# Posts

• ### 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.