Ciro Santilli @cirosantilli  40

Basically it is a larger space such that there exists a surjection from the large space onto the smaller space, while still being compatible with the topology of the small space.

We can characterize the cover by how injective the function is. E.g. if two elements of the large space map to each element of the small space, then we have a double cover and so on.

The key concept of topology.

We map each point and a small enough neighbourhood of it to ${\mathbb{R}}^n$, so we can talk about the manifold points in terms of coordinates.

Does not require any further structure besides a consistent topological map. Notably, does not require metric nor an addition operation to make a vector space.

Manifolds are cool. Especially differentiable manifolds which we can do calculus on.

A notable example of a Non-Euclidean geometry manifold is the space of generalized coordinates of a Lagrangian. For example, in a problem such as the double pendulum, some of those generalized coordinates could be angles, which wrap around and thus are not euclidean.

A metric is a function that give the distance, i.e. a real number, between any two elements of a space.

A metric may be induced from a norm as shown at: Section "Metric induced by a norm".

Because a norm can be induced by an inner product, and the inner product given by the matrix representation of a positive definite symmetric bilinear form, in simple cases metrics can also be represented by a matrix.

Host: Ben Krasnow.

Deals with materials, chemistry, microscopy, electronics.

Uber practical, well described setups deep science stuff, he is awesome and has been at Google since 2014: www.linkedin.com/in/ben-krasnow-6796a94/

Ben studied at University of California, Santa Barbara.

www.youtube.com/watch?v=qiMR6yAFfyA He obtained PhD from UCSB in Materials, exploring, among other things, growth of a semiconductor called lead selenide.

Aweseome dude filmed a bit of the high end univeresity equipment while doing his thesis.

Does deliver good concise ideas. Now we need something on the less concise side of it.

Great chemistry content. Well detailed setups, notably substance extraction from off-the-shelf products.

This dude is mind blowing. Big respect.

Some of the most impressive videos are the ones in which he goes and extracts metals from minerals himself all the way.

But God, the typography of the channel name is so insane! Why no space???

Shame the academic system wasn't compatible with him: www.reddit.com/r/codyslab/comments/f5531p/codys_qualifications/ Maybe there were safety issues involved though.