Priyam found me and said she and Katie were about to start. I got my notebook and a bottle of Pilsner and pulled up a chair to watch.

Priyam hoped to import Katie’s test — for detecting when graphs are hyperbolic — to her preferred mathematical setting, the study of infinite-type surfaces. Transplanting an effective technique into a different environment like this is a common way new mathematics gets made. To do it, mathematicians need to understand what, fundamentally, makes the technique work. The conditions of the original setting won’t all be present in the new one, but hopefully they won’t all be necessary.

“You have to dig deeper to see what something really relies on,” Priyam told me.

Later in the week I talked again with David Futer, and he arrived at an analogy: Transplanting a mathematical technique is like trying to bake your favorite cake for guests who are allergic to eggs. “You’re hoping that you can still bake the same cake without the eggs,” he said.

Katie wore a thick wool sweater and a serious expression. She knew that in her work, on finite-type surfaces, the hyperbolicity test relied on nine axioms, or conditions. And she had bad news for Priyam: She would have to get by without eggs.

“OK, maybe I’ll just tell you now, the axiom I don’t think works in your situation,” she said.

Priyam didn’t see it yet. “I’m confused about this condition.”

Katie went back a step. “Do you know what a median algebra is? OK.” And she started to explain it. Priyam was encouraged for a moment.

Katie wasn’t so sure. “I’m still a bit worried.” Priyam acknowledged she was, too.

After half an hour they’d at least clarified the picture. Of the nine axioms, they knew for sure that one of them wouldn’t be available to Priyam. Yet even without it, they hoped to prove that a different set of conditions, called the coarse median axioms, still applied. Get those conditions, and you still get the hyperbolicity test.

But whether that was possible was more than they could figure out that night. In the background, another round of laughter shook the center table.

Lost in the Woods

On Thursday afternoon the postponed hike into town for Schwarzwälder Kirschtorte set off, under mostly clear skies. Before we left, Schleimer, the co-organizer, announced that by popular demand there’d be a bonus round of five-minute talks that evening, my last of the event.

About 20 mathematicians went on the outing. We split into two groups: One took the road into town, the other climbed up a forested hill onto logging trails, even though their condition was uncertain following the previous day’s storm. I joined the overland group and fell in beside Anna Parlak, a graduate student at the University of Warwick. We’d talked over Skype before the workshop, but I hadn’t seen much of her since I’d arrived.



Now we walked beneath tall pines down a muddy trail riddled with tire tracks. I searched for drier ground along the trail’s edge while trying to keep up with Anna, who seemed unperturbed by the slop. I told her I’d enjoyed the five-minute talk she gave the afternoon before. She’d mentioned that under particular conditions, two different things — types of polynomial expressions — seemed like they should always be equal. Yet she knew of examples where they weren’t. After the talk another mathematician had asked her to explain why she thought the equality should be inviolable; in answering the question, she realized why it failed.

“Sometimes you don’t ask yourself the correct questions, but people can ask them for you,” she said.

The group paused at an intersection and studied a cluster of trail signs. No one was confident which way to go. Someone made a joke about topologists having a bad sense of direction, which got less traction than in-group jokes sometimes do. Finally a few people started trickling up the leftmost trail, so that’s the way we all went.

I saw Yair up ahead and hurried to catch up with him. I asked how he and Autumn were doing controlling the volume growth in their problem. He said it had been up and down: On Tuesday, when I first sat down with them, they thought they had a way to do it, but by Wednesday they realized it wouldn’t work. He said they were now looking for a new approach, and when I circled back to Autumn later that night she thought they might have found one.

As we started the descent out of the woods, I asked Yair about that baking metaphor with the eggs. He said that, for him, doing math was more like wandering in the fog. Sometimes you get where you want to go; other times, the air clears just enough to confirm that you’re going the wrong way.

Metaphors abound when people talk about doing math. You’re exploring a continent, or building a treehouse, or baking a cake. Or wandering in the fog. That’s partly because the primary experience of high-level mathematics is just about impossible for non-mathematicians to perceive directly. But even among experts, effective communication often requires allusion. Mathematical ideas are subtle and complicated. Expressing those ideas is like trying to put a powerful emotion into words or, to draw yet another analogy, like narrating a dream you’re rapidly forgetting.

“They are these vague, inchoate thoughts that are not even formulated at the level of language,” Futer told me.

Given the difficulty of the communicative task involved in collaborating on new mathematics, it helps to have time together in the woods to sort things out.

Impressions and Next Steps

The Schwarzwälder Kirschtorte was dry, as I heard it would be. But I ate it all. I’d had my fill of the woods, however, so for the return trip I joined the road group. We walked out of town along the river Wolf. The lowering sun glowed off the steep green pastures on the opposite hills. Beyond, the sky was so blue it felt like an overwrought reproduction of our own planet.