Chapter 16: The Conversation
I told Mum and Dad on Sunday morning over breakfast. I'd been awake since six trying to figure out how to say it and in the end I just said it.
"A professor from Cambridge emailed me last night. I think she wants to set up a time for me to visit."
Mum put her toast down. Dad looked up from his tea.
"Hold on," Dad said. "A professor. From Cambridge. Because of your maths paper?"
"Yeah. She read the proof and she wants to talk to me about it. I think they want me to come down and see the department, see if it's a good fit. She said I can bring family."
Mum was staring at me. She hadn't moved. Her mouth was open but nothing was coming out.
"Mum?"
She tried to speak and it came out in bits. "Liam, I..." She pressed her hand over her mouth and her eyes went red all at once, not slowly, just red, and then she was crying. Shoulders shaking, the works.
"Mum, come on, it's not even confirmed yet, they just want me to—"
"I'm so... I can't... after everything that happened..." She was trying to talk through it but the words kept breaking apart. "My son... you were in that hospital and we didn't know if you were going to... and now you're sitting here telling me Cambridge wants you."
Dad moved his chair closer and put his arm around her. She leaned into him and kept crying into his shoulder. He rubbed her back with one hand, slow, steady, like he'd done this before. He probably had. He looked at me over her head and gave me a small nod.
She pulled back after a bit and wiped her face with both hands. Took a breath. Tried again.
"I don't understand how this happened," she said. Her voice was still shaking. "I don't. You went into that hospital and you came back and everything was different. You're smarter, you're more focused, you're doing things I can't even understand. And I know I should just be grateful and not ask questions." She sniffed hard. "So for my own sanity I'm going to say it's because of the coma and we're going to leave it at that. Alright?"
"Alright, Mum."
"Good." She wiped her eyes again. Then she started crying again and Dad just sat there holding her and I sat across the table watching my mum fall apart in the best possible way.
I emailed Shaw back that afternoon. Kept it short. Thanked her for reading the paper, said I'd be happy to visit at her convenience.
She replied within the hour. Could I come to Cambridge on Thursday? She'd arrange a meeting with the mathematics faculty and a tour of the department. I could bring family if I wished.
Thursday. Four days.
"Four days," I said to Biscuit, who was asleep on my desk chair. "I'm going to Cambridge in four days. What do I even wear to Cambridge? Do I wear a suit? I don't own a suit. Jake has a suit but Jake's suit has a cigarette burn on the sleeve and smells like Jägerbombs."
Biscuit didn't have an opinion on this.
Mum did. She'd been preparing for three days already, looking things up on her phone, writing questions in a little notebook, asking me things I didn't have answers to. On the third day she called me from across the house and I knew my afternoon was gone.
"LIAM! CAN YOU COME HERE FOR A SECOND!"
I could hear her from my room. The house was not that big. I walked over to her bedroom where she had three shirts laid out on the bed and was standing there with her arms folded like a general inspecting troops.
"Mum, you don't have to scream like that. This house isn't exactly a mansion."
She looked at me like I should have shut up about three words ago. "Don't get smart with me. This is important. You need to make a good first impression. Yes they like you because you're good at maths, fine, but this is Cambridge, Liam. People with a lot of money and probably a dress code. So you are going to stand here and try these on until I'm happy. Am I clear?"
"Yes Mum. Loud and clear." I gave her a little salute, the cheeky kind, and she picked up a shirt and threw it at my head.
"Ow! Mum, that actually hurt."
"Good. Put it on."
I tried on the first one. She made me tuck it in, untuck it, tuck it again, and then decided she didn't like the collar. I tried the second one.
"Mum, come on. These all look fine on me, don't you think? I could wear any of these and nobody's going to care."
She didn't even look at me. Just shook her head and went to the wardrobe to get more options. I stood there with my arms out like a scarecrow while she held shirts up against my chest and squinted at them.
"This one," she said finally. "But only if it's ironed properly."
"I can iron it."
"No you can't. I'll do it."
That was fair. The last time I'd ironed something I'd melted a patch on the sleeve of my school shirt and she hadn't let me near the iron since.
Dad took the day off work. He didn't make a big thing of it. Just said "I'll drive" on Wednesday night and that was that.
The drive was about two hours. I sat in the back seat watching the motorway go past. Mum was quiet for the first ten minutes, which was unusual, and then she couldn't hold it in anymore.
"What do you think she's like? The professor? She sounds strict. The email you showed me was very... efficient. Do you think she's going to be testing you? Because if she is I think we should have prepared more. We should have looked up what kind of questions they ask. Do they ask questions? Is it like a job interview? Liam, why aren't you more worried about this?"
"Mum. Breathe."
"I am breathing. I'm just saying we could have prepared."
"She invited me. She read my proof and she invited me. I don't think she's going to give me a pop quiz."
Dad glanced at the rear-view mirror. "Just be yourself, Liam. If the maths is good enough they're not going to care what shirt you're wearing."
"See, Dad gets it. The shirt doesn't matter."
