Category: Mathematics Page 1 of 7

Modeling the COVID-19 Roller Coaster

On January 3, 2023

*A Duke team looks at the math behind COVID’s waves as new coronavirus variants continue to emerge.* *Credit: @ink-drop*

DURHAM, N.C. — First it was Alpha. Then Delta. Now Omicron and its alphabet soup of subvariants. In the three years since the coronavirus pandemic started, every few months or so a new strain seems to go around, only to be outdone by the next one.

If the constant rise and fall of new coronavirus variants has left you feeling dizzy, you’re not alone. But where most people see a pandemic roller coaster, one Duke team sees a mathematical pattern.

In a new study, a group of students led by Duke mathematician Rick Durrett studied the calculus behind the pandemic’s waves.

Published Nov. 2022 in the journal Proceedings of the National Academy of Sciences, their study got its start as part of an 8-week summer research program called DOmath, now known as Math+, which brings undergraduates together to collaborate on a faculty-led project.

Their mission: to build and analyze simple mathematical models to understand the spread of COVID-19 as one strain after another popped up and then rose to outcompete the others.

In an interview about their research, project manager and Duke Ph.D. student Hwai-Ray Tung pointed to a squiggly line showing the number of confirmed COVID cases per capita in the U.S. between January 2020 and October 2022.

*The COVID-19 pandemic has unfolded in waves. Adapted from The New York Times, July 18, 2022*

“You can see very distinct humps,” Tung said.

The COVID pandemic has unfolded in a series of surges and lulls — spikes in infection followed by downturns in case counts.

The ups and downs are partly explained by factors such as behavior, relaxation of public policies, and waning immunity from vaccines. But much of the roller coaster has been driven by changes to the coronavirus itself.

All viruses change over time, evolving mutations in their genetic makeup as they spread and replicate. Most mutations are harmless, but every so often some of them give the virus an edge: Enabling it to break into cells more easily than other strains, better evade immunity from vaccines and past infection, or make more copies of itself in order to spread more effectively.

Take the Delta variant, for example. When it first started going around in the U.S. in May 2021, it was responsible for just 1% of COVID cases. But thanks to mutations that helped the virus evade antibodies and infect cells more easily, it quickly tore across the country. Within two months it had outcompeted all the other variants and rose to the top spot, causing 94% of new infections.

“The natural question to ask is: What’s going on with the transition between these different variants?” Tung said.

For their study the team developed a simple epidemic model called an SIR model, which uses differential equations to compute the spread of disease over time.

SIR models work by categorizing individuals as either susceptible to getting sick, currently infected, or recovered. The team modified this model to have two types of infected individuals and two types of recovered individuals, one for each of two circulating strains.

The model assumes that each infectious person spreads the virus to a certain number of new people per day (while sparing others), and that, each day, a certain fraction of the currently infected group recovers.

In the study, the team applied the SIR model to data from a database called GISAID, which contains SARS-CoV-2 virus sequences from the pandemic. By looking at the coronavirus’s genetic code, researchers can tell which variants are causing infection.

Study co-author Jenny Huang ’23 pointed to a series of S-shaped curves showing the fraction of infections due to each strain from one week to the next, from January 2021 to June 2022.

When they plotted the data as points on a graph, they found that it followed a logistic differential equation as each new variant emerged, rose steeply, and — within six to 10 weeks — quickly displaced its predecessors, only to be taken over later by even more aggressive or contagious strains.

Durrett said it’s the mathematical equivalent of something biologists call a selective sweep, when natural selection increases a variant’s frequency from low to high, until nearly everyone getting stick is infected with the same strain.

“I’ve been interested in epidemic modeling since the end of freshman year when COVID started,” said Huang, a senior who plans to pursue a Ph.D. in statistics next year with support from a prestigious Quad Fellowship.

They’re not all typical math majors, Durrett said of his team. Co-author Sofia Hletko, ’25, was a walk-on to the rowing team. Laura Boyle ’24 was a Cameron Crazie.

For some team members it was their first experience with mathematical research: “I came in having no idea what a differential equation was,” Boyle said. “And by the end, I was the person in the group explaining that part of our presentation to everyone.”

Boyle says one question she keeps getting asked is: what about the next COVID surge?

“It’s very hard to say what will happen,” Boyle said.

The teams says their research can’t predict future waves. Part of the reason is the scanty data on the actual number of infections.

Countries have dialed back on their surveillance testing, and fewer places are doing the genomic sequencing necessary to identify different strains.

