#taught myself to draw mech specifically to draw these

[gggoldfinch]

brrrrrrr art dump for the tfp self insert oc my childhood self could only dream of 🤯 I drew these back in July but have been thinking about them again recently. This is so embarrassing but I'm so proud of how these character sheets came out, but I can't post them without context so here we goooooo (oc info at the bottom!!!!!!😭)

Embarrassing au & oc info time!!!!!! (tw for vague discussion of non-human self-harm in 10th bullet point):

Okay so basically to preface: in my wip fic (wip is a gross exaggeration), everything remains canonically accurate to TFP except for the fact I use my Magic Fanfic Writer Powers to incorporate ridiculous Cybertronian mysticism canon into it for the sole purpose of furthering my self indulgent plot armor via cyberforming (cyberforming being when organic material becomes that of Cybertronian-make through means of mysticism and/or science)

Marian (unabashed tradgoth self insert) starts off as human. She gets picked up by the Cons while smashed drunk one night bc they think she has info on the Bots (found her bc she was lurking on online forums asking too many questions about big robot aliens bc she once saw them brawling and wanted answers), then she just ends up being kept alive and kept around as a pet/team mascot/ emotional support human, because hey if the Bots have one then maybe humans can be of some use

After a while Marian ends up forming a bond with Starscream (and KO to a lesser extent) after they both end up treating each other with compassion and respect (wow! trauma-bonding!). She kinda definitely falls desperately in love with him (and thinks it's unrequited but jk!). Angst & hurt/comfort abound! Gratuitous usage of mass-displacement device for nsfw purposes! You didn't hear that from me...

She is accidentally killed during the Battle on Cybertron (ca. season 3) by being hit with a stray plasma blast.

Here is where AU material comes in lol!!!!!!!

With the Well of AllSparks alive once again, in an act of desperation SS leaves her body at the edge of the well and actually prays for once in his miserable life. Through a mystical act of pity or mercy or whatever, a fresh spark combines with her own approximation of a soul and cyberforms her corpse and resurrects/ reincarnates her. She's herself, with all her old memories— but also something new, with all-new potential. No one knows wtf is going on lol

She becomes the first mech created on "New" Cybertron. "Cyberform-forged" is the term used for her, making her something of a new race (in the same way the Terrans are a new race), and is more of a mystic anomaly than anything (largely because there is no opportunity to recreate the event).

She's formed with a Vosnian Seeker frame and Cybertronian alt mode. Her frame is weather resistant: built to tolerate and fly through high winds and dangerous weather (Cybertronian and Terran) and relies more on brute force than grace in aviation. This means she is bulky rather than slim and aerodynamic like SS. She's a revival of the (near?) extinct class (if we're going by Prime!canon then Starscream is the only confirmed Vosnian Seeker left). Why Primus decided that cranking out new Vosnians would be a good idea is beyond everyone; everyone is too busy wondering how tf cyberforming works and what the consequences of it are to really question it.

Physically, she's not overtly femme— more androgynous, which is on brand for her Vosnian build. She's top-heavy with large pauldrons, shoulders double the width of her hips. Her new frame reflects her old human body in a very rough, vague way— only enough to be noticeable to those who really knew her before. She applies face paint by hand like her old makeup, and paints one servo red to match how she used to wear her nail polish. She's shorter than SS, around 30-ish feet tall (whereas he pushes 35'). She doesn't have a very good grasp on how her wings emote, therefore anyone who can read Seeker body language can always tell exactly how she's feeling.

She suffers with severe ptsd, depression, and body dysmorphia/ dysphoria as a result of the cyberforming and her human death, and semi-often has moments of panic and distress wherein she self-harms in an attempt to undo what's been done. In a potent mix of lingering human neurodivergency & the jarring biological shift, she often gets overwhelmed by her new body, notably her biomechanical functions: her optics cause a significant amount of stress on a regular basis, to the point where visual "notifications" and all other miscellaneous visual obstructions have to be disabled in order for her to function. This means no scanning data, etc, without purposefully reactivating the internal procedures required for the task. She also has a lot of trouble coping with the lack of sexual dimorphism. Shortly after her "awakening," she brutalized herself due to severe confusion and psychological distress (see the second to last image :( ) and KO had to sedate and mend her.

Every who encounters her and knows her story kinda assumes she just "came back" with a processor malfunction (or if they don't know her story, think she was just traumatized by the war), which isn't totally wrong. Knockout is one of her major supporters thru this and professionally thinks she probably needs a mnemosurgeon to fix her, but can't find any to contact so soon post-war.