"Your father wore his work boots to our wedding," Mum said without missing a beat. "He's not the authority on dressing for important occasions."
"They were clean boots," Dad said.
"They were boots, Paul. At a wedding. In a church." She turned around in her seat to look at me and I could see her face doing that thing where it was trying to be three emotions at the same time. Worried and proud and something else underneath that she wasn't going to name. "Are you nervous?"
"A bit. Yeah. But the good kind, I think."
She reached back and squeezed my knee. "You're going to be brilliant."
We arrived just before eleven. Dad parked in a pay-and-display and spent three minutes reading the parking sign because he didn't trust it. Mum stood on the pavement looking at the buildings across the street with her mouth slightly open.
I stood next to her and for a second I didn't say anything either. I'd seen pictures. Everyone's seen pictures of Cambridge. But pictures don't give you the scale of it.
"Liam. This is... this is very..."
"Yeah. It is."
"These buildings are older than our whole street. Older than our whole town, probably."
"Some of them are medieval, Mum. Like, properly medieval. People were studying here before they invented the printing press."
She looked at me. "How do you know that?"
"I looked it up. I'm not completely useless."
Dad came back from the parking meter. He stood between us and looked at the college across the road. The stone, the windows, the gate with the coat of arms above it. He didn't say anything. Just put the ticket on the dashboard and locked the car and stood there for a few more seconds. That was Dad being impressed. You had to know what to look for.
The mathematics faculty was in a modern building on Wilberforce Road. A receptionist checked my name and made a phone call and five minutes later a woman came down the stairs.
Late fifties. Grey hair cut short. Cardigan over a blouse. No makeup, no jewellery except a plain watch. She looked like someone who had stopped caring about first impressions approximately thirty years ago.
"Liam Reed." She said it like she was confirming something she already knew. Her eyes moved to my parents. "And you must be Mr and Mrs Reed."
"Paul," Dad said. "And Sarah."
"Eleanor Shaw. Thank you for making the trip." She shook Dad's hand, then Mum's. Mum was gripping her handbag with both hands like she was worried someone might take it.
Shaw led us upstairs and through a corridor lined with photographs of old professors. She walked fast. Mum was almost jogging to keep up. Dad walked at his own pace.
She gave us a tour. Lecture halls, the library, the common room, the offices. She talked about the department like someone describing their own house. This is where you'd study. These are the people you'd work with. This is the kind of research we do here.
Mum touched everything. The bookshelves, the doorframes, the back of a chair in the library. Like she needed to confirm it was real.
After the tour Shaw took my parents to a meeting room and introduced them to someone from admissions and someone from the finance office. "They'll go through the practical details with you while Liam and I have a conversation." She looked at Mum. "Everything about funding and housing will be covered. Please ask whatever you need to ask."
Mum nodded. She looked terrified. Dad put his hand on the small of her back.
"We'll be fine," he said. To Mum, not to Shaw.
Shaw POV
I took Liam to my office. He sat in the chair across from my desk and looked around at the bookshelves and the papers covering every surface and the mug collection on the windowsill.
He was nervous. Trying not to show it but I could see it in the way he sat, too straight, too still.
"Relax," I said. "This isn't an interview. If I didn't already want you here I wouldn't have called."
He let out a breath and some of the stiffness left his shoulders.
"Your proof is being discussed in every combinatorics department I'm in contact with," I said. "Calloway at MIT. Weber in Bonn, who worked on the conjecture himself. They all want to know the same thing. How did you get to the sign-flip? The paper shows the result. I want to know how you found it."
He thought about it. Not performing thought. Actually going back through whatever had happened in his head.
"Everyone who'd tried to solve the conjecture used the same matrix," he said. "The standard one. And it never gave you what you needed. The numbers you got out of it were too small. That's where every attempt got stuck." He shifted in his chair. "I kept thinking, what if the matrix itself was the problem? Not the graph, the graph was fine. But what if you changed the matrix, flipped some of the values, and that changed what came out the other side? I tried a few versions that didn't work. Then I tried one where the changes followed the shape of the graph itself. And that one worked. The numbers shifted to exactly where I needed them. After that it was just applying a known theorem and it all fell into place."
No showing off. Just walking me through it step by step.
"How long did it take?"
"About a week, start to finish. But the actual moment where it clicked was one evening. I'd been thinking about it for days and it came together while I was writing notes at two in the morning. I was supposed to be going to sleep. My mum would've killed me if she'd known I was still up."
A week.
I opened my desk drawer and took out a sheet of paper. I'd written a problem on it that morning. A question about colouring in graphs and the boundaries you can put on them. I'd been circling it for a few months without finding a clean way in. I wanted to see how his mind worked.
"I want to show you something," I said. "This isn't a test. I'm curious how you'd approach it."
I handed him the paper. He read it. Read it again. Leaned back and looked at the ceiling.
"Can I have a pen?"
He turned the paper over and started writing. Not equations. Words. Fragments. Working out what he knew and what he didn't.
After about two minutes he frowned, crossed out a line, wrote something else, and started on the maths.
I sat and watched. The office was quiet.
