Originally by Springer, but later moved to the European mathematical society.
A wiki that gathers mathematical proofs.
MediaWiki-based.
This appears to be the creator: github.com/externl "Joe George".
A generalized definition of derivative that works on manifolds.
TODO: how does it maintain a single value even across different coordinate charts?
Harvard University + MIT combo.
As of 2022:Fuck that.
- can't see course material before start date. Once archived, you can see it but requires login...
- on free mode, limited course access
Also, they have an ICP.
November 2023 course search:
- Condensed matter: 4 hits, so not too bad
- quantum field theory: no hits
It is a shame that they refocused to more applied courses. This also highlights their highly "managed" approach to content creation. Their 2022 pitch on front page says it all:they are focused on the highly paid character of many software engineering jobs.
for as few as 10 hours a week, you can get the in-demand skills you need to help land a high-paying tech job
But one cool point of this website is how they hire tutors to help on the courses. This is a very good thing. It is a fair way of monetizing: e-learning websites must keep content free, only charge for certification.
Examples:
- a 2x2 matrix can represent a linear map from to , so which is a linear operator
- the derivative is a linear map from to , so which is also a linear operator
Applications of Lie groups to differential equations by 
Ciro Santilli 35 Updated 2025-03-28 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-03-28 +Created 1970-01-01Solving differential equations was apparently Lie's original motivation for developing Lie groups. It is therefore likely one of the most understandable ways to approach it.
It appears that Lie's goal was to understand when can a differential equation have an explicitly written solution, much like Galois theory had done for algebraic equations. Both approaches use symmetry as the key tool.
- www.researchgate.net/profile/Michael_Frewer/publication/269465435_Lie-Groups_as_a_Tool_for_Solving_Differential_Equations/links/548cbf250cf214269f20e267/Lie-Groups-as-a-Tool-for-Solving-Differential-Equations.pdf Lie-Groups as a Tool for Solving Differential Equations by Michael Frewer. Slides with good examples.
Ciro Santilli believes that molecular biology technologies will be a large part of the next big things as shown at: Section "Molecular biology technologies".
Bibliography:
- www.youtube.com/watch?v=mS563_Teges&list=PLQbPquAyEw4dQ3zOLrdS1eF_KJJbUUyBx Biophysical Techniques Course 2022 by the MRC Laboratory of Molecular Biology. Holy crap that playlist is a tour de force of molecular biology techniques in 2022!
Lebesgue integral vs Riemann integral by Ciro Santilli 35 Updated 2025-03-28 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-03-28 +Created 1970-01-01Advantages over Riemann:
- Lebesgue integral of is complete but Riemann isn't.
- youtu.be/PGPZ0P1PJfw?t=710 you are able to switch the order of integrals and limits of function sequences on non-uniform convergence. TODO why do we care? This is linked to the Fourier series of course, but concrete example?
Riemann integral vs. Lebesgue integral by The Bright Side Of Mathematics (2018)
Source. youtube.com/watch?v=PGPZ0P1PJfw&t=808 shows how Lebesgue can be visualized as a partition of the function range instead of domain, and then you just have to be able to measure the size of pre-images.
One advantage of that is that the range is always one dimensional.
But the main advantage is that having infinitely many discontinuities does not matter.
Infinitely many discontinuities can make the Riemann partitioning diverge.
But in Lebesgue, you are instead measuring the size of preimage, and to fit infinitely many discontinuities in a finite domain, the size of this preimage is going to be zero.
So then the question becomes more of "how to define the measure of a subset of the domain".
Which is why we then fall into measure theory!
The laplace operator for Minkowski space.
Can be nicely written with Einstein notation as shown at: Section "d'Alembert operator in Einstein notation".
Ciro Santilli's favorite music genre.
Ciro's 2020 perfect Friday evening: jazz fusion + study quantum field theory on an Amazon Kindle. Ahhhhhh.
If any of you ever read this, do send me an email to Ciro Santilli saying hi and we can agree on a clear separation of usernames.
Although if you are just starting out, maybe you should just go from scratch with a unique Internet alias.
A younger unrelated Argentinian homonym who likes soccer that can be found through Google:Ciro used to like playing soccer too! :-)
- twitter.com/ciro_santilli
- www.facebook.com/profile.php?id=100009065024069
- www.instagram.com/ciro.santilli/ contains a few cool-sexy-boy-selfies as of 2025 when he hit that stage
- www.youtube.com/@cirosantilli9603
- www.linkedin.com/in/ciro-santilli-bb77b72b7
www.ancestry.com.au/genealogy/records/ciro-santilli-24-bkmssg documents a "Ciro Santilli" born 31 Jan 1887 at Castelvécchio in Subéquo, L'Aquila, in the Abruzzo region, just like Ciro Santilli's ancestors. Parents Francesco Santilli and Anna Silveri. The page also mentions:
- Ciro Santilli found in New York, Passenger and Crew Lists (including Castle Garden and Ellis Island), 1820-1957
- Ciro Santilli found in Oregon, Naturalization Records 1865-1991
Important discrete mathematical group by Ciro Santilli 35 Updated 2025-03-28 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-03-28 +Created 1970-01-01 There are unlisted articles, also show them or only show them.