#topological space

[shamelessnightlight]

How could they not be? Look at all the differential structure! The smoothness! The metric! So curve, much geodesic, Morse functions. I think I had better go lie down and perhaps fan myself.

fun fact: Boston Museum of Science calls their evening lecture series “SubSpace”, which would be a totally innocuous math term except for the fact that, to make sure you know these lectures are higher-level and not aimed at their usual audience (kids), they chose to subtitle it “SubSpace: Adult Experiences”

😶

[bundbunds]

Let S be a bounded, convex, compact, well-defined set, tightly compressed, seasoned, smoked, strictly marinated, and covered with a layer of Worcestershire sauce. Then S is homeomorphic to any well-made dish.

[bubbloquacious]

I've posted about this before but it's still crazy how the convergence of a sequence does not depend on the ordering of the sequence. Any reordering has the exact same convergence properties.

[more-mathematics-mor-lam]

Screaming because my friend the Alexander Horned Sphere made it into a differential geometry book

[mindblowingscience]

Physicists have discovered a strange twist of space-time that can mimic black holes — until you get too close. Known as "topological solitons," these theoretical kinks in the fabric of space-time could lurk all around the universe – and finding them could push forward our understanding of quantum physics, according to a new study published April 25 in the journal Physical Review D.

Black holes are perhaps the most frustrating object ever discovered in science. Einstein's general theory of relativity predicts their existence, and astronomers know how they form: All it takes is for a massive star to collapse under its own weight. With no other force available to resist it, gravity just keeps pulling in until all the star's material is compressed into an infinitely tiny point, known as a singularity. Surrounding that singularity is an event horizon, an invisible boundary that marks the edge of the black hole. Whatever crosses the event horizon can never get out.

But the main problem with this is that points of infinite density can't really exist. So while general relativity predicts the existence of black holes, and we have found many astronomical objects that behave exactly as Einstein's theory predicts, we know that we still don't have the full picture. We know that the singularity must be replaced by something more reasonable, but we don't know what that something is.

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[isomorphismes]

Hadamard knew in 1898 that negative curvature and simply connectedness for surfaces embedded in 3-space force uniqueness of geodesics joining two points—implying that any segment of geodesic is also a shortest path.

But there is a long way toward the modern statement: “on any complete abstract Riemannian manifold of ≥0 curvature of any dimension, curvature is the quotient of its universal covering by a discrete group of isometries.”

Marcel Berger, Riemannian Geometry during the Second Half of the Twentieth Century

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Hadamard, 1898 being Les surfaces à courbure opposées et leurs lignes géodésiques

[kaleysian]

I went to a lecture by Dr. Arturo Pianzola on Applications of Galois Cohomology to Infinite Dimensional Lie Theory. Such a fascinating piece of research!

[enbymaths]

no tips about sheaves tbh but just here to offer moral support as someone who accidentally fell into algebraic geometry and keeps asking himself whether he likes it or it's just stockholm syndrome (it's both <3)

I'm screaming. This is probably going to happen to me too, gotta blow up something

[nostalgia-tblr]

What do English people call a close? You know, the stairwell bit where all the flats are in a tenement? If you go to visit someone at their flat, what do you call the bit where you wait for them to answer their door? That communal stairs… area?

("Modern AUs don't require research" MAYBE IF YOU'RE ENGLISH THEY DON'T 😭)

[tokinanpa]

today I have discovered that there is a topological space called the Hawaiian earring

this fact has made my whole day

[sdmsims]

take me to cockroach bug city

buildbuy stuff???? on sdmsims?????? no way man

some deco stuff i converted from the ultra desert map in USUM! since i'm using TOOL to recreate some key ultra space locations for my save, i figured the iconic rocks would convert nicely.

new mesh, from pokemon ultra sun/moon

12 objects, all labeled in catalog with [ Ultra Desert ] under the rocks category

these are 3ds models man. the highest polycount is the large structure with is like 1.2k

feel free to recolor or mesh edit, but don't put any edits behind any form of paywall

DL : SFS (free, no ads, home grown, may contain cockroaches,)

[lipshits-continuous]

Bought myself a nice notebook to take my notes in for the studying I'm planning on doing over summer :)))

[greenouillee]

Cauchy Convergence in (Q, d), infinitely close yet never quite irrational,,, this too, is yuri

[bubbloquacious]

Why yes I am still thinking about graphs as presheaves. Another thing you can do with functors from small categories is think about their limits and colimits. We usually think of limits and colimits as objects of the codomain category, so in this case the limit of a graph-as-functor would be a set. However, we can turn it back into a graph by then applying the diagonal functor Δ from the category of sets to the category of graphs.

Let's think about what this functor does. It maps a set X onto the constant functor from Q^op at X. Such a functor maps every object of Q onto X and every morphism of Q onto the identity function of X. So, the vertex set and the arrow set of Δ(X) are equal, and the source and target functions are the identity. The graph we get consists of one vertex for every element of X and each vertex has a single loop.

A cone under or cocone over a functor F: C -> D is exactly a natural transformation from a constant functor to F or from F to a constant functor, respectively. The limit of F is the terminal object in the category of cones under F, so in addition to the limit object we have a natural transformation from the constant functor at that object to F. Dually for colimits. Natural transformations in Q^hat are graph homomorphisms, so we can do some fun interpretation here.

The limit of a functor with domain Q^op (or Q, as they are isomorphic) is an equalizer. In Set, the equalizer of functions f,g: X -> Y is the set { x ∈ X : f(x) = g(x) } (up to a canonical bijection). It follows that the limit of a graph is exactly the set of arrows of G whose source vertex coincides with their target vertex. The set of all loops of G. Applying Δ to this we get the graph that has a vertex for every loop, and a loop on each of these vertices. The universal cone from this graph to G is the graph homomorphism that maps each loop in this graph onto the corresponding loop in G.

The colimit of a functor from Q^op is a coequalizer (surprise surprise), and in Set the coequalizer of f and g is the quotient set of Y by the smallest equivalence relation ~ such that f(x) ~ g(x) for all x ∈ X. In a graph G we have that Y is the vertex set, X is the arrow set, and f and g are the source and target functions. This means that if two vertices v and w are connected by some arrow, then they will be equivalent under ~. Applying transitivity and symmetry, we find that the equivalence classes of ~ are exactly the connected components of G. The universal cocone over G is the graph homomorphism that maps all vertices in a connected component of G onto the corresponding vertex of the graph-that-has-a-vertex-for-every-connected-component, and all arrows in that connected component onto the unique loop on that vertex.

[theoretical-mutant-4734]

Moebius strip cut in half , and then cut in half again. So weird

[shrimp-milf]

👀 you're also a math srimp

math shrimp!! 🤝🤝