She's rather clumsy, and takes a while to acclimate to such a different body (it also doesn't help that Seekers tend to have disproportionally long limbs, as well as cumbersome wings). She smacks things and other mechs with her wings, crushes things in her servos, basically she severely underestimates her own strength and size

One perk to her new body is that she can stream music directly into her processor, which is a function she abuses often to drown everything else out. There are functions she couldn't even dream of before: she can disable pain processors, turn off sight and hearing like throwing a switch, disable various biomechanical functions. She doesn't need to breathe like a human, or expel waste. Energon tastes like battery acid and firecrackers, but hey, she can turn off "taste" processors too.

She's afraid of herself at times, and by extension is afraid of intimacy too. She's afraid she's too different now, in a foreign body she can't seem begin understanding. She feels burdened by expectations; she's not a real Vosnian Seeker, she's not even a real Cybertronian in the way that counts— she is a freakish amalgamation of human memories and a soul stuffed into a shell made new especially for her, despite her having had no say in the matter... Or, at least, that's what she thinks. Those closest to her think she's a miracle and are thrilled to still have her, indefinitely now. Before when she was human, SS had been frustrated and genuinely afraid of feeling anything remotely positive towards her, because of how tragically short human life spans are (and his fears had been proven valid when she was killed). Now he has Marian forever, and while coping with his own problems post-war, he cares for her and teaches her how to be a noble Seeker.

YIPPEEEEEEE

[megatronswaifu]

wrote a fanfiction and wanted to draw something to go with it!!

if you want to read some sickeningly sweet ooc overlord with nightlight please take a look under the cut. i’m happy with how it turned out because i really didn’t take it seriously like i’ve tried in the past.

“Nightlight’s Shadow” <2k words rating: GEN tw: mild canon-typical violence

-------------

“What’s wrong?”

Overlord peered through the door, hunching over significantly. The phase-sixer was incredibly bored, having killed everyone he felt like and no missions assigned to him for a long while. When Overlord was bored and without those to maim, he went to go bother Nightlight.

The minibot sat in the shape of a ball on her berth. Her helm was tucked between her knees, and her little horns poked out, the only thing clueing Overlord in that “Oh, that part of the purple dot is her head.” When she looked up at Overlord, he thought to himself that this was the most pitiful face she had made to-date. 

The towering bot squeezed into the room, scraping the doorframe with his shoulders as he stomped in (he wasn’t stomping on purpose, it was simply a feature of his size), his pillar audials threatening to pierce the ceiling. Nightlight’s accommodations were definitely made with efficiency and budget in mind. Why would anyone waste shanix and space on the ship just in case someone larger than a pea wanted to visit this room? Overlord thought he should carve out the wall and ceiling so his visits weren’t so difficult.

“What’s wrong?” Overlord asked again, and Nightlight turned to face away, “Why the long face?” 

“I’m not a good Decepticon,” the tiny bot started. Her voice was quivering and hoarse, like she had been crying. Little cheekplates having subtle remnant streaks of tears confirmed this. “I’m not scary. Everyone keeps being mean to me. The bigger bots push me and call me names, even when I don’t do anything to them.” 

“Why don’t you kill them?” replied Overlord like it was obvious. He had slithered onto the berth, laying sideways lazily behind Nightlight, his legs curled so he could fit. Luckily the slab was medium size, having not been made specifically for Nightlight, but it still creaked horribly under Overlord’s weight.

“I can’t do that! I’m not strong enough like everyone else. I’m not big. I can’t beat them up....I...every time I try to fight back,” her face scrunched up and her voice became even higher pitched and even wobblier, “I...I get my- my tailpipe kicked!”

Nightlight choked and whined, stifling a staticky sob in her forearms. She clearly felt so strongly about these simple tussles that it pained her enough to cry. The poor thing. The blue mech brought her into his chest with his big servo like a hockey stick to a puck.

“Don’t cry,” Overlord cooed, “I’ll give you advice.” Nightlight peeked at him from her arms. “You can’t beat them up and you can’t grow any. But that doesn't have to stop you.” 

“When I walk around, everyone moves. As if I have a force field. Nobody gets in my way.” He gestured to Nightlight with his chin. “Why do you think that happens?” She looked away again, not in an attempt to hide her tears, but in thought.

“Um….because you’re really tough,” the moped said, “And, um, you’ll, maybe you’ll beat them up if they’re mean to you.” Nightlight always said things like “maybe” when talking about if Overlord would do something violent or not. Like she wasn’t sure if he was a bad mech, or she didn’t want to accuse him of anything. How kind.

“Yes, that’s true. I’m very tough.”

“But I’m not...it wouldn’t work for me. I’m not really-”

“Yes it would.”

