Mathematics

12:10 AM

have a new favourite term

In homological algebra, the hyperhomology or hypercohomology of a complex of objects of an abelian category is an extension of the usual homology of an object to complexes. It is a sort of cross between the derived functor cohomology of an object and the homology of a chain complex. Hyperhomology is no longer used much: since about 1970 it has been largely replaced by the roughly equivalent concept of a derived functor between derived categories. == Definition == We give the definition for hypercohomology as this is more common. As usual, hypercohomology and hyperhomology are essentially the...

"yeah so for a variety the hypercohomology spectral sequence gives the Hodge-de Rham spectral sequence for algebraic de Rham cohomology"

real tongue breaker

I'm still convinced that the person that coined the terms "orientation preserving/reversing" did this with the specific purpose of letting lecturers trip up

dc3rd

Hi @TedShifrin. You said something particular in one of your lectures that is still wracking my head. This was the point in the lecture where you said it: https://youtu.be/pcHFb8V5U_4?t=531

You said the "point" $x$ is in the plane, but the "vector" $x$ is not. How is that possible given that starting from the origin to get to the point, that would be a vector.

Thorgott

@user2103480 Balarka did some of that shit earlier this year

that man's too far gone

@dc3rd Look at the picture he drew. Does the vector x, meaning the arrow with tail at the origin and tip at the point x, lie in the indicated plane?

dc3rd

12:29 AM

@Thorgott, in his picture the vector he drew is not in the plane, I agree. I also agree the $A$ and $x$ are not orthogonal. But if $x$ is a "point" in the plane, it means that $A \cdot x = 0$. Since this is the equation of the plane.

hang on a second, that last idea doesn't actually make sense.....

well the "point" $x$ does satisfy the equation of the plane $A \cdot x = 0$. But the vector does not.....so our we treating a "vector" and a "point" as different objects? I ask because if we "sub in" the values of my "vector" $x$ into the equation $A \cdot x$, I will get $0$. But if I take the dot product of my "vector" $x$ with $A$, I will not get zero........so we are doing different operations on the same object then?

Thorgott

I haven't seen the rest of the lecture, so maybe I'm messing up the context, but isn't the equation of the plane in which $x$ lies $A\cdot x=2$?

user2103480

@Thorgott lmao

I'll never leave the comfy zone of probability, analysis, point set topology/measure theory and classical logic again

Mike Miller

12:45 AM

@user2103480 It's totally unnecessary

To use the name hyperhomology

user2103480

@MikeMiller that's why this needs an especially impressive-sounding name

Mike Miller

Hyperhomology is just homology

Of a more complicated thing

user2103480

spicy homology

Thorgott

In any case, mathematically $x$ is always just one thing: a tuple consisting of 3 real numbers. But we can choose to visualize this mathematical object geometrically in different ways: sometimes we think of it as specifying a point in 3d-space and at other times we think of it as specifying an arrow in this space. The thing is that a geometric picture involving one of these visualizations need not be comparable when you substitute it with the other visualization. They're simply different visualizations.

user2103480

The one girl in my cohort that was taken under one professor's wing already from like semester 3, with a lazer-focused path and top grades, was broken by equivariant cohomology

dc3rd

12:54 AM

@Thorgott, oh yes you are right about $A \cdot x = 2$ and not $0$. I had written some more in my comment, but you made it null by stating that it need not be the case that one visualization be comparable to another. As well as we know what the "point" is doing but that need not say anything about the "vector". THanks for your assistance.

user2103480

She was into algebraic combinatorics - and not at all into topological methods - and her prof gave her a master's thesis topic on equivariant cohomology of grassmannians and algebraic K-theory. In her words, "a few isomorphisms make it easier", but apparently that topic was the nail in her maths-coffin. Handed her thesis in as quick as possible and started a comfy insurance job

(at some point, she just felt she'd rather study something with a more tangible impact/applicability)

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$Mathematics$ Mathematics