Summer Haag and Clyde Kertzer made major news in the math world while working on a summer research project

在2022-2023学年结束之前, graduate student Summer Haag and junior Clyde Kertzer were looking for summer research opportunities in mathematics, 他们的研究课题.

It was an REU (研究 Experience for Undergrads) with 凯瑟琳·斯坦奇他是博彩平台推荐的副教授 数学系, 詹姆斯·瑞卡兹, 同一系的博士后研究员, 这引起了他们的注意, as it dealt with a topic in which they both had an abiding interest: number theory.

“I knew in undergrad that number theory is what I wanted to do,哈格说。. “When I saw Kate and James were doing a number theory REU, I said, ‘That one! 我想要那个!’”

“I’ve taken a bunch of number theory courses here at CU that I’ve really enjoyed,Kertzer说, who withdrew his applications to other REUs when he was accepted into the one with Stange and Rickards. “我超级兴奋.”

The REU would explore a branch of number theory called Apollonian circle packings, 哪些是分形, 或者永无止境的模式, made up of infinite circles just touching each other but never overlapping.

Neither Haag nor Kertzer had much experience with circle packings.  

“我以前见过二次型, 我也见过莫比乌斯反转, 但我从未见过它们与圆形包装有关,哈格说。. “我很兴奋能学到这些东西.”

“我去图书馆找了一本书, 这是我能找到的唯一一本博彩app推荐圆形包装的书, 然后开始阅读,Kertzer说.

探索的空间

在REU的头几周, Stange and Rickards gave Haag and Kertzer the background information they’d need for the project and taught them how to use code that Rickards had developed to gather data on circle packings. After that, they gave Haag and Kertzer room to explore.

“We set out with a fun project idea that would give students a chance to experience research by collecting data, 寻找模式并证明它们,斯坦奇说。. “博彩平台推荐没有一个非常明确的目标.”

“We had a long list of possible problems to explore,” Rickards adds. “博彩平台推荐看不到真正的最终目标.”

改变了, 然而, when Haag and Kertzer’s explorations produced data that called a well-known math conjecture into question.

局部-全局猜想, 在过去的二十年里被广泛接受, predicts the curvatures of the circles inside a circle packing. 根据这个猜想, if a researcher knows the curvatures of a few circles in a packing (the “local” circles), that researcher can then predict the curvatures of the circles in the rest of the packing (the “global” circles).

一次又一次, evidence seemed to support the local-global conjecture, to the point that pretty much everyone familiar with it assumed it was true.

“Even though it hadn’t been proven, it was almost guaranteed to be true,哈格说。.  

两个数字而不是一个

但后来, 一边往里卡兹的密码里输入数字, Haag and Kertzer decided to do something that hadn’t yet been done. Instead of entering one number into the code, they entered two and looked at the resultant packings.

事情开始变得有趣起来. 数字, 根据局部-全局猜想, should have appeared together in the same packings didn’t.

斯特奇把这种情况比作监狱. It was as though the numbers that were supposed to be locked up had dug a tunnel when no one was looking and escaped.

Haag, Kertzer, Stange and Rickards all knew what this data meant for the local-global conjecture, which is why Rickards’ immediate reaction was to double-check his code for errors.  但是没有. 代码是正确的. 局部-全局猜想, on the other hand, was not.

在接下来的几天, Stange and Rickards put together a proof of their findings, 工作太快, so feverishly and so precisely that Haag and Kertzer couldn’t help but be inspired.

“这真的令人印象深刻，”Kertzer说. “That’s the point where we want to be as mathematicians.”

这四个人在预印本服务器上发表了一篇论文 arXiv with a title as unambiguous as its content is eye-opening: “The Local-Global Conjecture for Apollonian Circle Packings Is False.”

对于一个暑期研究项目来说还不错.

数学有趣的一面

But what Haag and Kertzer found even more gratifying than disproving a major outstanding conjecture was experiencing first-hand the creative side of mathematics research. 它不全是公式和规则. 这是直觉、探索和游戏.

“Some advice Kate gave me will stick with me for a while,” Kertzer recalls. “如果你不确定，就跟着你的鼻子走.’”

Math research, Stange explains, “often feels like exploring a jungle. 你不确定你会发现什么, but the creativity comes in deciding what leaf to turn over, 走哪条路, 你在回答什么问题啊, 你要怎样回答他们呢. Some of the deepest insights in mathematics come from creative leaps connecting apparently unconnected ideas.”

Luckily for Haag and Kertzer, there is plenty more jungle to explore.

“Some of my students are so thoroughly confused that I want to do research in math,” Haag says. “他们会说，‘数学不是已经完成了吗? How many questions could possibly be unsolved in math?’”

哈格笑着回答:“这么多.”

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