Nightlight stuttered a few syllables of denial before resorting to looking at Overlord with a tipped helm in confusion. Overlord couldn’t help but laugh.

“When I walk around places where nobody knows me,” he said, “Where nobody knows I could mash them to a slurry, my force field still works. That is because I hold myself a certain way. I hold myself with an expectation that everyone fears me,” the duocon puffed out his chest plating a little, and it made a “clink” sound when it tapped his tiny companion, “With confidence. Confidence in myself and that the force field will work no matter what.” He smiled triumphantly. “Lo and behold, the seas part.”

Nightlight looked at him like he was the coolest mech on Cybertron. She had uncurled and instead was facing him, sitting with her knees forward and her pedes behind her. “So,” she spoke with a bit more pep in her voice, “they don’t know you’re strong...but they still kind of know you’re strong because you walk so confident.”

“Exactly.”

“But...but I don’t think I could do that.”

“Why not?”

“Cause what if I pretend to be strong and then they figure out I’m not and they beat me up?”

“If you walk with enough confidence, they won’t challenge you. And if they do, you threaten them. Then they run off like little glitchmice, with not a finger lifted.” Overlord waved his free servo as he talked, and Nightlight rubbed her fists on her optics and cheeks as he spoke, scooting closer to him.

“If I was injured in a way that left me unable to fight, but able to use my words, I would still win. In that moment, when I threaten them, it is not pain they are afraid of,” he explained, half-lying. Overlord loved to taunt, but he rarely threatened. If someone challenged him, most of the time he smashed their head in immediately. “I’m not touching them. They aren’t experiencing it. What they fear is the prospect of pain. All you have to do if you want to scare them off is make them believe you’ll rip them apart.”

Overlord had a feeling he was losing her, given her big optics staring at him. Or maybe that’s just how she looked. He poked her in the chest with acquired gentleness. “I can teach you. I can make you like me,” he said, “You can be intimidating. And nobody will ever bother you.”

“Really?”

“Really.”

-------------

Nightlight had such a tall stack of datapads in her arms that the top of the pile concealed her face. She had to stumble slowly in the hall and occasionally, carefully glance to the side of the stack to see that there was nobody in front of her, and issue out “excuse me”s and “I’m sorry”s accordingly.

Overlord’s lessons on being braver and more intimidating had not yet been put into practice. Fortunately so, in Nightlight’s optics. The two wheeler hadn’t come across anybody who felt like bullying her for a whole week, and she was hoping her lucky streak would last forever. She didn’t want to try out what she had learned from the phase-sixer, even though out of everybody on the ship, he was probably the best person equipped to teach it. Nightlight didn’t want to mess up.

Turning a corner like an old bot driving slow on the highway, she scooted forward on her pedes and shifted her grip on the datapads. She felt the datapads sliding, and she “eep!”ed as she steadied them, and they settled back in place. “Whew,” she whispered.

Despite her dearest wishes on shooting stars, Nightlight didn’t get much time to be relieved. Just as she found her footing again, some mecha decided to sneak up beside her and stick their pede out. The minibot made a brief yelping sound before landing on her front, some of the datapads breaking her fall, layered like a deck of cards. The rest scattered and clattered around them. The floor and the pointy edges of the datapads poked and scraped her chassis and hands.

“Oops,” said the bot above her, laughing. She recognized him from his voice. She didn’t know his name. “Heh, watch where you’re going, squirt.”

Nightlight stayed on the floor for a moment, facing down, steeling herself and her urges to cry. It was action time. She got up, pushing herself with her tiny servos, whipping herself around with gusto and pointed her finger right in the mech’s face. 

“How about YOU watch where YOU’RE putting your STINKY PEDES, BUSTER!!”

The Decepticon stood with his mouth agape. He stared at Nightlight like she had grown another helm. Her being any bit of aggressive was pretty equivalent, really. “Wha-” he snorted, before barking out laughter that scraped Nightlight’s audials from being too loud. “What’s your problem, Autobot model? You think you can just waltz up in here and get sharp with me? You lookin’ to get pummeled?”

“You’re the one asking for a beating, stupidhead!” Nightlight yelled back with surprising volume, looking up at the considerably taller mech, even stepping towards him with gritted teeth. She stomped at him and almost jumped towards him doing so, looking like a dog trying to chomp at a chewtoy placed above it. “Get out of my way or pick these up,” she pointed at the datapads now, “and take them to room L2400! Or I’ll rearrange your face so much you’ll have to get your whole head replaced!!”

Nightlight, venting hard, felt equal amounts proud and equal amounts terrified. She had used the strategies Overlord taught her! Nightlight had tried her hardest and her best, put on her scariest face with her scariest voice. Hours of practice with Overlord, of him showing her how to be unabashed and angry, were coming to fruition.

