homeowmorphism

I graduated!

Fri, 24 Oct 2025 19:28:00 -0400

I graduated! It’s been a long time coming. Those who have been here before might have concluded I was due to graduate in 2021, or 2022 at the latest. As I began writing my thesis, my timeline was interrupted: first by bereavement, then by medical leave, and later by caregiving.

I am so, so relieved (and proud!) to announce I have finally finished, graduating with cum laude distinction (the highest honors awarded to a thesis defense in Spain). An arXiv link to my thesis and video of a practice recording of my defense are soon to follow.

Update: they are up!

Last updated: 2025-10-24.

Reading List for Aspiring Mathematicians

Sat, 08 May 2021 20:21:00 -0400

My first college major was neuroscience. 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.

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.

Readings that helped me really understand math

Note: 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.

Better Explained - 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.

Gilbert Strang’s Calculus Book - 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.

A Book of Abstract Algebra by Charles Pinter - 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.

Readings that nourished my soul

A Mathematician’s Lament by Paul Lockart - an essay dissecting of how mathematics is misrepresented in pre-college education.

Mathematics for Human Flourishing by Francis Su - an essay discussing who is mathematics for and what can we gain by learning mathematics.

On Proof and Progress in Mathematics by William Thurston - an essay by one of the greats of mathematics about what mathematical progress is.

Peak by Anders Ericsson - a book dispelling the myth of born geniuses in various fields, including mathematics.

New Talk About Research Motivations Available

Mon, 19 Apr 2021 20:21:00 -0400

A new talk 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.

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.

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.

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, which I am now able to accept as a good thing.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.

If you are interested in my line of research and would like a gentle starting point, I would recommend this talk to you.

PS: I think the audio should be better in this recording than in the previous ones also.

Second Paper on arXiv

Mon, 12 Apr 2021 16:21:00 -0400

My second paper has been uploaded on arXiv. It is titled “Regular left-orders on groups” and is joint with Yago Antolín and Cristóbal Rivas. 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 YouTube.

Looking For My Next Job

Sun, 28 Mar 2021 16:21:00 -0400

I am on the lookout for a job 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!

PS: The Lambda Calculus post’s align environment should be fixed.

MathJax Errors Fixed

Sun, 17 Jan 2021 19:21:00 -0500

The broken math should have been fixed. 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.

Two New Talks

Thu, 03 Dec 2020 19:21:00 -0500

Two new seminar recordings 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.

You can download the slides for the above talk here.

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’’.

You can download the slides for the above talk here.

The talks have a slight overlap at the beginning of about ten minutes.

Dealing with Math Anxiety

Sun, 06 Sep 2020 20:36:00 -0400

I still deal with math anxiety regularly as a math PhD student. Here’s my advice on overcoming it.

Step 1

Take a deep breath. Stop staring at those symbols for a bit. Make yourself tea.

Step 2

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.

Step 3

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.

Step 4

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.)

Step 5

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.

Step 6

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 completely 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.

Remember: This has to make sense somehow.

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.

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.

Step 7

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!

Step 8

Wow, this took longer than you thought! Don’t worry, that’s normal. You’re not bad at this.

Extra tip that’s been crucial for me

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 terrible 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 subjector 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. . I don’t like wasting my time on things that aren’t written with my situation in mind if I can avoid it. Some say it’s good to bang your head against the wall. I say that hurts.

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!

This post was originally posted in Recurse Center’s internal forum, but made it to my blog at the suggestion of Wesley Aptekar-Cassels.

How to Make a YouTube Video Out of Your Beamer Seminar

Mon, 02 Mar 2020 19:36:00 -0500

I posted my one-hour seminar on YouTube. Here’s why I thought it was important for me to do so, and how you can make your own using iMovie.

Update from 2026: These days I just use screen capture and speak into noise cancelling headphones. The mouse allows me to point at the things I want to speak about. I record about a section of the talk at a time to allow me to practice several times and settle into my presentation choices.

Why make talks available online
- Why slides
- How time consuming is this?
Let’s make a video in two minutes!

Why make talks available online

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.

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)!

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.

Why slides

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.

How time consuming is this?

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.

Also, it’s really super easy to make this video, I promise!

Let’s make a video in two minutes!

I’m using iMovie 10.1.14 for these instructions.

Step 1 : Take screenshots

Open your beamer slides and put them in fullscreen. On Skim, I use the Option+Command+P shortcut.

Take a screenshot of this fullscreen. I like to use the Command+Shift+3 shortcut.

For example, this is a slide I would like to add to my talk.

Step 2: Getting slide into iMovie

Open iMovie and click the + Create New button and select Movie.

In a new project, simply drag the screenshot you took into the timeline.

Now it might be that the background is in “Ken Burns” (the weird zoom mode by default). Change this by clicking the crop icon and then choosing fit as your option. In general, you can fix this default mode by going to the Preferences… menu.

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.

Step 3: Getting the audio

To get audio, click the mic button, and make sure the audio feed is connected to your built-in microphone. Record what you’d like to say about this slide.

Step 4: Sync the audio and the slide

Simply drag the audio file (in green) to time the audio input with the visual input.

Next, drag the visual file so that your slide ends at the same time as your audio.

That’s pretty much it! Repeat step 2-4 until you have your talk in video form.

Step 5: Export

Choose share > file. This may take a while.

They’ll ask for some settings for the quality of your exported video. I just chose these ones.

Once you have the file, upload it on YouTube! If you have issues uploading the video because it is longer than 15 minutes, you can follow this tutorial to verify your YouTube account.

By the way, here is the finished product from our little project!

Happy sharing!

Conference Notes Posted

Sun, 08 Dec 2019 14:36:00 -0500

It’s been a while. Lots has happened since I started graduate schoolwhich I may or may not elaborate on at some point . 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!