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The Equation

Three problems are to be solved by the students of the math lecture. Only the third one seems to be especially hard, impossibly so. Will it ever be solved?

Monday was a rush, they had moved the lecture to a new hall and because she got lost in a book about Fourier, she found out too late, she had to run, but still only made it to the last minute, just as the professor was finishing up.

Hastily, she scribbled down the homework problems, two from the left, one from the right side of the blackboard, then checked her schedule and cursed - she would have to run all the way across campus to catch the next lecture.

This semester was killing her, it was the third and instead of slowly getting the hang of study, she felt it got worse every day. The summer she had spent mostly holed up in her dorm room or in the library, reworking, rechecking the last semester that was mostly a blur, always trying hard not to fail. Not that her grades were that bad, but she had just gotten lucky, right? The others, she was sure, had gotten all As, because come on, maths really isn’t that hard, is it?

She forgot about the math homework all the way until Wednesday evening, when she cleared out her backpack and found the paper, somehow crumbled under a book. She straightened it out on the desk and peered at the three equations. One was different, she spotted it immediately, although she didn’t know why it had come up. Surely as preparation for the next lecture, so it should be ok to handle.

It took her about two hours to solve the first two problems, she had to look up and go back, but in the end she felt pretty good about it. The last one, though, she had a vague idea, connected to something from a book she had been reading earlier this week, or maybe over the summer? She scanned her calendar and it seemed OK to postpone the problem to Thursday, or maybe even the weekend, after all, the lecture was only Monday and one more hour would be reasonable.

Thursday night, she lay in bed, thoughts from experimental physics swirling through her head, theorems and concepts. She could see them with her eyes closed. Next door dorm held a party and the thumping music surely didn’t help. She could go over and grab a beer, she vaguely remembered her dorm mate inviting her. No, it would be terrible, she felt so out of energy and interest that she saw no reason to drink, sit alone (like most of the times) and watch others kiss or do stupid drunken bets. But to occupy her mind, she grabbed the math homework and tried to get as far as possible just using her head.

What was it about this problem that was so hard? The first steps seemed easy, she connected, reconfigured, rethought, but suddenly there was a … gap. She tried different approaches, what had she learned so far? At one point, the numbers blurred and she couldn’t hold it in her memory any longer.

Friday, after she had woken up with the math notes in her hand, she casually asked her dorm mate if she had finished the homework and received a confident reply - done and dealt with on Monday night. She sighted. What was she telling herself - that anyone could study math? Clearly, she was not meant for this. Too proud to ask to see the solution, she gritted her teeth and gripped the pencil.

Saturday she was the first in the library, tackling this problem with support from the old masters. She gave a triumphant “Yes!” that earned her a dismissive look from the librarian when she overcame the first barrier and now had something quite elegant on the sheets. She was determined to finish this homework, it had become her nemesis. If she could do this, she could do the whole course. She adjusted her glasses. Well, now she had a beautiful next step, but after a couple of minutes of staring at the numbers, she felt all energy drain from her body. She had written herself in an alley without exit. This was unsolvable. Maybe she had copied the numbers wrong? No, she thought, all those years of diligently learning with her tutor, she didn’t make that kind of mistake, not her, not anymore.

On Sunday, she was invited to a sports event and dinner, so she had little time to work on the problem, at least not in writing, and several people asked her if she was alright, because she seemed absent and didn’t contribute to the conversation. At night, back in her dorm, the problem seemed as hard as ever, as if all her work had been for nothing. She couldn’t allow herself to fail, she had even found old study notes from the lecture to see if the professor had given out hints, but there was nothing. Everything else made perfect sense, and it helped to solve the first two problems. The third, while related, had no explanation anywhere. 

At 3 am, she awoke from the sound of her dorm mate noisily climbing into bed. It would be so easy to ask… just ask for the solution. But the snoring not thirty seconds later told her this chance was over. Tomorrow, Monday, again, packed with lectures and study group, all the other courses taking their toll, and even though focusing would be hard, she had to try to pull the switch in her mind.

Closing her eyes, she could suddenly see it. Euphoria ran through her body like someone had showered her with cold water. She dropped the pen twice as she tried to get the numbers on the paper. Of course, she didn’t see all the steps, but she found the crucial one. It came down to almost not more than one page in her tidy and erect handwriting. She didn’t bother to double check, she knew it to be 100% correct, but she smiled, because the key idea had been in a book that she read over the summer. Even though she didn’t remember which one.

Monday math and this time, she was so early, she even got a seat, having left the previous lecture before its official end. Everything was well as the professor started to solve the homework problems on the blackboard, he seemed bored today, his voice droned on. As he finished the second problem, he put down his notes and started the lecture. She was confused. For a few seconds, she hesitated, looking around at the others. They didn’t seem overly attentive, but no one was bothered that the third problem was missing. She inhaled and raised her hand. The professor caught the movement. He nodded, his face sour.

“What about the third homework problem?” she said, her voice small in the large lecture hall.

“What third problem?” he said, brows furrowed together.

She read the equation out loud. Even more confusion on the professors face. He shook his head.

“I’m not sure I understand you correctly. The equation you’ve read just now…” He sighed and rubbed his eyes. “I think I said it all last time, but maybe you had better things to do than attend my lecture.” She felt herself getting red from his attention, his admonition, his annoyance. “It’s called the Hodge conjecture, virtually every higher mathematician on this planet for the last eighty years has cracked their heads on it…” he paused, “...and failed.”

Suddenly, the paper under her fingers seemed to burn.