“We don’t know the nature of future mutations,” Durrett said. “Changes in people’s behavior will have a significant impact too.”

“The point of this paper wasn’t to predict; rather it was to explain why the waves were occurring,” Huang said. “We were trying to explain a complicated phenomenon in a simple way.”

This research was supported by a grant from the National Science Foundation (DMS 1809967) and by Duke’s Department of Mathematics.

CITATION: “Selective Sweeps in SARS-CoV-2 Variant Competition,” Laura Boyle, Sofia Hletko, Jenny Huang, June Lee, Gaurav Pallod, Hwai-Ray Tung, and Richard Durrett. Proceedings of the National Academy of Sciences, Nov. 3, 2022. DOI: 10.1073/pnas.2213879119.

Anyone Can Be a “Math Person”

By Sophie Cox

On December 8, 2022

In Art, Humanities, Lecture, Mathematics, Science Communication & Education

Dr. Francis Su, a mathematician and professor at Harvey Mudd College and the author of “Mathematics for Human Flourishing,” wants you to know that math can be beautiful. As these “infinitely quartered” squares show, 1/4 + 1/(4²) + 1/(4³) + … = 1/3. Image attribution: Tdadamemd, via Wikimedia Commons, under Creative Commons CC0 1.0 Universal Public Domain Dedication

Francis Su, Ph.D., visited Duke to talk about math. He began by talking about art.

Su, a mathematician and professor at Harvey Mudd College, displayed “Hope,” an 1886 painting by George Frederic Watts. He asked the audience to look at it, really look at it, and think about what’s happening in the painting. At first glance, it shows a blindfolded woman holding a wooden object. She seems to be in pain. But the more time we spend looking, the more we notice. We might notice that there’s a single star above her. We might notice that the wooden object is a lyre with only one string left attached. We might notice, too, that the woman is plucking that final string and straining to hear its music.

If we take the time to explore the history of the painting, we might learn that Martin Luther King, Jr., talked about the same painting in a sermon. Su quoted a line from that sermon: “Who has not had to face the agony of blasted hopes and shattered dreams?” We find beauty in art, and often we find it relatable as well. Art invites us to look closer, to wonder, to feel, to ask questions, to imagine.

“Why,” Su asks then, “don’t we approach mathematics the way that we approach art?”

Professor Francis Su’s book, “Mathematics for Human Flourishing.”

Whether we consider ourselves “math people” or not, we rarely if ever hear mathematics discussed as an affirmation of human virtues and desires—love, beauty, truth, the “expectation of enchantment.” Su wants to change that. In his book “Mathematics for Human Flourishing” and in his talk at Duke, he envisions mathematics as beautiful, inclusive, and accessible to anyone.

Along with the painting “Hope,” Su’s first slide shows a quote by Simone Weil: “Every being cries out silently to be read differently.” Simone Weil, according to Su, was a “French religious mystic” and “widely revered philosopher,” but she also had a deep interest in math. Her older brother, André Weil, was an influential mathematician whose mathematical achievements often overshadowed her own. In a letter to a friend published posthumously in the book “Waiting for God,” Simone Weil wrote: “I did not mind having no visible successes, but what did grieve me was the idea of being excluded from that transcendent kingdom to which only the truly great have access and wherein truth abides.” Su sometimes wonders how Simone’s relationship to mathematics would have been different if André had not been her brother. Again, “Every being cries out silently to be read differently.” According to Su, when Simone Weil speaks of “reading” someone, she means “to interpret or make a judgment about them.”

Su has a friend, Christopher Jackson, who is an inmate in a high-security prison, serving a thirty-two year sentence for involvement in armed robberies as a teenager. When you think about people who do math, Su asks, would you think of Chris? “We create societal norms about who does math,” and Chris doesn’t fit those norms. And yet he has been studying mathematics for years. After studying algebra, geometry, trigonometry, and calculus while in prison, he sent a letter to Su requesting help in furthering his mathematics education. The two men still correspond regularly, and Chris is now studying topology and other branches of mathematics.

“Every being cries out silently to be read differently.”

Why do math in the first place? Just as you can take your car to a mechanic without fully understanding how it works yourself, we might think of math as “only for the elite few” or perhaps as “a means to an end,” a tool “to make you ‘college and career ready.’” Su sees it differently. He views math in terms of human flourishing, “a wholeness of being and doing.” He points to three words from other languages: eudaemonia, a Greek term for “the overarching good in life”; shalom, a Hebrew word often used as a greeting and roughly translated as “peace”; and salaam, an Arabic word with a similar meaning to shalom.

