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luveline · 10 months ago

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Hi I have a request for zombie Steve! I’d love to hear the story of how they met 🫶🏻

zombie!au —You rescue Steve at the start of the end of the world. fem, 2.4k

The sound of them makes your throat close up. Fear like a knife held too tight in unwilling hands, the heat. It’s the hottest summer Hawkins’ has had for years, and you’re overdressed. You couldn’t fit your favourite hoodie in your bag so you wore it but it doesn’t matter, you lost your bag somewhere in the school gymnasium. You’re lucky you didn’t lose your leg when that thing grabbed you. What were they calling them on the radio? Something starting with G.

Those… things, they can kill people. You saw it just ten minutes ago, your algebra teacher turned to a creature, Maisie Lewinsky from your homeroom stained with gore under her hands.

You press the back of your hand to your mouth to stifle a hot gasp. What are you supposed to do now? The Hawkins station said everyone would be waiting here, a repeat radio message, but by the time you heard it the sun was going down and there was nobody left. No cars, no promised convoy, nobody.

You’re the last living girl left in Hawkins.

You’re gonna die in here.

Terrified of breathing to loud but spooked that staying will seal your fate, you lift yourself up from the floor of the science lab to peer over a high table. There’s no signs of life. No signs of the dead, either. You’d thought this might be a good place to hide, the thick fire door unbreakable, but you can’t stay here. It’ll be dark soon.

You wish you had your stuff. They’ve for sure taken anything worth eating from the cafeteria kitchens and Bradley’s has been empty for days. You aren’t sure where your next meal is coming from. Fuck, you’re already dead—

“Fuck!” a voice echoes, boyish and terrified from somewhere outside of the door. “Fuck! Get the fuck away from me, holy shit!”

He sounds scared but firm at the same time. Your own fear is like the insufferable heat, riding the back of your neck as you creep toward the door. There’s gotta be more of them outside. That must be why whoever it is that’s shouting sounds so depeserate. But fuck, there’s relief too. There’s someone still here.

“Fuck! Jesus, help me!”

“Okay,” you say unsurely.

You wedge open the door to the science lab and poke your head out cautiously. There’s a dull thudding sound somewhere to the left, metallic screeching further down, but the panicked shouting (and now panicked yelping) is coming from outside.

You look around for a weapon. There’s nothing to take.

“Holy fuck I do not wanna die!”

Me neither, you think, sniffling back your worried tears. You don’t wanna die, you just want your bed. You want to be home, and safe, but there’s no one to look after you anymore, and you can’t just let people die ‘cos you’re scared. You run from the science lab to the fire escape door full pelt, arm in sudden hot pain at the collision, but the door gives and swings hard into the adjourning wall. You look around frantically for the source of the shouting as it bounces off of the exterior walls of the school and the stone floor of the courtyard, heart suddenly afloat in your chest.

“Hello?” you shout. “I’m here, I’m–”

“What the fuck!”

It’s said with such horrified anger that you give pause, even as your hands shake, cold sweat wetting your lip and colder in the rare afternoon breeze. You dart toward the shouting a moment later, and maybe you’re too late, you can’t save anybody, your shoes pinch as you race down the few concrete steps that lead to the parking lot.

Snarling curdles the air. Your neck snaps left, away from the cars and open territory and toward the subject of your nightmares these last few days. You’ve seen glimpses of these things, always too scared to stay and help, always too stupid, too weak, and seeing them now cements it.

A group of geeks grab at a boy where he hangs from the bars of a metal staircase leading up to the roof of the building. You run toward it on instinct but stop before they hear you, eyes wide. His hands are white-knuckled, his hair falling down into his face, but you know who it is now you’re close enough to see him. You could recognise Steve Harrington a thousand feet away.

“Hey!” you shout. “Hey! Over here!”

Why did you say that? Why are you yelling? The geeks turn their heads to easier pray and you’re done for —they start to run. You stumble back in terror.

“My bag! Get my bag, get the knife!” Steve shouts.

You swing yourself around in a huge circle. There, further into the lot, lies a bag. Further past it lies a wooden baseball bat spiked with fifty silver nails.

