 1
\end{pmatrix},
\quad
B =
\begin{pmatrix}
0 & 1 \\ -1 & 0
\end{pmatrix}.\)</p>
<p>$A$ sends by linear fractional transformation the point $\tau$ on the upper half-plane to $\frac{1 \cdot \tau + 1}{0 \cdot \tau + 1} = \tau + 1$. $B$ sends $\tau \mapsto \frac{0 \cdot \tau + 1}{-1\cdot \tau + 0} = - \frac{1}{\tau}$.</p>
<p>Take your standard basis torus generated by ${(1,0),(0,1)}.$ This is identified with the point $i \in \mathbb{H}$.</p>
<p>Under the $\text{PSL}_2(\mathbb{Z})$ action of $A$, $i$ gets sent to $i + 1$, which corresponds to the torus generated by ${(1,0),(1,1)}$. But this is also exactly where the linear $\text{SL}_2(\mathbb{Z})$ action of $A$ sends our standard basis!</p>
<figure><figcaption>Shortest curves in blue on the pink torus, where only one correspond with the generators of the fundamental group (the edges of the pink parallelogram)</figcaption><img src="/assets/img/2017/hyperbolic/compatible1.jpeg" /></figure>
<p>Under the $\text{PSL}_2(\mathbb{Z})$ action of $B$, $i$ gets sent to $-\frac{1}{i} = - (-i) = i$. This is also exactly where the linear $\text{SL}_2(\mathbb{Z})$ action of $A$ sends our standard basis once we’ve rotated the image back to the upper half-plane.</p>
<figure><figcaption>Shortest curves in blue on the pink torus, where only one correspond with the generators of the fundamental group (the edges of the pink parallelogram)</figcaption><img src="/assets/img/2017/hyperbolic/compatible2.jpeg" /></figure>
<p><strong>The action of the Mobius transformation on the complex plane is compatible with the action of $\text{SL}_2(\mathbb{Z})$ on the the lattice!</strong></p>
<p>To Prof. Rafi, there is more to this than pure coincidence. This is one of the reason why we study mapping class groups, Teichmüller spaces and hyperbolic surfaces as as a trifecta of some sort!</p>
<p><em>Ouf! We are done. Give yourself a big tap on the back :).</em></p>
<p>Hope you enjoyed this, and that everything makes much more sense when you reread the definitions!</p>
<h2 id="sources-and-resources">Sources and Resources</h2>
<blockquote>
<p><a href="http://www.math.toronto.edu/~rafi/">Kasra Rafi</a> as recalled by me. Errors are mine. Please let me know of them!</p>
</blockquote>
<blockquote>
<p><a href="https://www.math.brown.edu/~res/Papers/surfacebook.pdf">Notes by Evan Schwarz</a> Chapter 20.</p>
</blockquote>
<blockquote>
<p><a href="http://www.maths.ed.ac.uk/~aar/papers/farbmarg.pdf">A Primer on Mapping Class Groups</a> by Farb and Margalit, sections 10.2 and 10.3; 2.2 for the Alexander method and the computation of the mapping class group of the torus. Here are <a href="https://www.dropbox.com/s/8etcwhwblqi1fwn/Alexander%20trick.pdf?dl=0">illustrated textbook pages</a> on that section (illustrations mine).</p>
</blockquote>
<h2 id="acknowledgements">Acknowledgements</h2>
<blockquote>
<p>Thanks to <a href="http://www.math.toronto.edu/~rafi/">Kasra Rafi</a> for his time and patience teaching this and for agreeing to proofread this!</p>
</blockquote>
<blockquote>
<p>Thanks to Felix Bauckholt (currently an undergrad at Waterloo), for his great questions and feedback which allowed me to significantly clarify this blog post. Always a pleasure to discuss math with you!</p>
</blockquote>
<p>Last updated: 22-10-2021.*</p>
Sun, 08 Oct 2017 10:29:00 -0400
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