The pattern on Romanesco broccoli is a fractal, common in both math and nature. Image credit: Francis Su

“What attracts me to music,” Su says, “isn’t playing scales over and over again.” But once you “experience a symphony,” you might see the value in playing scales. Can we learn to think of math the same way? Here, Su quoted mathematician Olga Taussky-Todd: “The yearning for and the satisfaction gained from mathematical insight brings the subject near to art.”

Beauty and awe probably aren’t the first words that come to mind when most of us think of math, but Su believes math can unlock “transcendent beauty.” He references a quote by C.S. Lewis: “the scent of a flower we have not found, the echo of a tune we have not heard, news from a country we have never yet visited.” That is what math at its best can do for us. It can help us see the big picture and realize that we’re “just scratching the surface of something really profound.”

“Math is not a single ‘ability,’” Su says. “In reality, math is a multi-dimensional set of virtues.” When learning or teaching math, we often focus more on skills like recalling facts and algorithms, factoring polynomials, or taking a derivative. But Su believes more important lessons are at play: virtues like persistence, creativity, a thirst for deep knowledge, and what he calls the expectation of enchantment. And, he says, employers are often much more interested in virtues than in skills. “If you want to be really practical about this—and I don’t, with mathematics, but if you do—then it’s actually the virtues that are more important than the skills,” Su says.

One basic human desire that Su believes math can help fulfill is the desire for truth, which, in turn, can help build virtues like a thirst for deep knowledge and the ability to think for oneself, which can help us figure out what’s true instead of just blindly trusting authorities. “Truth is under attack,” Su says. “Misinformation is everywhere.” Su wants to teach his students “to think, to be ‘that person who doesn’t need to look at the Ikea instructions.’” But he also wants them to view math as more than just a means to an end. “It’s my responsibility to help my students remember the beauty” in math and to understand that their dignity as human beings isn’t dependent on their grades.

Along with truth and beauty, he believes math can and should bring opportunities for exploration and discovery. “My role isn’t to be a teacher,” he says. “My role is to be a co-explorer.” He recalls his own excitement when he first saw a Menger cube, or Menger sponge, cut along its diagonal. The resulting cross-section is beautiful and, yes, enchanting. “What would it look like for classrooms to be like that?” During the pandemic, Su started adding more reflection-focused questions to his exams, questions like “Consider one mathematical idea from the course that you have found beautiful, and explain why it is beautiful to you.” Even more traditional math questions can be phrased in an “exploratory” way. Su gives the example of a question that asks students to make two rectangles, one with a bigger perimeter and one with a bigger area.

A visual representation of Nicomachus’s Theorem.
Image from Cmglee, via Wikimedia Commons, licensed under Creative Commons Attribution-Share Alike 3.0 Unported.

Another desire or virtue important in the field of mathematics is justice. Su wants math to be accessible to all, but not everyone has had positive experiences with math or feels like they belong there. As an analogy, Su talks about receiving dishes from a “secret menu” when visiting certain Chinese restaurants with friends who are fluent in Chinese. When he goes there on his own and requests the “secret menu,” however, he is sometimes turned away or told that he wouldn’t like those dishes. “Are people side-by-side in the same restaurant having different experiences” in math, too? “Who are you to say they do or don’t belong in mathematics?”

Even Su himself hasn’t always had wholly positive experiences in math. One of his professors once told him he didn’t “have what it takes to become a successful mathematician,” and he almost quit his Ph.D. program. Instead, he switched to a different advisor who had encouraged him to stick with it. Meanwhile, he surrounded himself with people who could remind him why he loved math. Math as a field can be competitive, but “if you think of mathematics as human flourishing… then that’s not a zero-sum game anymore.”

In Su’s words, “we’re all math teachers” because “we all pass on attitudes about math to others.” He says studies show that parents can pass on “math anxiety” to their kids. But Su encourages people to “believe that you and everyone can flourish in mathematics.” Simone Weil. Christopher Jackson. And you.

Who will you read differently?

Meet New Blogger Addie: A Recovering Advice Columnist

By Guest Post

On September 21, 2022

In Mathematics, Policy, Science Communication & Education, Students

“My name is Meg Stalter I’m 5’7 I’m living in LA and a fun fact about me is something bad happened to my cousin.”

As made evident by her Twitter profile, my favorite comedian, Megan (“Meg”) Stalter, knows how to make an introduction. Stalter is best known (as far as I know) for her role in the HBO comedy “Hacks,” in which she plays Kayla (whoever that is).