You sprint past the bag to the bat and try to grab it while you’re still running, knees grazing hot white fire on the tarmac and hands like acid as you force yourself up again, running further, putting space between you and the too fast footsteps that follow. When you’re sure you’ll have room you swing to see them, their maws dripping gore over white buttoned shirts and once prim blouses. There must be ten of them at least. Only two stay to snap their jaws at Steve Harrington where he attempts to climb up the stairs from the bottom, his foot dangerously close to bloodied teeth.

You pull the bat back as the first of the creatures reaches you. With a grunt more terror than exertion, you force the bat forward, wood arcing through the air, shiny nails catching the light of the setting sun and slamming downward into flesh.

Your eyes flare as wide as they’ve ever been. The geek stops cold and drops, your strangling grip on the bat forcing it up out of the mash of his brains. Another geek leaps over him as you scramble back.

“Run!” Steve yells from the stairs, stress stretching his voice thin and high. “Run away!”

You drop the bat and sprint for your life. Down into the parking lot, past a handful of locked cars and suitcases discarded. This must’ve been where everybody was before they left. There couldn’t have been room. Boxes and trophies, books, magazines and toys, all manner of possessions string like a breadcrumb trail down the road that you have to avoid. You run until your calves are burning over the road that will lead toward Hawkins middle, where you throw yourself into the woods, and hope without any real hope that they’re empty.

Grass folds under your feet. Your panting is as loud as your heart.

When the only shallow breathing you can hear is your own, you circle back to the High School, sticking to the shadows so as not to attract any more attention. A few geeks have collected to join the two you’d left behind, and for a second you’re sure Steve’s succumbed to fatigue and fallen into their blackened clutches, but you spot him balancing dangerously on a handrail between two sets of stairs, leg pulled back in preparation to kick any opposition away.

You sweep up the bat and try to make a plan. You were never going to be able to handle that many people before, not with their new mutations, but you can handle four. Maybe. Probably not.

“Steve, what do I do?” you call. “You have to tell me what to do.”

“You came back!” He swears and shimmies further up the railing as one of his attackers finally manages to traverse the blocked up staircase. “I don’t know what to do! Just hit at them until they die!”

It’s easy for him to say. They’re gruesome creatures, the faces of people you once knew but none of their humanity. They can run as fast as any person can. A human bite has alarming force behind it. The voice on the radio warned you that what you’re trying to do is a bad idea, and yet. You roll the bat in your hand. Your chest aches as hard as your dry throat.

The first geek goes down easy. Unsuspecting, you manage to whack it in the back of the head hard and break through soft skull. The second turns to see you just as you’re lifting the bat again, and it runs hard into it as it comes down, killing itself.

The third is where things get tricky.

“Fuck,” you mumble, lifting your bat to find a sloughing of cartilage and tissue stuck between the spines. “Oh, fuck,” you moan.

“Be careful!” Steve shouts.

You step back and trip, nearly falling. “Stay away from me!”

It snarls in response. Eyes clouded, the geek is a little slower than the others, and it follows you sluggishly away from Steve. The fourth remains, snapping, but you can’t keep watch.

“Stay away from me!” you warn again.

Steve swears on the railing, his cursing followed by a wet thunk.

The geek doesn’t listen, it bites.

You pull your arms to the side, hands wrapped tight around the base of the bat and ready to swing. With a huge, aching cry, you swing the bat to the side and knock the nails clean into its cheek.

It doesn’t die.

Fuck fuck fuck! You throw yourself to the floor by the geek’s feet and out of its reach, on knees, on your feet again, scrambling toward Steve’s bag. You glance over your shoulder as your knees slam down hard into the floor, never so scared in your life, horrified as the bat stays stuck between tendons and the geek takes a running jump toward you.

You pull the knife from Steve’s bag and hold it out in front of you, squeezing your eyes closed in terror.

“Fuck, hey!”

You scream as the weight of the geek lands on top of you. You scream like it’s taking bites of you, until your throat burns and there’s no sound left to make and you choke on it instead. A short, sharp sound.

Then the weight is pulled off of you. Someone lets out a massive gasp.

“Did it get you?”

You blink your eyes open against the glaring white sun where it meets the horizon. If you’re crying, it’s your business, water on your cheek and a dizzy hot feeling everywhere else.