The mech looked like he wanted to say something else, but instead he froze up, his expression contorting into a wide-eyed frown. He frantically vented, taking a few quick breaths before letting out what Nightlight could only describe as a “squawk”. His helm darted between facing forward and towards the datapads. He was shivering so hard his chassis rattled. Was it working?

“I’ll do it! I’ll do it, I’ll pick up the datapads!” he threw himself at them, scooping them up like they were shards of his spark. Nightlight couldn’t help but look on with her mouth in the shape of an O. “Just leave me alone, okay?! I don’t want any trouble anymore!”

“Really?!” asked the purple minibot, “I, I mean, yeah! And I don’t wanna ever see your dumb ugly face ever again, got it?!” She shook her fist at him, throwing in a growl to her intimidation tactics. It sounded like a baby cyberwolf.

Taking no time to look back or even respond, the mech rushed off with the datapads in his hands. He screamed and practically jumped in the air when Nightlight yelled “L2400!” to remind him of where he was supposed to be going.

When the bot disappeared in the hallway, Nightlight stood still. She seemed to start to gradually vibrate, before exploding in excitement, jumping around the hallway, squealing and screaming, dancing and throwing her fists all around. She did it! She did it! She was intimidating! She could stand up for herself! She didn’t have to be bullied anymore! She was a real Decepticon! 

Nightlight then felt a little guilty. She put her servo to her mouth and thought. The mech looked so scared. Had she been too mean? 

No, she hadn’t been. Overlord told her that she should stand her ground, go full force, and dish back exactly what her bullies were doing to her. There was nothing wrong with that. An optic for an optic, and then some. The moped bounced in place. Overlord would be so proud of her!

“...I have to tell him!” she said, out of breath. She dashed down the hallway despite this, giggling and cheering, back in the direction she came from.

As Nightlight skipped away, Overlord stood at the other end of the corridor, in direct line of sight of where her bully had been standing. He backed away into the darkness with a wicked smile.

[clericalkobold]

@tformers-secret-santa gift for @driftingstarryskies!

I basically ignored canon timelines and probably took a few liberties with characterization as well, but we do what we have to do, right? Hope you like this!

Of all the places Drift expected to find himself after being exiled from the Lost Light, a party hosted by Starscream (Lord Starscream, and wow, wasn't that a strange concept?) to celebrate a new planetary holiday wasn't on the list. And yet here they were.

So here he was, standing in a corner in an extravagantly decorated ballroom, watching his courtmate enjoy the festivities. Stopping on Cybertron had been Ratchet's idea, and Drift didn't regret agreeing. The medic's joyful reunion with his Amica had been more than enough reason to stick out any amount of tension with Starscream. Speaking of whom...

"You always were a wallflower." In his new frame, Starscream didn't loom over Drift quite so much. Drift didn't think that made him any less dangerous, but if Wheeljack's trust was good enough for Ratchet, it was good enough for him. "I would have thought Rodimus' influence would change that," the Seeker continued.

Drift's plating clamped down at the mention of his best friend. "I didn't want to distract Ratchet from this," he finally replied, if a bit stiffly, jerking his chin at the Amica pair chatting by the refreshment counter.

Starscream's posture relaxed the slightest amount. "Neither did I," he murmured, uncrossing his arms and crossing them again.

Drift gave him a searching look. "So," he said, "you and Wheeljack, huh?" He wasn't sure what surprised him more: the lack of death threats or the hint of a smile that passed over Starscream's face. "How long has that been going on?" he probed farther.

"A while."

"Really."

The faint smile became a smirk at Drift's dry tone. "About a year."

Huh. "It's going well, then?"

"Does that come as a shock?"

"A little."

Starscream barely bristled, but it was there in his EM field. Drift fumbled to smooth it over.

"I wouldn't have expected myself to get this far, either."

A disdainful huff signaled their conversation's return to normalcy. "And with the fearsome head of the Autobot medical division, no less."

Drift grinned. "Oh, so now you're turning this on me?"

"You must admit it's an unexpected coupling."

"That's the understatement of the day. We've only been together a week or so."

A serving drone stopped next to them in its path around the room. Starscream selected a flute of engex from its tray.

THANK YOU FOR *CHOOSING* SWINDLE'S CATERING SERVICE, the drone's screen flashed at them with a smiley emoticon before it zoomed off.

Drift waved after it. "What's with this "Chosen One Day" thing, anyway?"

The Seeker waved his hand airily. "Just a little thing I established last year. 'A day to express love toward each other,' that sort of drivel. It helps to keep the peace, I suppose."

"And you've been courting Wheeljack about a year?"