He worked for about fifteen minutes. At one point he stopped, said "no, that's wrong" to himself, scratched out half a page, and started again from a different angle. Talking under his breath. "What if you... no. Okay, but what about the other version of the graph... yeah, that's better."
He put the pen down.
"I can't solve it. Not here, not in fifteen minutes. But I think you're looking at the wrong version of the matrix. If you flip to the other version of the graph and look at what happens to the numbers there, the bound on the colouring should get tighter. It's similar to what I did with the hypercube but the shape is different so the same trick won't work directly. You'd need to change it." He paused. "But the idea is the same. Change the matrix, change the numbers, use the interlacing theorem to force the result."
He'd found in fifteen minutes a connection that I had spent two months failing to see. I hadn't considered it because I'd been thinking about the problem within the framework I already knew. He hadn't. He'd walked in from outside and seen a path that I'd walked straight past.
"That's a very interesting direction," I said.
He smiled. A bit embarrassed, a bit pleased.
"I should tell you that I've been working on a variant of that problem for some time. The approach you've just suggested is not one I had considered."
His eyes went wide for a second. "I'm sorry, did you just say you've been working on this? And I just... in fifteen minutes..." He sat back in his chair. "Right. Okay. That's... I don't know what to say to that."
"You don't need to say anything." I looked at him. "Liam. I don't say this often. You have an extraordinary mind. The way you approach problems, the way you move between ideas, the speed at which you see connections that others miss. I can teach technique. I can teach rigour. I cannot teach someone to see what you see."
His hands were flat on his knees.
"I want you at Cambridge. I want you in this department. We will arrange funding to cover everything. Tuition, accommodation, living expenses. Your parents will not need to worry about the cost."
"Everything? Like, all of it?"
"All of it. Shall we go and tell your parents?"
He was already out of the chair.
Liam POV
Shaw walked with me back downstairs to the meeting room. Mum and Dad were sitting at the table with the admissions woman and a stack of forms that looked like it had grown since we left. Mum had her notebook out. She'd been writing things down.
She looked up when we came in and I could see her trying to read my face. I couldn't help it. I was grinning.
"Mrs Reed," Shaw said. She pulled out a chair and sat down at the table with them like she did this every day. "I've just had a very good conversation with your son. I'd like to offer him a place in the mathematics department at Cambridge, under my supervision."
Mum's hand went to her mouth.
"The university will provide full funding," Shaw continued. "Tuition fees, accommodation, and a living expenses stipend. There will be no cost to your family."
"No cost," Mum said. Her voice was barely there. "None at all?"
"None at all. Liam's work is exceptional. The department is prepared to fund him fully. This is not something we do often, Mrs Reed. I want you to understand what that means."
Mum looked at Dad. Dad looked at Shaw. He hadn't said anything since we'd walked in.
"What does that look like, practically?" Dad asked. His voice was calm. Steady. The kind of voice he used when he was making sure he understood something before he reacted. "Room and board. Books. Travel home. All of that's covered?"
"All of it. The stipend covers living expenses including travel. Accommodation is provided through the college. He'll have a room, meals in hall, access to all facilities. The only thing we ask is that he does the work. And based on what I've seen today, I don't think that will be a problem."
Dad nodded once. Slow. Then he looked at me and something shifted in his face, not a smile exactly, more like something loosening behind his eyes.
"Well," he said. "Alright then."
Mum was crying. Again. She was pressing her notebook against her chest with one hand and wiping her eyes with the other. The admissions woman quietly slid a box of tissues across the table.
"I'm sorry," Mum said to Shaw. "I'm not normally like this. It's just... he was in hospital. Not that long ago. And we didn't know..." She stopped. Took a breath. "We didn't know if he was going to be okay. And now you're sitting here telling me he's going to Cambridge on a full scholarship. You have to understand what that sounds like to someone like me."
Shaw looked at her. Something changed in her expression. Not softer exactly. More like she was seeing something she recognised.
"Mrs Reed. Your son has a gift that I have not seen in thirty-four years of teaching. Whatever happened to bring him to this point, he is here now. And I intend to make sure he has every opportunity to become what he's capable of becoming."
Mum couldn't speak. She just nodded. Dad reached over and took her hand.
Shaw stood up. "Take your time with the paperwork. There's no rush today." She looked at me. "Liam. I look forward to September."
She shook my hand. Then Dad's. Then Mum's, and Mum held on for a second too long and Shaw let her.
We stayed for another half hour while Mum and Dad went through the forms with the admissions woman. I sat in the corner and pretended to read a brochure about college life but I wasn't reading it. I was thinking about the problem Shaw had given me. The other version of the graph. The numbers that changed when you changed the matrix. I wanted to go back upstairs and work on it right now.
We drove home. I sat in the back seat with my phone in my hand. Shaw had already emailed. Practical details, more paperwork, a reading list. One line at the bottom that wasn't about logistics.
"I look forward to working with you, Liam. Welcome to Cambridge."
I read it four times. Then I put my phone away and watched the road. Mum was crying. Dad was driving. And I was going to Cambridge.