Steve Harrington looks at you like you’re a ghost. “Did it get you? Are you okay?”

You look at your hand and the knife —his knife— where it rests on the tarmac. “I don’t think so. How do you know?”

“They bite! Did it bite you?”

“I don’t know.”

“How can you not know?”

“Because I’m not exactly uninjured, Steve!”

He frowns at you. Well, he glares. “You’d know if it bit you. Don’t be dense.”

“How am I supposed to know?”

“You’re telling me you don’t know what a bite feels like?”

“Some of us had homework.”

He wrinkles his nose. “Is that supposed to be funny?”

Well, yeah. It was supposed to be hilarious.

You look around the parking lot and the school courtyard for any outliers, but the school seems well and truly abandoned now. You can’t hear anymore huffing or screeching, no crying, not even the sound of a radio. Everyone’s been playing them nonstop for weeks, waiting for days like today. Suddenly the raptures here, and you aren’t part of the rescue.

But you saved Steve Harrington, at least. You’re accruing some good karma.

Steve doesn’t hold his hand out, he just grabs you under the arms and pulls you up into a standing position. You’re surprised he can do it, you aren’t light, but you remember his last skins game in the gymnasium and nod to yourself. Of course he can pick you up. Plus, you help, using your legs despite their stiffness to brace yourself on the ground.

“Doesn’t look like it bit you,” he says, quieter now, his hands sliding down to yours briefly before he stands back. “What are you doing here?”

“I thought this was the rendezvous point. I mean, it was, right? We missed it.”

“We missed it.”

“How’d you get here?” you ask.

“Bike. Car chose the worst possible time in the world to die. Not that I could’ve got gas.” He eyes you hopefully. “Tell me you drove here.”

“I biked too, but it’s gone.”

“Gone?”

“Tire popped.”

Steve rubs his eyebrows. His hands are clean where yours are caked. You stand unable to mask your heavy breathing now, and when you reach for him to steady yourself, he doesn’t move away.

“Sorry,” you mumble, licking your lips. You’re a map of little pains. “Are you okay?”

Steve’s hand reaches back to cover yours like he’s going to pull it off, though he doesn’t. “Are you alone?” he asks.

You wince. “Yeah.”

“Where’s your stuff?”

“I lost it.”

“Where?”

“I don’t know.” Your knees hurt. “It’s gonna get dark soon.”

It’s a question. You’re immediately thankful to have found him, because he’s a real living person, and you don’t think you can do this alone. You don’t mean to force him, but you need to know what he’s doing and soon.

“…Better come with me, then,” he says finally.

Steve walks out of your grasp, grabbing up his bag which you’d unfortunately ripped open and shoving the spilled contents back inside. He doesn’t stop to zip it closed, walking straight in the direction of the school.

“Where are we going?”

“Same place as everybody else.” You stumble. Steve, reluctant, frowning hard enough to etch a new wrinkle, holds out his hand to catch you by the elbow. “Where did you think?” he asks.

“I don’t know,” you say, half-indignant.

“You ask a lot of stupid questions, you know that?” He looks you up and down. “How’d you do that?” He points at your bleeding knees.

“I ask stupid questions?”

He grabs the bat from near the felled geek and stands tall. “Jesus. Let’s go find a car.”

It’s not as easy as his tone might suggest. You don’t find a car, you never do, and you never stop asking him obvious questions, but Steve says thank you for saving him eventually (nearly an entire year later, with a hand on your cheek).

#steve zombie!au #steve harrington x reader #steve harrington x you #steve harrington x y/n #steve harrington #steve harrington fic #steve harrington blurb #steve harrington drabble #stranger things x reader #stranger things fic #steve harrington imagine #steve harrington fanfic #steve harrington fanfiction #stranger things

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bbunnyyy · 1 year ago

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Coconut Pudding.

Hitoshi Shinsou x Reader slight E2L >:))))))

A/N: Not proofread. I apologise for any incoherent sentences/ incorrect grammar. I hope you enjoy this as much as I did :)

☁︎ Reader and Shinsou meeting through Aizawa. Aizawa offering reader extra training. ☁︎

Shinsou always rubbed you the wrong way- the both of you snarkily taunting the other every breathing moment you were in each other's presence. It was almost like one of you would bite off the other's head the next instant. Aizawa and Yamada were low-key scared you guys would kill each other one day.

