IMC [2]

I paid over £2,000 for that phone!

With that expense and the price I needed to pay for the lawyer, I was almost broke. $15,000 disappearing just like that.

I hope the deal goes through. Otherwise, I would be sad.

I walked alone around London for quite some time, visiting all sorts of places and trying to relax a bit before the 2nd day of the IMC would begin.

I ate some Fish and Chips at a local restaurant that had good reviews. It would be a huge loss if I didn't try some while I'm in London.

Though it wasn't the greatest thing I have ever tried.

Finally, I got back to my room around 12 PM and quickly changed my clothes to more comfortable ones before leaving for the IMC.

Same hall, same participants, but different professors.

This time I drew the number '86' and made my way towards the table.

This time around, we kicked off the competition without any jokes.

Surprisingly, it seemed like some folks were actually into the humor, given the noticeable disappointment etched across a few faces.

Silence.

We dove in.

The second-day problems held a similar level of difficulty, although calculus seemed to take the spotlight this time.

However, there was this one question, the final question. Most likely there to separate the great from the absolute best. The mathematicians from the math gods.

'Let G be a group and n ≥ 2 be an integer. Let H1 and H2 be two subgroups of G that satisfy'

'[G : H1] = [G : H2] = n and [G : (H1 ∩ H2)] = n(n − 1)'

'Prove that H1 and H2 are conjugate in G.'

Let me explain this real quick. Imagine you got this squad, right? Let's call them G. Now, G's got two crews, H1 and H2. They're like two dope teams with the same number of members.

And there's this chill spot where both teams can kick back, let's call it (H1 ∩ H2).

If you count how many ways we can split the whole squad into H1 or H2 you get the same number each time - n

Plus, if you check how many ways your whole squad can be split into just the chill spot you get - n times (n - 1).

Now, the big task is to prove that H1 and H2 are connected in some special way.

"Ok, cool". I mused under my nose.

If I could find this cool dude g in our big group G, it would be nice.

Imagine this dude doing some magic moves, like g⁻¹H1g, and it turns out to be the same as H2. That would be the sign they are conjugate...

I started by using a cool trick,

'n(n − 1) = [G : K] = [G : H1][H1 : K] = n[H1 : K],'

This was when I found out that the number of ways we can break down G into chunks with K is n - 1.

Well, now I could break down H1 into n - 1 subgroups, each with a different vibe, let's call them hiH1.

And when we do the same thing with H2, we see that it's also split into these n - 1 different sections, each represented by hiH2.

When H1 and H2 team up, forming this super duo called H1H2, this is when it gets interesting, it's like a massive mashup of n - 1 different sections, each represented by hiK.

Now, here's the cool part: these sections don't overlap. Each hiH1 section is unique, and the same goes for hiH2.

But because we can represent HiH2 as HiK...

We're saying that K, the hangout spot where H1 and H2 intersect, is actually a subset of H2.

I could finally conclude.

'H1H2 is a disjoint union of n−1 left cosets with respect to H2; hence L = G\(H1H2) is the remaining such left coset. Similarly, L is a right coset with respect to H1.'

So, for every cool dude g in L, you can say L is like gH2 or H1g. Every cool dude g in L connects in a special way, making H2 = g⁻¹H1g..

End of proof.

Puzzling through that one was quite a brain workout. But now, as the second day of IMC was coming to an end, it was time to kick back and chill on that chair.

I was feeling pretty confident that I aced all 10 questions. The results would come out next week, though.

...

In the evening, I joined the other participants from the USA to hop on the bus and check out the London Eye and Big Ben.

Now, I'm not exactly a heights enthusiast, but it was still a cool experience. Plus, it gave me some downtime to mull over a question I had in mind for Professor Terence Tao.

He was one of the greatest mathematicians of our time.

Whether it's geometric combinatorics, arithmetic combinatorics, analytic number theory, or partial differential equations... he was the go-to guy.

How the non-commutative geometry could alter the classical relationships between magnetic charge, electric charge, and Planck's constant.

That was the content of my question.

I knew that there existence of a magnetic monopole had not been proven yet, but the Dirac monopole quantization condition somehow matched the observations.