That smirk returned at full force. "What is this, an interrogation?"

Drift crossed his arms, prepared to wait Starscream out. "Just a friendly conversation. Surely Wheeljack's taught you about those by now."

"It was my impression that two mechs ought to be friends in order to call anything between them 'friendly.'"

Sporting a smirk of his own now, Drift gestured to their courtmates. The Amica Endurae had sat down at a table opposite Ironhide, where Ratchet was challenging the head of the CSF to arm-wrestling. Wheeljack had a datapad out and was taking bets from the nearest partygoers. "I think we can at least upgrade to acquaintances, being attached to a friendship like that. Call it relationship preservation."

Starscream actually laughed. Drift had never heard the former Air Commander laugh like that, in a short, soft sound without the harsh snicker of mockery. It was nice.

The Seeker took a sip of his engex before speaking. "To answer your—implied—question, yes, creating this holiday did have a role in starting our courtship."

Drift blinked. Starscream averted his gaze and continued.

"I haven't told many people this besides Wheeljack, but 'Chosen One Day' was meant to celebrate the titular 'Chosen One,' not the loved ones of individual citizens. Of course, I couldn't very well say that after the media flubbed my press release. And it was such a hit with the populace, my approval rating actually rose a percentage point." He smiled, a little ruefully. "So while I brooded at home, the whole of Cybertron and most of the colonies threw parties, exchanged gifts, and organized festivities without any input from their government beyond that one speech."

"Tragic," Drift couldn't resist saying.

Starscream snorted. "It certainly felt so at the time." His EM field flared just enough that Drift caught a hint of amusement and something else, something almost fluttery.

"So then what happened?" he asked, because that was a very specific kind of fluttery and because whatever was happening right now, whatever was the root of this openness in Starscream's demeanor, couldn't possibly happen often.

"Wheeljack showed up at my door." Starscream replied. "He brought others, too. Ironhide. Blurr. Rattrap. Others." His voice hitched. "They were so impressed by my moment of 'selflessness' that they'd decided to throw a little party just for me."

Drift tried to resist biting his lip in sympathy as the Seeker rebooted his vocalizer. He knew what that sort of acknowledgement meant to Starscream, at least a little. And finally getting it under false pretenses, well, that he wasn't sure of.

"It was... nice," Starscream said, optics fixated on the glass in his hand. "A little too nice. I slipped away a few hours in, went out on the balcony to clear my head."

"And?" The query escaped his vocalizer without a thought to precede it.

"He came and found me. We talked—to this day I couldn't tell you what it was we talked about, but we talked, and somehow," now he was smiling again, his tone shifting into amusement, "the slagger wheedled me into going on a date a few nights later. Obviously nothing really changed overnight, but Wheeljack has a way of... well, he just has an effect on me."

"A good one," Drift suggested.

Starscream ducked his head. "The best."

Drift was at a loss for what to say next. That kind of honesty, coming from Starscream... wow. He leaned back against the wall, crossing his arms.

"You know I was exiled from the Lost Light."

"I had heard something of the sort, yes."

"When the truth got out, Ratchet left them to bring me back."

Starscream set his now-empty flute on the returning drone's tray and picked up a new one. He held it out to Drift. "And yet here you both are."

Drift accepted it, taking a sip before shrugging. "It wasn't as far off course as we would have thought."

He thought back to that conversation on Ratchet's shuttle, when he still hadn't agreed to return and Ratchet was "getting real fragging tired of this slag," when the medic had finally snapped that no, it wasn't about guilt, it wasn't about justice, it was about Drift, it had always been about Drift, and he "slagging well should have gotten on that Primus-forsaken pod" when Drift first left because he— "Because you what?"

—and that kiss and every one after still tingled on Drift's lips, would always linger there along with the memory of Ratchet's reply that didn't come until they were sitting in the cabin the next morning, planning their course back to the Lost Light—

—because he loved Drift, he fragging loved him and he wouldn't ever let that go, would never fail him again, and that statement led to more conversations, some harder than others, until they'd hashed it out as best they could and everything else could wait until they were home.

Drift realized that Starscream was waiting for him to continue. "I wasn't quite ready to go back to the Lost Light," he confessed. "Stopping here was his idea, but that was the reason. And then Wheeljack was here, and seeing them together... It helps. More than I would have thought." It made him miss Rodimus, but that wasn't something he was ready to unpack. Not to Starscream, not to Ratchet, and not to himself. But something in his spark had lightened a little, and he was beginning to miss home in an anticipatory way. "Besides, it's just plain adorable," he added with a nod across the room. The Seeker turned in the direction of their partners again.