Hitoshi Shinsou. That name was enough to have anger surging through your veins. Everything about him ticked you off- his sarcastic remarks, his cold expression, his stupidity. Each sparring session was tense, Shinsou desperate to prove himself to Eraserhead and get into 1A and you fighting to prove yourself, to prove you deserved your spot in 1A. You and Shinsou got to know each other better whether you liked it or not- you were spending most of your free time with him.

The closer you got, the more time you spent in his presence, the more obvious it was- his bitter and cold demeanour was just an exterior. He was a small fluffy hamster at heart. The distance between you and him that felt like oceans between shores closed before you realised it. Closed by lingering touches and lingering gazes during training. Closed by the softness that seeped into the other's gaze when thinking of the next time you'd meet. In conclusion, He made your heart pound and your cheeks heat up.

Just why?

You asked yourself. Shinsou would be a bitter gourd if someone like you was pudding. You poked your tongue out at the thought, slinging your bag over your shoulder as you walked out the diner all alone. There was a chill in the breeze that made people walk faster than usual, wanting to get back to the warmth of their own homes. Couples were walking hand in hand and kids clung to their parents as the buzz of chatter emptied into silence as they walked past.

You thought about how nice it would be if Shinsou was nicer to you for once. How nice it would be if you could spend time alone with him, How nice it would be if he harboured the same feelings as you did for him, How nice it would be if he liked you- How nice it would be if his love was like Coconut Pudding- sweet. Not overly so. Just right. The kind that makes you melt. The kind that feels refreshing. Oh, how you wished he were with you at this very moment.

☀︎

You tucked your hair behind your ear [sorry to bald readers/ readers without ears] while standing up to leave. The school bell rang five minutes ago, the last of your classmates already gone- eager to get home after the hell of what they called a curriculum UA put them through. Sighing, you started packing your bag. You were late for training. Even thinking about how exhausting training would be was enough to bring tears to your eyes. Looking down at the last few books you needed to stuff into your bag, you thought of Shinsou.

You thought about how you could feel the waves of heat coming off him when he stood closer to you as he helped you with algebra. You thought about how his uniform smelled when it was freshly washed. You thought of how the bruised, red knuckles on his hands looked as he grabbed your hands to lead the way when you were being too slow. You thought of the soft monsoon breeze that slightly ruffled his hair, How his purple eyes looked when he was surprised- like when Sensei kicked him in the stomach which led him to be winded and on his knees hunched over for the next ten minutes.

Sharp snaps brought you back from your melancholic state. Annoyed, you looked at the source of the disturbance. "Earth to Y/N." Shinsou said, now snapping his fingers in your face. Swatting his hand away, you rolled your eyes.

"Piss off." You groaned. "You look like Uraraka's quirk sent you floating all the way to Jupiter. Whatcha thinkin' about?" Shinsou said, now leaning on the desk behind him. His arms flexed as he rested his weight on the wood behind him. "None of your business. If you're dying to know, information costs money. I do miss those macarons from the bakery down that street..." Putting a finger on your chin, you pondered exaggeratedly. Shinsou hummed, "You're late to training." "Right." Inhaling sharply, you looked down at your bag, which was still unpacked. "I'm joking, I'm joking. Sensei cancelled training today. Said something about sorting out legal stuff 'bout Eri."

Before you could get another word out, what Shinsou said next had you staring at him like he had not two, not three but twelve heads. "Now what is it with you and your love for exploring space while staring at someone? If you don't fancy the idea, you can just say so." Shinsou stated, his gaze lowering to the floor but his voice as monotonous as ever. You shook your head, wiping that silly lil expression off your face. "Pff, like I'd say no to free macarons." It was Shinsou's turn to ogle at you- "I didn't say I was paying, I asked you if you could give me company at the cafe if you didn't have any plans."

Shaking your head, you put your hands up. "It's a shame, then. I guess I have no choice but to go home," You pouted. Wordlessly picking up your bag and stuffing the remaining books inside, Shinsou pulled the bag from your hands and slung your bag over his shoulder while walking. You smiled at him happily at his compliance and pulled him out the classroom by his elbow.

