homeowmorphism

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!

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.

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.

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.

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!

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 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.

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

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.

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!

I asked Kasra Rafi 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.

Browsing around to get an answer about a technical question, I stumbled unto an unusually well-structured StackOverflow question by Martineau which had a corresponding unusually helpful and thorough answer. This made me go if only my questions were this well-posed, I would’ve gotten better quality help, so allow me to share it.

I was really confused about this, so I talked to Victoria Kirst who teaches Computer Science at Stanford and was a Google engineer in the past. She really elucidated unit tests for me! Here are my takeaways.

Unit tests are to make sure you haven’t broken anything in the program when you change it. In order words, unit tests are there to prevent regression in your code.

Forget the name calculus and maybe even lambda. A lambda calculus is the smallest Turing complete language using only one operation: evaluationAlso known as $\beta$-reduction. , and one object: functions.

The way the lambda and the calculus comes up is the following: we typically name the functions starting with a $\lambda$ (lambda) symbol, and we say it’s ‘‘calculus’’ in the sense of a particular method or system of calculation or reasoningSource: Google. .

So how do we write functions, and what is this way of reasoning?

Cryptographic hashes play a fundamental role in the structure of blockchains, which is what Bitcoin is built on.

Essentially, hashes act as digital fingerprints for data. If you have a person and a fingerprint, it is easy to verify that the fingerprint belongs to the person, but a fingerprint alone tells you very little about someone. We will later see that this property is useful for verifying the authenticity of a blockchain.

Let’s explore this more formally in this pre-introductory post to Bitcoin.

This is useful for Colemak users.

There’s a direct way to remap keys in Sierra from command line.

To remap caps lock to backspace, copy paste this command line (without the $ sign).

$ hidutil property --set '{"UserKeyMapping":[{"HIDKeyboardModifierMappingSrc":0x700000039,"HIDKeyboardModifierMappingDst":0x70000002A}]}'

That’s it.

My fundamental goal for RC is to learn how to think and work like a programmer in a professional environment.

My programming background has been coding for a few research projects in SageMath, which is basically Python with pure math library functions, and a couple foundational CS courses. I’ve never worked in industry or managed a large code-base.

Today I had the chance to learn about what constitutes a good programming setup from Saul Pwanson, who’s programmed his whole life (that’s over 30 years).

Someone asked me to put up a transcript of the speech I gave on organizing a satellite March for Science. Here it is.

This Math Overflow post answer about how to not forget old math by Timothy Chow is so relevant I’m posting it here on my blog.

Forgetting lots of stuff is inevitable. But there are ways you can mitigate the damage of this information loss. I find that a useful technique is to try to organize your knowledge hierarchically.

I’m currently reading the long overdue 1981 paper of Gromov showing that virtually nilpotent groups are precisely the groups with polynomial growth.

To my surprise, the breakthrough wasn’t mind-blowingly, spectacularly, unspeakably awesomely seminalDon’t get me wrong, it’s still very seminal, just not as much as I originally thought, which would have been hard to satisfy for any mathematical result. as I thought it was, in the sense that the literature was somewhat building up to itOf course, this is a fundamental theme in mathematical research, which is easy to forget because it is routinely contradicted by myths of lone mathematicians. As you can see, I’m completely guilty of forgetting myself! .

This actually took me a couple hours to figure outErm, more like a day if we take into account the frustration and the resulting procrastination. On the other hand, I got to listen to this amazing podcast with Kara Swisher, a fearlessly straightforward journalist. Life-changing. , so hopefully I save you some time!

I installed Sage from binary in my Cluster ssh account and documented it step-by-step.

For a sourceTakes more time to build and there may be debugging involved, but the end program will be better performance-wise. installation process, click here.

I installed Sage from source in my Cluster ssh account and documented it step-by-step.

For a binarySimpler installation, but the resulting program will be suboptimal performance-wise. installation process which takes less disk space, click here.

The growth of a group is an aesthetically pleasing large-scale property that is borderline geometric and algorithmic.

I will give an overview on the research on growth by historical and intuitive examples.

Although proofs are often thought as a way to show beyond reasonable doubt that a mathematical statement is true, this is not always the case in practice, even for mathematicians at the top of the field. Andrew Wiles’s proof of Fermat’s Last Theorem originally contained a gap, and Fields medalist Vladimir Voevodsky found one of his papers to be very wrong seven years after publication.

Mathematical publications are long, and their machinery convoluted. State-of-the-art proofs are by no means a certain way to establish truth, as these proofs are proof-sketchesThe great Dani Wise once said. “Unless you are in the first week of your first-order logic class, all proofs you are going to see are proof-sketches. Deal with it!” . For a long time, I have wondered what the role of a proof is, and where proofs are going with the advancement of computing power and formal verification.

As a visual person, this passage from Geometry, Topology and Physics by Nakahara Nakahara, Mikio. Geometry, Topology and Physics. CRC Press, 2003 blew my mind:

The Torus $T^2$ is a product manifold of two circles $T^2 = S^1 \times S^1$. If we denote the polar angle of each circle as $\theta_i \mod 2 \pi \quad (i = 1,2)$, the coordinates of $T^2$ are $(\theta_1, \theta_2)$. Since each $S^1$ is embedded in $\mathbb{R}^2$, $T^2$ may be embedded in $\mathbb{R}^4$. We often imagine $T^2$ as the surface of a doughnut in $\mathbb{R}^3$, in which case we inevitable have to introduce bending of the surface. This is an extrinsic feature brought by the ‘embedding’. When we say ‘a torus is a flat manifold’, we refer to the flat surface embedded in $\mathbb{R}^4$.

The bending of the torus is an extrinsic feature brought by the embedding! This makes a lot of sense: I couldn’t subconsciously associate the product of two circles with the torus in 3D, because they are in fact different: the 3D torus is an embedding of the cartesian product of two circles in 3D, not the object itself.

Another reminder that visualizing things, while helpful, can be misleading.