I do not have a Twitter account and I have never seen the show. While we are talking about me, I will explain that I do not really watch TV, with the one exception of West Wing.

Since we are still talking about me, you should know that I fibbed. There are two exceptions. The other one is Grantchester, a Masterpiece Mystery about a hot priest who solves crime (but that was sort of a given, no?).

I share Stalter’s bio for a few reasons. For starters, it makes me smile, and sharing a smile is a tried-and-true way to score a friend (cha-ching!).

*Meg Stalter once again proved her knack for making a first impression at her Emmy’s debut*

On top of that, it is a good example of someone who knows how to make a first impression. I expect to have made a great impression by the time I finish this, but to ensure things got started on the right foot, hedging my bets if you will, I thought it best to leave the preamble to someone at the top of the trade.

Stalter’s bio also proves a simple point; it is not merely what you say that counts, but how you say it.

I am something of a sub-par reader. I love to read, it is just not my biggest strength (doesn’t mean it can’t be (growth mindset)! Just facing today’s facts). I don’t think I read enough as a child, so now I am slow and I usually fall asleep.

But I get by. I power through my class readings, I keep a book on my bedside table, and I get my news through the radio (that and two free tickets to the Hoppin’ John’s Fiddler’s Convention–it pays to be tuned into WUNC on Saturday nights at 10. Cha-ching!).

This relationship with reading influences my writing style. When I write, I try to keep my readers awake. Not with what I write — I have full faith in the topic at hand’s capacity to speak for itself — but with the way I write it.

My past experience writing for a published paper was in high school, where I spent four years as co-editor of the “Hustle and Bustle” page. I authored a satirical advice column in which troubled high schoolers (me) could send their personal woes to someone who would publish them for the whole school to read (also me). I like writing as a secondary form of chatting.

*My senior year, I retitled my column “Dear Addy,” after the well-known advice column “Dear Abby.”*

And so it is with this laudable writing background that I report to you on the groundbreaking discoveries from one of the top research universities in the U.S.

Why write for a research blog? Research is interesting. Research makes the world go round. Just ask a freshman. They all came here for the “research opportunities,” as did all the other freshmen at all the other universities.

Before I sign off, I will let you know where you might catch me in my free time. This is a key element of the standard student bio, and I am prone to severe FOMO, so let me get right to it.

I am a sophomore from Hickory, North Carolina hoping to major in Public Policy and minor in Math. In my free time you might catch me listening to NPR, jogging, potting, singing to myself, making a smoothie, telling people about my smoothie, spamming my contacts for an ice cream date, or for the not-so-lucky, trying my best at Appalachian-style fiddle.

By Addie Geitner, Class of 2025

Modeling Biology: Ruby Kim’s Research on the Math of Physiological Cycles

By Cydney Livingston

On May 25, 2022

In Biology, Mathematics, Students

What lies at the intersection of mathematics and biology? Freshly-minted math PhD Ruby Kim and her work on mathematically modeling human dopamine cycles.

Kim’s work has centered around her creation of a math model to predict how a person’s dopamine levels ebb and flow over the course of a day “to understand the general mechanics of how disruptions in the (biological) clock lead to disruptions in dopamine.”

She said there is a pretty long history of mathematicians using differential equations to see how different clock genes and proteins change over the course of a day’s circadian cycle. Yet, no previous models have connected the circadian clock – controlled by the brain’s suprachiasmatic nucleus – to dopamine levels. And Kim tells me that work suggesting dopamine changes throughout the day are likely controlled by the internal circadian clock itself is “relatively new.”

The first step in Kim’s work was validating scientists’– or “experimentalists,” as Kim dubs them – hypotheses about dopamine and dopaminergic enzyme cycling.

Many physiological processes are controlled via circadian rhythms and the internal clock in humans, as well as other organisms.

“But I’d like this work to help experimentalists go one step further and be able to test out hypotheses more easily.” For example, Kim says that her model has the potential to reveal other fascinating phenomena, such as how drug treatments or different genetic mutations may impact circadian rhythm or dopamine. This is thanks to the multifaceted layers of Kim’s model.

“From a mathematical perspective, the math model is very interesting. It has a lot of interesting dynamics,” she says. “Not only does it show nice, 24-hour rhythms, it shows both steady state behavior… but then also behavior that’s really wild – something called quasi-periodic behavior, where the internal clock is significantly different than the external 24-hour light-dark cycle.”

“This leads to oscillatory behavior that’s not periodic,” she says. These sorts of quasi-periodic behaviors have been observed in experiments and misunderstood, but they can be computed.