Ah, if only you hadn't missed the blush on his face as you walked through the door of the classroom.

#shinsou x reader #shinsou hitoshi #hitoshi shinsou #hitoshi #shinsou #bnha #bnha x reader #mha x reader #x reader #hitoshi shinso x reader #hitoshi x reader #shinsou hitoshi x reader

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sonar-creature · 4 months ago

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Vector Space Alchemy

I can't get this idea out of my head but like.

Alchemy in homestuck is really cool, I like the whole pseudo-computer science, but I saw a reddit post that was like "what if alchemy was based on real vector spaces" and as a chronic nerd I really like that.

I want to do a more comprehensive write up but I came up with a 7 dimensional vector space (so that a 2-vector cross product is defined, I don't want to learn exterior algebra rn) that I think I can build a system of alchemy out of. Each of the 7 dimensions is related to one of the Platonic or Kepler solids to play into that theming as well.

Probably will effortpost about this with some graphics over the weekend.

#alchemy #homestuck #linear algebra

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coming-up-with-usernames-is-hard · 4 months ago

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PhD Blog Week 4

Reading

Kac and Raina bombay lectures, read half of chapter 4, it seems to make more sense than Date's book but that might just be because the notation is closer to what my supervisor uses and I've already seen the explanation once now

Courses

Lie theory: Proved Engel's theorem and looked at Lie's theorem, introduced the Killing form

CFT: SO MUCH QFT. Supposedly QFT is not a prerequisite for this course but if I hadn't already spent a whole year studying QFT I'm not sure how I would have followed anything in this lecture. We're taking an axiomatic approach which is essentially "correlators exist and have these nice properties", and then we introduced path integrals and showed they had the same nice properties, so I guess they can be the same thing. Of course path integrals are badly defined and don't make any sense, but it's fine, you just believe that path integrals and functional derivatives work like normal derivatives and it all works out fine. No actual CFT this time, the conformal symmetry should come in the next lecture

Differential Topology: Painfully detailed calculation of the tangent space to S¹, defined a vector field. Really wish we were using derivations as the basis of our definitions, alas I have to deal with equivalence classes of elements of a disjoint union. It's a weird definition of the tangent space because it presupposes that it's a vector space of dimension equal to that of the manifold. Plus, it's so notation heavy. We're explicitly working with equivalence classes of pairs of a label of a chart and a vector in ℝⁿ and everything is just nested brackets. When I've seen this material previously the goal has always been to drop as much notation as possible and forget any equivalence classes that may be taking place and just work with a representative

Talks

Example showcases started this week, we each have to give a 30min talk about an "example" relevant to what we're researching. We saw four this week, all of which were good. The first was on moduli spaces of flowlines, I followed what was happening but I didn't quite get the point of what we were doing. The second was on surgery theory, I think I got the idea but I've got absolutely no clue on any of the details. The third was on dual Artin groups, which are related to Artin groups (although the "dual" is misleading, there's no duality) and Artin groups are related to Coxeter groups, which I know a little about because Coxeter diagrams are related to Dynkin diagrams and Dynkin diagrams are related to Lie algebras which are nice. The fourth was on CAT(0)-spaces and CAT(0)-groups. I went into this with no idea what a CAT(0)-space is and now I feel I have a good idea, and there was a nice way to think of the free group as moving different parts of its own Cayley graph into focus which reminded me of zippers in type theory, where a zipper is the derivative of the type, which feels like some black magic where you start differentiating objects formed from (co)products in a category as if they were normal polynomials.

