student research

CU students follow their noses, disprove math conjecture

Anonymous — Thu, 30 Nov 2023 18:03:21 +0000

CU students follow their noses, disprove math conjecture Anonymous (not verified) Thu, 11/30/2023 - 11:03

Daniel Long

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

Prior to the end of the 2022-2023 academic year, graduate student Summer Haag and junior Clyde Kertzer were looking for summer research opportunities in mathematics, their subject of study.

It was an REU (Research Experience for Undergrads) with Katherine (Kate) Stange, CU �鶹ӰԺ associate professor in the Department of Mathematics, and James Rickards, a postdoctoral researcher in the same department, that caught their eye, 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,” says Haag. “When I saw Kate and James were doing a number theory REU, I said, ‘That one! I want that one!’”

CU �鶹ӰԺ students Clyde Kertzer and Summer Haag disproved a longstanding conjecture in mathematical number theory during their summer research experience.

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

The REU would explore a branch of number theory called Apollonian circle packings, which are fractals, or never-ending patterns, made up of infinite circles just touching each other but never overlapping.

Neither Haag nor Kertzer had much experience with circle packings.

“I’d seen quadratic forms before, and I’d seen Mobius inversions, but I’d never seen them pertaining to circle packings,” says Haag. “I was excited to learn that stuff.”

“I went to the library and got a book, the only book I could find on circle packings, and started reading,” says Kertzer.

Room to explore

For the first few weeks of the REU, 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, looking for patterns and proving them,” says Stange. “We didn't have a very definitive goal.”

“We had a long list of possible problems to explore,” Rickards adds. “There was no real end goal in sight.”

CU �鶹ӰԺ scholars Katherine Stange (left) and James Rickards research number theory, an aspect of which includes Apollonian circle packings.

That changed, however, when Haag and Kertzer’s explorations produced data that called a well-known math conjecture into question.

The local-global conjecture, widely accepted for the better part of two decades, predicts the curvatures of the circles inside a circle packing. According to this conjecture, 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).

Time and again, 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,” says Haag.

Two numbers instead of one

But then, while entering numbers into Rickards’ code, 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.

That’s when things got interesting. Numbers that, according to the local-global conjecture, should have appeared together in the same packings didn’t.

Stange likens the situation to a jail. It was as though the numbers that were supposed to be locked up had dug a tunnel when no one was looking and escaped.

Katherine Stange partnered with engineering PhD graduate Daniel Martin to create a pattern for an Apollonian circle packing puzzle laser cut from wood; the pattern can be found here.

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. But there were none. The code was correct. The local-global conjecture, on the other hand, was not.

Over the next few days, Stange and Rickards put together a proof of their findings, working so fast, so feverishly and so precisely that Haag and Kertzer couldn’t help but be inspired.

“It was really impressive,” says Kertzer. “That’s the point where we want to be as mathematicians.”

The four published a paper in the preprint server arXiv with a title as unambiguous as its content is eye-opening: “The Local-Global Conjecture for Apollonian Circle Packings Is False.”

Not bad for a summer research project.

The playful side of math

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. It wasn’t all formulas and rules. It was intuition, exploration, play.

“Some advice Kate gave me will stick with me for a while,” Kertzer recalls. “‘If you’re not sure, just follow your nose.’”

Math research, Stange explains, “often feels like exploring a jungle. You aren't sure what you'll find, but the creativity comes in deciding what leaf to turn over, which path to take, what questions you are trying to answer, and how you will go about answering them. 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. “They’re like, ‘Isn’t math done? How many questions could possibly be unsolved in math?’”

Haag smiles when she answers: “So many.”

Did you enjoy this article? Subcribe to our newsletter. Passionate about mathematics? Show your support.

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

Off

Traditional

White

Undergrad research opens up a field of streams

Anonymous — Tue, 17 Dec 2019 18:57:42 +0000

Undergrad research opens up a field of streams Anonymous (not verified) Tue, 12/17/2019 - 11:57

Clay Bonnyman Evans

CU �鶹ӰԺ environmental studies student gets a taste of ecological research via independent project

As she approached the end of her time at �鶹ӰԺ High School, Lydia Jones figured she’d soon be attending college in another state.