They'd missed the end of Ratchet's arm-wrestling challenge, and the pair had returned to the refreshments table. Wheeljack was speaking animatedly to a small gathering of mechs. Ratchet stood next to him, resting an elbow on his shoulder while downing engex from... was that a stein?

"Of course he brought that thing. Ever since Swerve's opened..." Drift's exasperated groan trailed off into a fond chuckle.

A strange expression passed over Starscream's face. His attention remained on the scene across the room, but when he spoke, it was quiet and directed at Drift. "I'm starting the ritus tomorrow."

Drift goggled. "The Conjunx Ritus?"

"Can you think of another I would bring up?" Despite the sarcastic wording, his voice was soft, serious, completely unlike any side of him Drift was used to, even after the day's surprises. "Yes, the Conjunx Ritus."

He tossed back the rest of his engex to stall for time to think of a reply. It backfired when the fizzy liquid accidentally flowed into his ventilation system. He sputtered as highgrade mist shot from his vents.

"Scrap," he wheezed, "sorry, sorry, let me just—" Starscream made a face, beckoning another service drone over, this one with microfiber towels hanging from outstretched arms. He handed one to Drift and took another to wipe off the side of his chassis that had been facing the speedster.

"Could you possibly draw a little more attention to our conversation?" he sighed, but the corners of his mouth seemed unable to stay turned down.

Drift sheepishly handed his glass and towel to the drone. "Um. So, what are you thinking you'll do for your three Acts?"

The Seeker set down his own towel and shooed the drone away. "I'm taking him to the new observatory in the morning. The place hasn't been opened yet, so we'll have it to ourselves." He gave Drift a wry smile. "The first and second Acts are precisely none of your business, but I'm rather proud of the third. I just haven't figured out a way to unveil it while I'm occupied."

"You need a wingmech?" Drift returned the smile. This day was taking all sorts of unexpected turns. "Isn't there anyone else you'd rather have help?"

"You seemed invested enough. I may have let Wheeljack into my confidences, but I wouldn't trust Windblade with anything along these lines. She's got some pretentious ideas when it comes to my personal life. At least you, I know, can follow my instructions rather than interfering with them."

Drift grinned. "Well, I think I do know the value of letting things happen at their own pace." He held out a hand. "I'd be honored to help out any way I can."

Just they shook on it, a familiar hand latched onto Drift's shoulder.

"Mind if I cut in?" his favorite voice purred in his audial.

Drift released Starscream's hand and turned to face Ratchet. "You want to dance?"

His courtmate offered a lopsided smile that set his face on fire. "If you can stand to tear yourself away from your new friend."

Starscream was too busy rolling his eyes to notice Wheeljack sidling up next to him. The engineer snagged one of his hands, and his wings shot up in surprise. Wheeljack captured the other one and tugged him down to touch their forehelms together—then abruptly pulled him out onto the dance floor.

"Doesn't waste any time, that one," Ratchet murmured, slinging his arm around Drift's waist.

Drift snickered. "It's pretty flattering that you can step away from your Amica just to spend time with your courtmate."

"I'm a lucky mech to have both."

"Come on, you sentimental spark, are we going to dance or what?"

As Ratchet let him guide them both out to join their friends (what a weird, if not unpleasant, thought that was to include Starscream in), Drift caught up one of his hands and signed a quick message.

"Thank you."

[thousandmaths]

how to write a talk: a case study

I found out that I would be giving this Thursday’s combinatorics seminar talk on the Friday before... so I had very limited time, and I decided to try something different. That process left me with a lot of scrap writing, so I decided to use it to sketch my entire process from “know the results and only idle thoughts on the presentation” to a finished product. This is in part for other people who might be interested in the talk-writing process, and in part so that I can reflect on how to do better.

Parameters of the talk:

We have the room reserved for 80 minutes. 

But the talk doesn’t start until 4:40pm so it’s considered polite to finish in the usual 50 minute timeslot that an ordinary class would take. That leaves tons of wriggle room, so usually a talk can take the full 50 minutes and a bit more, with questions going for as long as seems productive.

Most of the audience are combinatorialists, but almost none of them are graph theorists (myself included, frankly). 

There are usually 1-2 undergrads, half a dozen grads, a few postdocs and occasionally a professor.

The basic structure of this post: after the introduction, there are three main parts. The first is fairly technical, where I give brief summaries of the talk in increasing length, commenting in-between on how I might expand for the next one. The second is much less so, since I am now at the stage where I’m preparing orally, and the notes are just records of key points of improvement.  The third section is similar, but focuses specifically on the spit-and-polish stuff carried out <36 hours before giving the talk.

------

The talk in one sentence:

We determined the expected matching polynomial for random lifts of cycle graphs (also go check out GRWC).

