You can't have a single edge (or corner) flipped in the #Rubik's #cube (unless you have disassembled it) because after every legal move the sum of all the edges/corners rotations is zero (module 360).

#group theory and #cohomology

https://ruwix.com/the-rubiks-cube/how-to-solve-the-rubiks-cube-beginners-method/

https://puzzling.stackexchange.com/a/24

#TodaysMath 2/2: The appropriate #Floer #cohomology theory to associate to (composable) Lagrangian correspondences is *quilted Floer cohomology*. I like this paper by Wehrheim and Woodward to learn about it:
https://arxiv.org/abs/0905.1368

Introduction Time!
Well hello there stranger! 37 year old #dad , omninerd and general #art fan here.
I like #Maths : specifically #cohomology for the thrills and #data , for the bills (also some thrills: please refrain from kink shaming).
I like #philosophy : I read mostly #Levinas adjacent stuff these days, but I’ll rant on #analytic stuff any day.
Other sources of joy: #Boardgames #Beethoven #Bach #Rap #Prog #Zelda #roguelikes #Dnd #scifi #breadtube #leftism #spirituality #christianity #hiking

'Computing Cohomology Rings in Cubical Agda' by Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg: "extends previous developments by providing the first fully mechanized definition of cohomology rings... The formalization is constructive so that it can be used to do concrete computations, and it relies on the Cubical Agda system which natively supports higher inductive types and computational univalence". #Agda #Cubical #HoTT #Cohomology

#ExplainingMyResearch 23
To start, I am working on 2-dimensional "universes" like the disc with boundary, which are not strictly speaking GC, but similar enough to be useful.
My #preprint
https://arxiv.org/abs/2207.06894
describes how to define #Floer #cohomology for so-called log-#symplectic surfaces. I am currently finishing off the full description of the category of branes in this setting, so keep your eyes open! 🙂

#ExplainingMyResearch 17
Now, in this particular setting, this does not seem very useful - you can tell what the minimal number of intersection points is by just looking at the picture.
But this is actually just an extremely simple example of a complicated invariant called #Floer #cohomology, which is central to the #math description of #MirrorSymmetry!

#ExplainingMyResearch 16
HOWEVER, there is a #cohomology associated to this setting which essentially counts the *minimal number of intersection points* given a particular arrangement of endpoints, no matter what the actual lines look like.
Computing it for a particular arrangement of lines involves counting the number of intersection points that is actually present and substracting a certain number of them again (those that are "exact", a distinct property).

#ExplainingMyResearch 10
Today's thread is about #cohomology. Almost no matter which area of pure #maths, theoretical #physics and also many more #applied fields you work in, you will encounter some form of cohomology. While these are defined in very different ways depending on context, they sometimes turn out to still compute the same thing; and their core properties are always the same.

something I don't really understand is why people try to give intuition for group #Cohomology by describing it as the singular cohomology of a classifying space. the derived functor description is not exactly thrilling but it's quite acceptable to me since the base functor is inherently motivated and there's always a resolution that uses basically transparent objects (instead of, say, injective modules or sheaves). cohomology of a K(G, 1) doesn't seem to have either of those properties to me

Well, actually, it’s a bit annoying that \(\delta\) returns lots of n-1 simplices for any n-simplex: it is a multifunction. If we instead map the other way around, from the boundary to the inside, we simply get a function! And that’s called cohomology, and is what the cool kids do all the time!

4/end

#homology #cohomology

#Progress this week,

- putting the final touches on an application for a #phd position. this is the first for me, so the hardest to write. yes it took me a while, there's been a lot going on.

- finally learning #cohomology in a way that sticks. I grew tired of half remembering definitions.

- saw @ColinTheMathmo 's excellent "colors can compute" talk.

- installed #zotero since I see guides to integrating it with org-roam, and hope it makes better #bibtex than #calibre.