“All the friends I’d made in Colorado were going to school in different places,” says Jones, 22, a senior majoring in Environmental Studies. “That’s what I thought I was supposed to do, too.”

But before launching her college career, she decided to take time off to attend a National Outdoor Leadership wilderness-education program in Baja California, Mexico. And it was there, more than a thousand miles from home, that she began seriously considering attending the �鶹ӰԺ.

We need to be looking at different dams and diversions, to assess whether they make sense for river ecology, but also for the culture of the area and to see whether benefits outweigh the costs. … It’s really important to realize that there are pluses and minuses; it’s really not a black-and-white issue.”

Lydia Jones, a senior majoring in Environmental Studies, takes a break while climbing. Photo courtesy of Lydia Jones.

“One of guides there had gone to CU �鶹ӰԺ for environmental studies and was talking about how great it was,” Jones says. She looked into it, decided to stay home and has never looked back. “I’ve really loved it. I’ve really found my passion and it’s been great.”

She’s particularly pleased to have discovered the opportunity to undertake an independent undergraduate research project exploring the impact of dams on river ecology with instructor Ryan E. Langendorf. Teaming up with a faculty member, undergraduates can design projects that allow them to get a taste of researching their given topic in depth.

“A lot of undergraduates don’t even know this is possible,” she says.

In fact, when she approached Langendorf, he didn’t, either. But he liked the idea, and starting last summer, Jones was hard at work learning about river ecology and consulting professionals in the field and a laboratory to design a small field project.

“My own research has involved gathering water samples above and below two dams, two diversions that are part of (the city of) �鶹ӰԺ’s water supply,” she says.

Using sampling and recording techniques gleaned from the professionals, she tested water for pH levels and then sent the samples to a laboratory to analyze them for dissolved nutrients like phosphorus and nitrogen to gauge the health of the water beneath the dams.

The project, while too small to be statistically significant, has given Jones valuable field experience and the data will be available to future researchers. She hopes to publish a piece on her research in a popular-science journal or website and give presentations about the work to non-governmental organizations.

“Pretty much every river today is dammed, and it can be an extremely controversial topic,” she says, noting that Proposition DD, a Colorado ballot measure to allow and use revenues from sports betting on water projects, narrowly passed in November.

With science continuing to accumulate about the impacts of global climate change, issues involving water, water supply and dams are only going to become more critical, Jones says.

“This is a very legitimate issue. We need to be looking at different dams and diversions, to assess whether they make sense for river ecology, but also for the culture of the area and to see whether benefits outweigh the costs,” she says. “It’s really important to realize that there are pluses and minuses; it’s really not a black-and-white issue.”

Interested in learning about undergraduate research?

Students interested in undergraduate research can get contact the CU �鶹ӰԺ Undergraduate Research Opportunities Program, which offers consultation, workshops and research grants to undergraduate researchers. Other opportunities for support are available through the university’s Biological Sciences Initiative.

CU �鶹ӰԺ environmental studies student gets a taste of ecological research via independent project

Off

Traditional

White

Grad students’ capstone project aims to help Rocky Mountain National Park

Anonymous — Mon, 30 Sep 2019 19:25:48 +0000

Grad students’ capstone project aims to help Rocky Mountain National Park Anonymous (not verified) Mon, 09/30/2019 - 13:25

Jeff Thomas

As park usage rises, CU �鶹ӰԺ grad students join effort to manage the growth

Four �鶹ӰԺ graduate students spent their summer working at one of the nation’s premier national parks—Rocky Mountain National Park—as part of their capstone project in the Masters of the Environment Graduate Program.

John Hannon, management specialist for business programs at the park, has managed student groups, including those from the National Park Service’s intern program and Colorado State University, but this is the first time he’s worked with CU �鶹ӰԺ students.

Park visitors at the Alpine Visitor Center on Trail Ridge Road. Photo courtesy of NPS. At the top of the page, Longs Peak, the highest point in the park, is framed by fog and a sharp ridge.

