Posts

Introducing Growth of Groups
March 14, 2017
The growth of a group is an aesthetically pleasing largescale 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. Stateoftheart proofs are by no means a certain way to establish truth, as these proofs are proofsketchesThe great Dani Wise once said. “Unless you are in the first week of your firstorder logic class, all proofs you are going to see are proofsketches. 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 ThreeSpace
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.
Another reminder that visualizing things, while helpful, can be misleading.