Sabbatical leave…

It has been quite a dramatic year. COVID-19 has made quite a substantial amount of impact on everyone’s life including mine. For some, it must have been a dreadful year. However, for some, it must have been a year full of chances.

In a new environment away from home, I found myself in a new home with a new hope. Life is not easy, but is definitely beautiful because we all have hope. I now have my own sense of “Life is beautiful.” I am once again beginning to dream big. (For those who haven’t watched “Life is beautiful,” it is a great one. Please watch it.)

Friends, especially STUDENTS, please dream big and keep up the hard work. It will change the way you view the world and the future.

AI can solve PDEs faster than ever before.

The following is an article on a new neural network architecture that solves a family of PDEs once trained much faster than traditional PDE solvers. Instead of learning hidden functions, the new neural nets learn operators. This allows for solving not only a particular PDE problem, but also a family of PDEs without retraining. There is a deep neural network called DeepONet, which learns operators. The new neural network in the article is FNO (Fourier Neural Operator).

https://www.quantamagazine.org/new-neural-networks-solve-hardest-equations-faster-than-ever-20210419/

The Continuum Hypothesis

The continuum hypothesis is that there is no cardinality between \aleph_0 and \aleph_1. Cantor proved 2^{\aleph_0}>\aleph_0 and conjectured 2^{\aleph_0}=\aleph_1. However, Gödel thought that the continuum hypothesis was false and predicted 2^{\aleph_0}=\aleph_2. In fact, results by Gödel in 1940 and by Cohen in 1963 confirm that the continuum hypothesis is independent of the ZFC axioms. This means that we need additional axiom to resolve the continuum hypothesis. More can be found in the following quite entertaining article.

https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/