My example showcase is next week, so I've started planning that. Currently planning to focus on how we can take inspiration from physics to discover things in maths, but that might all change once I write the talk and work out if it fits in 30min

Supervisor Meeting

Met with my supervisors, most of the meeting was spent looking at different ways to represent the Clifford algebra of fermions. One way was as an infinite tensor product of copies of ℂ² on which Pauli matrices act, the other was as the exterior algebra of an infinite dimensional space where the basis consists of single-fermion states and the wedge product gives us all of the required anticommuting properties. Still have some sign ambiguity between different soruces that I've yet to track down. Also did an example of the boson algebra acting on the Maya diagrams which clarified some misconceptions I had from the previous meeting

Reading Groups

Complex Geometry: We finished off the proof of the Bruhat decomposition and I followed most of it, it's just when it comes to putting it all together that I get lost. I decided I wouldn't be able to give the talk next week, I just don't know enough algebraic geometry and don't have the time to learn it

Infinity Categories: Looked at infinity groupoids (I understood these ok) and how they're the same as Kan complexes (don't understand these). So far it's a lot of drawing simplices and then filling them in. Similar to complex geometry, each step follows but putting it all together I get lost in the details

Categories: Normal 1-categories seem easy in comparison, after missing the first week which clashed with infinity categories the second week we looked at duality and functors

Teaching

TA'd two first year tutorials this week. The problem sheet this week was much harder than the first two, mostly because the questions weren't written very clearly.

I was assigned marking from two courses this week. Marked about 40 assessments, which took about 6 hours. I could go faster if I wrote less feedback but I think the feedback is useful while I have the time to give it. That said it would be nice not to have to write "underline vectors" ever again

#phd blog

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ageofhearingloss · 2 years ago

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15 questions for 15 mutuals

THIS IS SO FUN! thank you sweet sweet @gold-mines-melting for the tag💕✨💕

were you named after anyone?

i was named after my tia🤩

when was the last time you cried?

lol last night, i was already on the brink of a meltdown and then i spilled a ton of water on the floor and that's what broke me HA

do you have kids?

absolutely not! i may have an education degree, but kids freak me out!

do you use sarcasm a lot?

i think?

what's the first thing you notice about people?

i would say energy/vibe, but that's based on if people smile or laugh or if they have a hard or standoffish exterior

what's your eye color?

like a light greenish brown

scary movies or happy endings?

happy endings 100%, i don't need external things to scare me as a form of entertainment SORRY

any special talents?

uuuuh im a musician? i went to college for music education so i can play every string, woodwind, brass, and percussion instrument taught in western public schools at a very very veeerrrry basic level

where were you born?

new york

what are your hobbies?

i've gotten back into playing music again after not wanting to touch an instrument again after i graduated, and i've been writing again (thanks to you all💕)! i also dabble in film photography!

have any pets?

MY SWEET BABY KITTY, TIMMY! also my childhood cat, monkey, and german shepherd, adi, who both live at my parents

what sports do you play/have you played?

i was a swimmer for a long time and was a certified lifeguard in my teenage years but never kept up with it

how tall are you?

6'1" baybbbeeeeeee

favorite subject in school?

i LOVED school, especially algebra and english but i also really liked any type of science. but i was a music nerd so i was always involved in orchestra, band, and choir and if i ever had a free period i was in the band room practicing

dream job?

i've always wanted to get into concert photography but i am so intimidated and nervous and also im still very much an amateur sooooo it's truly just a dream lol

no pressure tag (so sorry besties if you've already done it!): @joopsworld @mydarlingdanny @watchingovergvff @ourlovegrows @anthemofgvf @malany-gvf @sacredjake @sulkyrie @iheartjakekiszka @jmkho @tripthelightfandomtastic @sacredthefran @demolitionndann @im-derty-dan @gray-gvf11

#this is so fun!#i love reading about you all #i LOVE YOU GUYS #about iz #tag game

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then-ponder · 9 months ago

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I think it’s interesting how the real umbers show up in at least geometric and quite possibly exterior algebra. Like I don’t know how to describe it, but I always feel like î should actually represent [what I’m going to call a perpendicular algebra] and what the reals always represent was the parallel part.

Kind of just spit balling here. Not anything serious.

#math #mathblr #linear algebra #geometry algebra

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stuart-boehmer · 1 month ago

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APROPOS KERR METRIC AND INTERIOR SOLUTION

APROPOS THE VACUUM KERR METRIC AND INTERIOR KERR SOLUTION

Stuart Boehmer

According to a formula of Tolman (“Relativity, Thermodynamics and Cosmology,” Clarendon Press, 1934), with which apparently most authors are unfamiliar, and which can easily be reproduced with a little careful thought, the relationship between the spatial metric, gmn, and the space-time metric, Gmn, is not gmn = Gmn, but gmn = Gmn – G0mG0n/G00.