Many of the programs crucial to the visitor experience at Rocky Mountain National Park, such as alternative transportation and commercial services, have recently been brought to the forefront by the strain of increasing park usage, according to Hannon. And it’s an opportunity he thinks these students can help fix.

“It’s really good because they bring an outside perspective and a new perspective,” said Hannon.

The Masters of the Environment (MENV) program requires students to complete a capstone project (which can be completed as a team), instead of the thesis or practicum requirements of many traditional masters’ programs.

Some of the students pick a team then decide on a program, but Kirsta Tortorice, Stephanie Reifenberg, Alison Dietze and Bridger Tomlin picked the project before the team.

The program routinely holds networking events to meet with potential partners for grad students, which is where the team met Hannon and learned about his project.

“For us, we met John (Hannon), and the rest was history,” said Reifenberg, a Michigan native who is concentrating on land management and wildlife.

“Some teams are formed by who wants to be in the project, and some by who wants to work together,” said Dietze, who is focusing more on sustainability. “We all just very lucky we work so well together.”

“The park has seen a 1.5 million increase in (annual) visitors since 2012,” said Hannon. “Naturally, the park didn’t get any bigger, and the roads didn’t either.”

“Given the career interests of this team, working alongside agency staff at Rocky Mountain National Park is an unparalleled opportunity to learn about the challenges and opportunities facing public lands managers while also developing professional skills,” said Instructor Lydia Lawhon, the advisor for the project.

This experience is a little different for each student, but they all share one thing: their interest in learning new skills from those that make the Rocky Mountain National Park a great place, and figuring out how they can use established skills to make it even better for visitors.

The project “is super interesting because I have a love of national parks, and it’s interesting to experience from both perspectives (as a park visitor and manager),” said Tortorice, a graduate of Endicott College in Massachusetts who came to CU �鶹ӰԺ specifically to work on policy and public land management issues.

Tomlin, a �鶹ӰԺ native who graduated from Maine’s Bowdoin College with a degree in environmental studies and history, agrees.

“I wanted to go to grad school in urban or environmental planning, and it’s hard to find the unique blend of policy, hard science and social science that’s here (at CU �鶹ӰԺ),” Tomlin said. “What got me excited about this project is the opportunity to work with a project that’s close to home, and being able to visualize the impact to the location.”

“Working with asset management is transferable to any field,” Tomlin said. “This is a very useful experience, especially in project scoping.”

The work has just begun for the project, according to Hannon, and any policy suggestions will go through National Park Service staff and its extensive public input process.

“I think we have some pretty specific goals, and we’ve designed this project to where they have the right amount of time to reach them (policy suggestions),” he said.

As for the grad students, finding themselves in a team doesn’t present any problems as the program makes them work together frequently.

“To be involved in this (program at the national park) is really awesome,” said Reifenberg.

Four CU �鶹ӰԺ graduate students spent their summer working at one of the nation’s premier national parks—Rocky Mountain National Park—as part of their capstone project in the Masters of the Environment Graduate Program.

Off

Traditional

White

Student uses the stage, journalism to shine a light on veterans

Anonymous — Wed, 03 Apr 2019 19:21:48 +0000

Student uses the stage, journalism to shine a light on veterans Anonymous (not verified) Wed, 04/03/2019 - 13:21

Kenna Bruner

**The result is a play, How to Leave a Battlefield, which will be performed on campus**

Wars tend to be remembered for their battlefield glory or decisive military action, not for soldiers’ recovery and reintegration into civilian life. A CU �鶹ӰԺ senior, however, is using theatre and veterans’ own words to change that.

Sean Guderian takes a break during rehearsal. At the top of the page, he works with a student actor. CU �鶹ӰԺ photos by Patrick Campbell.

When Sean Guderian was a freshman, an English class writing assignment propelled him on a four-year search for answers he found by interviewing military veterans. He recorded many hours of interviews with veterans of wars and conflicts. As he listened to their combat stories, he asked them what happened after.

“I wanted to know how soldiers come home as veterans and live out the rest of their lives after living through the most extreme dangers and responsibilities life can throw at you,” said Guderian, who is majoring in theatre performance and journalism.