And get this, it's like the golden ticket when it comes to explaining why the charge of any object is throwing back an integer multiple of the elementary charge.

When we were getting back from Big Ben, Isaac got lost in the parking lot. This guy was something else...

We got back to the campus, and I made my way to the office where Professor Terence Tao was stationed.

Luckily, no one was there at this late time. The only soul in sight was Professor Terence, kicked back in his chair, catching some videos on his monitor.

"Evening, Professor."

"Good evening, what's on your mind, student?"

"My name is Max. Mind sparing a moment to answer a question I've got?"

"Well, what kind of question we're talking about?"

"It's more of an opinion than a question, actually", I said sheepishly

"Now that I think back, I saw your paper. It looked good at first sight... OK. Hit me with the question."

"Can I sit down?"

"Make yourself at home. This isn't even my office!", he said and laughed.

I sat down on the chair opposite to the Professor and began,

"I know that this is not Professor's specialization, but lately I've been going through a paper on magnetic monopoles in a non-commutative space-time."

Professor nodded like he knew everything about it.

"I was thinking about the consequences of controlling the quantum world on the Dirac monopole quantization condition. If we could correctly describe the non-commutative geometry, we could theoretically alter the relationships between magnetic charge, electric charge, and Planck's constant."

"That sounds interesting, but only at first sight. There are other physics laws that would prevent us from using this to our advantage. Think about the conservation of energy. Even if you somehow change the relationships, what would you accomplish?" - "This non-commutative space-time is a tough sell."

It seemed that the Professor didn't see any worth in what I was trying to preach.

I continued with passion, "By manipulating these fundamental constants, we might gain unprecedented control over energy conversion processes. More efficient energy transfers without violating the law of conservation could be possible"

The Professor arched an eyebrow. "Hold on, kid. You're suggesting we guide energy transformations with pinpoint precision. But you can't just conjure up energy from thin air. In reality, wouldn't that precision end up wasting more energy than it would save?"

I paused, considering it. "Fair point, Professor. The precision does come with its costs. But maybe we could come up with a way to balance between the costs and the benefits."

Professor Terence Tao leaned back. "It's an ambitious tune you're playing, kid. Like tuning an instrument. Precision with a purpose. But remember, the laws of conservation are non-negotiable..." - "But I dare you to try it", he smiled

"Challenge accepted Professor"

...

The next morning, I was on the flight back to New York.

We landed safely and took the bus back to Massachusetts. The team was clearly exhausted and everyone went straight to their beds, but this was not the end of the day for me...

Late in the evening, I got a call from Oliv

"Hey, love, are you back at MIT?", she asked

"Yes, I came back like 3 hours ago. Went straight to sleep, but now I feel a bit better..."

"It's good that you feel better because I wanted to invite you to my new apartment!"

"Oh? Have you moved already? That's cool. Send me the address and I'll be there soon. Is it far from MIT?"

"Nah, it's extremely close. You could even go on foot. It's less than 1 Mile away"

"I'll be right there then..."

"Nice! I'm gonna prepare then..."

"Prepare for what?", I had a weird feeling about this.

"You'll see", she said and disconnected the call.

I quickly put on some comfortable branded clothes and took off.

Just in case I bought a pack of condoms on the way. Not that I was expecting something...

I got to the apartment building she lived in and, of course, it was one of the expensive ones. Her parents would not buy her a cheap crib after all.

Walking up to a door on the second floor, I rang the doorbell.

"Come in!", Oliv shouted from the inside.

I opened the door, stepped inside, and proceeded to take off my sneakers.

"I'm here, Max", she said from inside one room.

I walked up to the slightly open doors and right away; I saw a cool fireplace that made the room feel super comfy and chill.

I walked further in and I saw Oliv with her beautiful long blond hair and slim stature.

She glanced at me with an intense look. I was worried I had done something wrong, but she just began to feverishly work at the buttons on her blouse.

She quickly peeled off her blouse and pushed her skirt down her legs so that she lay on the comforter, only in thin black panties and a black bra.

I could see the sparkle in her eyes and the hormone-fed desires were clearly showing as the black panties that she wore had wet spots and were getting half-transparent.

She turned her head towards me and said softly, "Come on Max..."

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