Words to define:

matching polynomial

graph lifting

“expected” in this context

The talk in one paragraph:

The matching polynomial of a graph $G$ is $\displaystyle \sum_{k\geq 0} (-1)^k \mu(G,k) x^{n-2k}$ where $\mu(G,k)$ is the number of matchings with $k$ edges. Instead of trying to compute matching polynomials, we can do something harder: for a graph $G$ we say that a graph homomorphism is $H\to G$ is a $d$-sheeted covering graph if the preimage of every vertex in $G$ is $d$ vertices in $H$, and if it is a local isomorphism. We were interested in computing the “average” matching polynomials for $d$-sheeted covering graphs, and we succeeded in doing this when $G$ is a cycle.

(also go check out the Graduate Research Workshop in Combinatorics)

Places to expand:

classical matching polynomials of the cycle and path

a bit of background on covering graphs

a little less terse definition of a covering graph

still pretty ambiguous what “average” means

explaining the formula for $\mathcal M_d(C_n; x)$

The talk in one page:

Recall that for any (loopless) graph $G$, a matching is a collection of edges where no two edges share an endpoint. We denote by $\mu(G,k)$ the number of matchings with $k$ edges, and we define the matching polynomial to be a variant on the generating function:

$$\mathcal M(G; x) = \sum_{k\geq 0} (-1)^k \mu(G,k) x^{n-2k}.$$ In general, matching polynomials are fairly difficult to get our hands on: the function problem of computing $\mathcal M(G; x)$ is #P complete. However, for some of our favorite graphs, we get some of our favorite polynomials; in particular we recover the Chebyshev polynomials of both kinds via $\mathcal M(C_n; 2x) = 2T_n(x)$ and $\mathcal M(P_n; 2x) = U_n(x)$.

Because of some concerns in spectral graph theory, matching polynomials were generalized by Hall, Pruder and Sawin in 2015 to $d$-matching polynomials, which have a fairly straightforward-looking definition: 

$$\mathcal M_d(G;x) = \Bbb E_\lambda \mathcal M(G^\lambda; x).$$

where $G^\lambda$ ranges over all $d$-sheeted covering graphs of $G$. These are essentially covering spaces in the topological sense that also respect the combinatorial structure. Formally, we say that a surjective graph homomorphism is $H\to G$ is a covering graph if it is a local isomorphism (that is, $f:N(f(v)) \to N(v)$ is a bijection for all $v$).

[ Technically, we cannot average over the (proper) class of all covering graphs; so we should instead average only over those on a fixed vertex set— we do not consider these graphs up to isomorphism. ]

I worked on questions related to $d$-matching polynomials with Garner Cochran, Corbin Groothuis, Andrew Herring, Jamie Radcliffe, and Ranjan Rohatgi during the summer last year at the Graduate Research Workshop in Combinatorics. We were able to explicitly compute the $d$-matching polynomial of cycle graphs in terms of Chebyshev polynomials:

$$ \mathcal M_d(C_n; x) = \frac{\mathcal M(P_{nd+n-1}; x)}{\mathcal M(P_{n-1};x)} $$

Rough draft of outline:

Introduction

Graphs and matchings

GRWC advertisement

Matching Polynomials

Definition and examples

Classical orthogonal polynomials

Graph Lifting

Definition

A model for graph lifts ($S_n$)

Generalizing to other groups?

$d$-Matching Polynomials

Definition

Background: Ramanujan graphs

“Warm-up problem”

Main result

Statement of result

Chebyshev aerobics

Statement of key lemma

Proof of key lemma

Things to learn:

Can you get Legendre polynomials? Well? Can you?

What’s up with the other groups?

What’s up with Ramanujan graphs?

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Outline and notes after 3 preliminary runs:

(75-80 minutes on third run. Intro/1 should be shortened.)

Pre-written Text + Introduction

GRWC advertisement 

Define matchings and $\mu(G,k)$

Define matching polynomials

Matching Polynomials

Examples $C_6$ [pic], $C_n$, and $P_n$

Where are we going?

Covering graphs

Covering maps: Definition & Example [pic]

$d$-labelings; state space of covers [extend pic]

$d$-matching polynomials: Definition (& Motivation?)

Topological proposition & remarks

Results

“Warm-up problem” [pre-drawn pic]

Statement of result

Chebyshev aerobics and key lemma [pic]

Proof of key lemma [extend pic]

Comment on non-combinatorial aspects

Consider proving $A\cdot B = A\coprod B$. Do not mention isomorphism classes as an alternative state space. Do not do an example in Covering/3. Use the $\sigma$ in the old writeup for Results/4.