The culmination of his work is How to Leave a Battlefield, a stage play Guderian wrote using transcripts from some of the recorded interviews in the hopes of shining a light on soldiers transitioning to civilian life. His play will be presented April 16 and 17 at 7 p.m.; and April 20 at 2 p.m., in the Loft Theatre, in the CU �鶹ӰԺ Theatre Building.

"I think that what most people don't realize is that for war vets, the real war doesn't start until after they leave the battlefield. Which is the fight for their own existence."
Erik Lincoln Stevenson, infantry machine gunner and Marine scout sniper, United States Marine Corps, 1996–2000
"Yeah, I find that people who go through trauma and heal and help repair themselves are really . . . awesome human beings."
Robert Sampson, infantry, 196th and 198th Light Infantry Brigade, Vietnam, 1971–72

Guderian was studying English when the project began. His writing group at the time, the Literary Buffs, visited Frazier Meadows, a �鶹ӰԺ retirement community, to entertain the residents with essays and short stories. During the visit, he met with two World War II veterans who shared stories about their experiences in combat. These stories gave Guderian a glimpse into the extent of what they encountered—“both incredible and horrific.”

Hearing their wartime experiences gave Guderian the idea for a theatre project that focused on the veterans’ return to civilian life rather than their time spent in combat. Not everyone was willing to have their raw stories open and bare on the stage. After speaking to about 30 veterans, he wrote a play using the testimonials of seven.

“I didn’t want to focus on the pain of war,” he said. “And while it did come up, and there’s some of it in the play to place things in context, it was important that we didn’t dwell on that. The message in the show is healing and how the veterans felt they were supported after they came back.”

“How do you leave a battlefield?” was always the primary question, and the answers Guderian received varied. Many of the veterans he talked with will never leave the battlefield in their minds. Others were more optimistic.

“It’s so easy to generalize a soldier,” Guderian said. “Here’s someone who signed up, gone to war, and when they returned are hurt by that. I was interested in the relationship with their service and how they live today.

“For as bad as things were for them, I’ve had veterans say they don’t regret joining and serving, because of how much they learned through the experience. In juxtaposition of that, there are situations where they carried out orders they regret,” he said. “They have to find a way to live with that.”

Guderian is framing his play based on the style of playwright and actor Anna Deavere Smith, widely recognized for developing a form of theatre dedicated to social change.

Guderian was introduced to this style of testimonial theatre in a class taught by Cecilia Pang, associate professor in theatre and dance at CU �鶹ӰԺ. Pang, who is Guderian’s faculty advisor, served as Smith’s assistant on the 1994 premiere of Twilight: Los Angeles 1992 about the Los Angeles riots.

Drawing on her own experience with this specialized type of performance, based on the testimonials of real people, but presented by actors on a minimal set, Pang coached Guderian on how to conduct interviews for this documentary-type theatre performance. She also taught him how to “conduct, transcribe and enact the interviews” in Smith’s style.

“It’s fascinating to see how the veterans opened up to Sean,” Pang said. “His script is genuine. It’s about authenticity. This show is a new technique for our students, because it’s not covered in CU’s curriculum. The experience has been a truly worthy endeavor for Sean. What has impressed me so much about Sean is that he has a vision, initiative and followed through with it.”

How to Leave a Battlefield, by Sean Guderian

When: April 16 and 17 @ 7 p.m. and April 20 @ 2 p.m.
Where: Loft Theater (C370) CU �鶹ӰԺ Theater Building
Cost: Free and open to the public

For his project, Guderian received a grant from CU �鶹ӰԺ’s Undergraduate Research Opportunities Program. UROP helps students develop their own projects under the mentorship of a faculty member. Guderian’s grant will go toward paying the student actors and crew.

This will be Guderian’s first stage show that he produced and directed.

“Sean is very clear on this show,” Pang said. “It’s his single-minded pursuit of a dream.”

The culmination of his work is How to Leave a Battlefield, a stage play the student wrote using transcripts from some of the recorded interviews in the hopes of shining a light on soldiers transitioning to civilian life.

Off

Traditional

White