Oliver Knill

The Perron Frobenius Theorem has a nice application for Markov chains. There exists a unique equilibrium if the Markov chain is a positive matrix. #shorts #linearalgebra #probability youtu.be/TIaDy-DWd_4?si… via @YouTube

Sarnak: "Proving that something is random is probably one of the hardest things in mathematics!" He also considers the probabilistic statement as the most suitable RH explanation for "a person on the street Talk of April 15, 2026 at Harvard. See the clip youtube.com/watch?v=JmqaTl…

20 years math in movies. people.math.harvard.edu/~knill/mathmov… youtu.be/ZnKXen-Qxr4?si… via @YouTube

If the digits of pi were random, then by Borel Cantelli, pi would write every possible text infinitely often. For Piday 2026, we build "Pi Monkey" and see how long it takes to write "piday". #shorts #piday youtu.be/wNupF4cBem4?si… via @YouTube

One minute Probability Review of the first half of the course. We tried to get the 10 most important focus points in 10 slides. people.math.harvard.edu/~knill/teachin… #shorts #probability youtu.be/gbGoOX8ZXM0?si… via @YouTube

Characteristic Functions of Random Variables #shorts #probability youtu.be/mJDmwnxcG5I?si… via @YouTube

The Caratheodory Extension Theorem youtube.com/shorts/Mbls5bU… via @YouTube

The Lebesgue Integral. We also show the short 4 page paper of Lebesuge from 1901. No journal today would even consider publishing such a short note these days. The paper is almost comically simple but it is a fantastic frame work. #shorts youtu.be/CvLgzCK2Poc?si… via @YouTube

The Probability approach to the Riemann Hypothesis #shorts youtu.be/yCUfTKI2YbY?si… via @YouTube

Volumes of higher dimensional spheres and balls in 58 seconds with proofs #shorts youtu.be/_YkeauGpCm4?si… via @YouTube

Spherical Averages youtu.be/q37p8OMdn1g?si… via @YouTube

A neat trick on how to lie with statistics youtube.com/shorts/AaqD-Uc… It is based on a famous paradoxon, but you can be almost certain that your manipulation will not be detected.

My bikeride to work this morning. youtu.be/1Vrf9wShVSE?si… via @YouTube

@rfleury Even if it would work, it is a bad idea to have an operating system do "random". Doing random might work at some point, and not later, maybe in a crucial moment of your workflow.Modern AI technology by nature is non-deterministic and does not belong into the operating system.

This is not a real company

T[x_]:=Total[IntegerDigits[x]^2]; HappyQ[x_]:=Nest[T,x,x]==1; Print["Happy 2026! ", HappyQ[2026]];

A bit more abou the Density of Wave front Conjectures treated here arxiv.org/pdf/2501.14611 with Emily Kang (we currently revise the paper and make more experiments) youtu.be/lt76dc5hUJI?si… via @YouTube

Descartes Theorem on Polyhedra. One of the amazing theorems in geometry. It is astounding that Descartes already got the main point of the Gauss-Bonnet theorem. #shorts #geometry youtube.com/shorts/kd4JrLj… via @YouTube

here is the correct link to the source of the C program which produced the wave animation. The program was written in April 2000 and is in the public domain. people.math.harvard.edu/~knill/various…

Quantum Wave Animation #shorts #quantumphysics youtu.be/v9uLJPrtYYk?si… via @YouTube As for the source code for this C program (which runs on linux using Xwindows for example), see this page: (the C program was written in April, 2000 (at UT Texas). people.math.harvard.edu/~knill/various…

Wave Interpolation. How to get from a wave localized at one point to a wave localized at an other point. youtu.be/Zkhvm4bzlLQ?si… via @YouTube