Regarding Covering/3/Motivation: I think it will be fine just to mention that this stuff gets used in spectral graph theory. You do need to know some of those details, though, since it should come up in questions if you do it right.

Intermediate stages:

Time spent 

Saturday: 9 hours. (Everything up to this point)

Monday: 2 hours. (Second timed run; talk is <65 minutes; a long interruption made it impossible to determine more precisely)

Tuesday: 3 hours. (Could not give full talk today; rooms were busy! Very nervous about the whole thing.)

We can completely remove the covering maps section. This could eliminate the pre-writing.

I’ve learned enough about the other groups to answer reasonable questions from Theo. It turns out that the finite complex reflection groups do indeed have a special role in the HPS proof. Also, I’ve learned enough about Ramanujan-ness to make a passing comment about them somewhere. I still have no idea if it’s possible to make the Legendre polynomials.

Also, any Cayley graph and any generalized Petersen graph can be written as graph covers! Wow! (Maybe do the ordinary Petersen graph as an example? Not the primary example though; you want a cycle with several edges.) Suddenly very excited about algebraic graph theory ;)

OH SHIT YOU FORGOT: You need to know how the “acyclic” polynomials came up in stat mech and then like chemistry or w/e.

Title and abstract:

Covers of Cycles and their Matchings

The matching polynomial is a variant on the generating function for matchings on a graph. Following new developments in spectral graph theory, these were generalized to “d-matching polynomials” in 2016 by Hall, Puder, and Sawin. In this talk, we will describe how to compute these polynomials for cycle graphs, stopping occasionally to observe the broader world of algebraic graph theory.

(submitted on Tuesday afternoon)

[ The title is kind of boring, but I couldn’t come up with anything better that was at all descriptive. ]

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Final Push:

Time spent

Wednesday: 7 hours. (Ended on a high note!)

Thursday: 0.5 hour.

I wrote a mathoverflow question and Godsil himself said he doesn’t know of any answer to the realizability of Legendre polynomials. So I’m not going to sweat it.

I gave a full presentation (for the first time since Saturday, yikes!) and discovered that my Intro/1 was terribad. Polished that up. Also smoothed over some other joints, including the reveal in Covering/3. Could be happier with my concluding sentence, but it’s serviceable.

Finalized timings on Wed. night (44–60 minutes):

Pre-drawing & pre-writing: 5 minutes

Intro/1 – Covering/2: 20 minutes

Covering/3 + comments: 10 minutes (can be shortened to 4 or 7 min)

Covering/4 – Results/2: 15 minutes

Results/3 – Results/5: 15 minutes (can be shortened to 5 min)

Replicated twice with error ±1 minute.

This gives three natural points in the presentation to take a breather: evaluate speaking speed and facetime with audience. Ask for questions if you need a longer pause.

Woke up late on Thursday and ate a big breakfast. Taught my classes and then spent a half-hour doing a run-through of the talk “sans writing”: just stood in front of a blank board, recited the words, and tried to focus on eye contact.

Final Outline:

Each number represents approximately one blackboard. The breather points are denoted by **.

Preliminaries

Introduction; GRWC advertisement [pre-written]

Define matchings, $\mu(G,m)$, matching polynomials

Examples: $C_6$ [pic], $C_n$, and $P_n$

Main Definition

$d$-labelings; definition & example [pic] **

Remarks on covering graphs

$d$-matching polynomials: definition & motivation **

Results

Topological proposition; remarks

“Warm-up problem” [pre-drawn pic]

Two theorems for cycles **

A proof in three lemmas

Main bijection [pic]; combinatorialization progress

The Talk Itself:

In the end, I went through the longest version of the talk that I practiced, and it still finished in about 47 minutes O.O [ Maybe I missed a few remarks from Main/3/Motivation, and I definitely was less verbose in Results/1 than in practice. ]

We had a smaller crowd than usual. One of the postdocs (who I’ve talked with, and who I respect a lot) made quite a few comments during the talk and asked many pointed questions afterward. This was the first time that I’ve ever given a talk where I felt challenged by an audience member. I think I kept composed, but it did shake me quite a bit, especially since the smaller turnout was already was working at my anxiety that maybe the material wasn’t so interesting. But I was thinking about it afterwards, and it occurred to me that if he really thought it was trivial and boring, he probably wouldn’t have been so vocal.

Still felt like I botched the conclusion. But the Q&A went very long and maybe served as a de facto conclusion— I hope the audience interpreted it that way, since it was a much better ending, frankly :P

Concept to Q&A: 22 working hours.

Afterwards I went shopping, bought a bottle of rum, drank with my roommate and played Starcraft late into the night. Good day, all things considered :)