Therefore, a singularity coincident with the ergosphere is found to occur in the g33 component of the spatial metric (where u3 is the longitude), thereby rendering the standard vacuum Kerr metric theoretically useless as a practical model of a rotating black hole (see my prior missive, “Theory of Black Holes,” apropos my thought on singularities occurring in nature—the mathematical trick I used there to render impotent the singularity in g11 at the Schwarzschild radius doesn’t seem to work here).

Thus, in order to find a practical working model, there seems to be no shortcut except to do the hard work of solving the full, interior problem, including the consequent vacuum solution for the region of space exterior to the rotating dead star. This remains an open problem, but with the assistance of machine computation it is conceptually trivial, as we shall describe presently.

Define the problem in this way for specificity: use spherical polar coordinates where r is the radial distance from the center along a path of constant co-latitude and longitude (therefore g11 := 1). I see no reason to complicate matters by using the hyperbolic elliptic coordinate system chosen by Kerr. The black hole or dead star is assumed to be spherical (density a nonzero constant inside a sphere of radius r = R) and rotating with constant angular velocity w := du3/dt. Because, as we are about to describe, the solution is in terms of Taylor series, there is no a priori reason we cannot use general functions d(r,u2) and w(r,u2) expanded as Taylor series with known coefficients).

At this point, allow me to parenthetically describe the process of “Involution” (W. Seiler, Springer, 2009) for solving any differential equation or system of differential equations in terms of Taylor series and justify it as being just as good (and, for purposes of practical calculation in no way inferior to) finding a solution in terms of “elementary” functions—the obsession for which no doubt contributes to the fact that this conceptually trivial problem has remained open so long. Indeed, this method could be used to solve any problem in any theory of physics and no “open problems” should remain anywhere in the entire discipline of physics, conceptually.

The method is this: expand all known and unknown functions in terms of Taylor series; the known functions have known coefficients, and the unknown functions have unknown coefficients which can be derived recursively by equating the coefficients of like powers of the coordinates, by the standard procedure. See what I mean by “trivial?”

Now some old-fashioned people may object that any sound theory must be construed in terms of “elementary” functions, which are in some sense “known.” Of course, the only elementary functions except for polynomials are the trigonometric, hyperbolic trigonometric and exponential functions—all of which can be reduced to the exponential function, which in turn can be accurately calculated in terms of—guess what?—Taylor series or some equivalent infinite recursive process.

These days, we might regard elliptic integrals as elementary functions and there is an elaborate algebraic theory reducing the evaluation of an arbitrary elliptic function to those of the first, second and third kinds, but no one is interested in this theory any longer—it is simpler to just evaluate in terms of Taylor series by machine computation (the “NI” in UNIAC and ENIAC stand for “Numerical Integrator”—that is why computers were invented!).

Conclusion: computation by machine is just as respectable as any reduction to elementary functions—and there is no escaping the use of machine computation when calculating numerical values of “elementary” functions anyway!

The method of involution is often described as reducing calculus to algebra, because, of course, machine computation must terminate in a finite number of steps and the Taylor series just turns out to be polynomials of high degree. Polynomials are, ultimately, the only functions whose numerical values can be computed in a finite number of steps.

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brickwall2 · 1 month ago

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APROPOS KERR METRIC AND INTERIOR SOLUTION

APROPOS THE VACUUM KERR METRIC AND INTERIOR KERR SOLUTION

Stuart Boehmer

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k-star-holic · 2 years ago

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"My Little Old Boy" Park Sung-woong is from a prestigious law school.

#My Little Old Boy #Shin Eun-jung #Exterior algebra #Laughter #Park Sung-woong #DinDin #Tony Ahn

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positively-knotted · 8 months ago

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Dissertationposting 4 - Finally, Topology!

We're done enough analysis for now, it's time for some nice topology!

Let's think about how the torus argument really worked. With a lot of analysis, we showed that in positively curved spaces, stable minimal hypersurfaces are PSC. Then using homology classes, we induced that smaller and smaller tori are PSC, reaching a contradiction.

