# Posts

• ### Conference Notes Posted

December 8, 2019

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!

• ### Intro to Mapping Class Groups, Teichmuller Spaces and Hyperbolic Surfaces.

October 8, 2017

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.

• ### On Organizing the March for Science In My City

May 6, 2017

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

• ### On Memorizing Research-Level Math

May 4, 2017

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.

• ### Gromov's Seminal Breakthrough (short)

April 21, 2017

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! .

• ### How to Run A Python Script in Slurm-Based Cluster in Five Minutes

April 18, 2017

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!

• ### Installing Sage in ssh/Cluster from Binary

March 23, 2017

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.

• ### Installing Sage in ssh/Cluster from Source

March 22, 2017

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.

• ### Introducing Growth of Groups

March 14, 2017

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.

• ### The Social Role of Mathematical Proofs

October 1, 2016

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.

• ### The Torus vs Its Embedding in Three-Space

July 3, 2016

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.