But really it was fluke that we got a conclusion about tori - in practice we really cared about homology classes with PSC representatives. More formally, define

So M is PSC iff H^+_n(M) = H_n(M), and Proposition 4 says that then H^+_{n-1}(M) = H_{n-1}(M) as well. Gauss-Bonnet on the other hand says that H^+_2(M) consists only of "spherical" classes. [1]

So how do we link these different H^+'s? Suppose we have some homology class in H^+_k(M) represented by a PSC submanifold N, and any homology class in α ∈ H_{n-1}(M). We can find a "generic" hypersurface Σ representing α, in the sense that Σ ∩ N is a hypersurface of N. But then Σ ∩ N is PSC, so [Σ ∩ N] ∈ H^+_{k-1}. [2]

In short, for each homology class α ∈ H_{n-1}(M), we have a map from H^+_{k}(M) -> H^+_{k-1}(M) by intersection. Let's just write this map as ∩, so we can introduce the following definition.

A closed manifold is SYS (short for Schoen-Yau-Schick) if we can find homology classes β_1,...,β_{n-2} ∈ H_{n-1}(M) such that β_1 ∩ ... ∩ β_{n-2} ∈ H_2(M) is not represented by a union of spheres.

Theorem 5.

Any SYS manifold is non-PSC.

This is the most general situation where our argument for the torus works! To review:

By Proposition 4, β_1 is represented by a PSC manifold. In the torus case, each β_i was the meridian T^{n-1} that you get by forgetting a different S¹ factor time.

Each intersection gives PSC hypersurfaces of smaller and smaller submanifolds by Proposition 4 and the intersection argument above. This is the induction argument.

By the time we reach dimension 2, the only possible PSC surfaces are unions of S², so if we can ensure that the intersection is something else (eg T²) then we have a contradiction.

From an algebraists point of view, this is a fantastic result, as PSC manifolds can't have too much homology going on. [3] On the other hand, it's a bit annoying to check, and most "nice" manifolds don't satisfy it - for example, no Lie groups are SYS. The easiest way to build them is by starting with Tⁿ and modifying carefully [4], but it's easier in 3 dimensions, where being SYS is just saying that you have homology classes than can't be represented by spheres. Here's one example I came up with.

Let K_1, K_2 be knots in S³, with exteriors X_1, X_2. Glue these along their boundary T by any map, to get a closed manifold Y. Then Y is SYS, and so cannot be positively curved (even though S³ can be! weird). This is a bit technical again, [5] but boils to (i) by a geometric argument, Y contains no spheres that aren't trivial in homology; and (ii) by Mayer-Vietoris, H_2(Y) is non-zero.

But the real utility of the SYS condition that it behaves well under connected sums.

Proposition 6.

If M, N are closed n-manifolds, with M SYS, and f : N → M has deg f = 1, then N is SYS. In particular, if M is SYS and X is closed, M#X is SYS.

This is huge, as it captures the idea that if some part of a manifold can't admit positive curvature, then neither can the whole thing, something which seems obvious but most theoretical methods miss. The proof is trivial with the right algebraic set up, basically saying that the preimages of the β_1 in M witness the condition in N.

Ok this is already longer than I wanted, so next time we'll see what happens if you want non-compact manifolds! It turns out you need a whole new type of submanifold, and a crazy long topological construction.

[1] Equivalently, H^+_2(M) is the image of the Hurewicz homomorphism π_2 -> H_2.

[2] More formally, the cap product with any α ∈ H¹(M) preserves H^+_*(M) by Poincaré duality, and we phrase everything in terms of cohomology classes and cup products.

[3] "PSC manifolds have small cohomology rings"

[4] Gromov "showed" that certain surgery constructions on a torus yield SYS manifolds. In particular, if c is a coordinate curve, k > 1, and γ a curve with [γ] = k[c] in homology, then surgery along γ yields and SYS manifolds. This isn't too hard to check, but does yield interesting examples. For instance, you can use it to find manifolds that are non-PSC but the usual spin-theoretic methods don't detect it; and if M_1, M_2 are as above with k_1, k_2 distinct odd primes, then M_1 × M_2 is PSC!

[5] Have another screenshot.

#dissertationposting #maths #math posting #cognisance #topology #algebraic topology #differential topology

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