Kim emphasized the experimental and clinical implications of her work. Dopamine is involved in learning and motivation and is also linked a plethora of psychiatric conditions like Parkinson’s, ADHD, and schizophrenia. “Patients with these conditions often also experience circadian disruptions,” Kim says. “That’s a pretty big symptom.”

Kim began her academic career in her home state of California at Pomona College as a pre-med math major. “I had always been intrigued by human physiology. And math was one of the subjects I was also pretty drawn to. I just didn’t appreciate it much because throughout high school and the beginning of undergrad, I didn’t see any direct applications,” Kim told me.

The marriage of her love for math with her intrigue in biology actually began at Duke when Kim attended a mathematical biology workshop during the summer after her sophomore year. “I had never heard of math biology before that.”

After working on a brief project to model sleep apnea in infants at the workshop, Kim returned to California and took up math modeling courses in her following semesters of undergrad. One of her professors, Ami E. Radunskaya, PhD, was extremely supportive and introduced Kim to a lot of “cool biological problems.” Kim went on to do research with Radunskaya, modeling tumor-immune interactions. This experience, Kim says, “kind of just threw me into academia.” The project gave way to an undergraduate thesis with Radunskaya that analyzed the long-term behavior of this tumor growth and treatment model.

Radunskaya then suggested that Kim pursue grad school. “I kind of applied on a whim,” Kim said, “It wasn’t something I had specifically imagined for myself.” Kim mentioned how no one from her home community had really ever gone to grad school and so it was not something she had ever “explicitly” thought about before.

Using mathematics to fight cancer | Department of Mathematics | University of Washington — An abstract of Radunskaya’s work on mathematical modeling and understanding tumor-immune interactions to address cancer.

In her search for a graduate program, Kim applied to math programs, as well as those that were interdisciplinary. “I ended up choosing Duke because I really liked my advisor,” Kim told me. While Kim’s advisor Michael Reed, PhD “does a lot of interesting math,” Kim wanted to work with him because math isn’t his focus – understanding “really complex biological systems using mathematical language is.”

“A lot of times you see people who do things at the intersection of math and biology that are more motivated from a mathematical standpoint … that’s just not what I’m interested in personally. I’m very interested in finding an interesting biological problem and then applying whatever mathematical tools I have.”

While at Duke, Kim was foundational to founding the university’s chapter of the Association for Women in Math (AWM). During her undergrad, Kim “had a really great experience with AWM,” finding both a community of women mathematicians and a network of women professors who were involved in the chapter.

At Duke, there wasn’t a chapter “but quite a few people who were interested in starting and being part of one.” This organization, which is open to people of any gender identity, heads mentorship programming that brings undergrads, grad students, postdocs, and professors together, organizes conferences, and contributes to their central focus of community building in math.

Events – Association for Women in Mathematics (AWM) — Example flyers of the events organized by the Duke chapter of the Association for Women in Mathematics.

Outside of her research, Kim spends most of her free time taking care of foster pups, which she describes as “extremely rewarding but also very tiring.” Her most recent foster, a four-month-old puppy, eavesdropped on our interview as he took a nap.

This fall, Kim will begin a post-doc with the University of Michigan’s math department as she “wanted to keep studying circadian rhythms with faculty who are really great in that area.”

Post by Cydney Livingston, Class of 2022

“Brains are Weird… and the World is Difficult”

By Sophie Cox

On March 30, 2022

In Behavior/Psychology, Business/Economics, Computers/Technology, Mathematics, Science Communication & Education

Institute for Consumer Money Management, and Duke University’s Center for Advanced Hindsight.

Intending to do the right thing doesn’t always lead to actually doing it, a tendency formally known as the “intention-behavior gap.” We can intend to go to bed early and still go to bed late. We can want to exercise and still choose not to. We can recognize the importance of saving extra money and still choose to spend it instead. So why is it so hard to change our behavior? Because, says Jonathan Corbin, Ph.D., “brains are weird” and “the world is difficult.”

Corbin is a senior behavioral researcher at the Center for Advanced Hindsight at Duke University. The Center for Advanced Hindsight recently partnered with NOVA Labs, Thought Cafè, and the Institute for Consumer Money Management to create the NOVA Financial Lab, a group of financial literacy games targeted at adolescents and emerging adults. In each game, players practice managing money while taking care of a pet. You may never have to sneak a cat into a concert or prepare a retirement plan for a dog in real life, but you will need to understand concepts like budgeting, interest, and debt. “What we hope people start to do,” Corbin says, “is really think about, ‘What decisions should I make now to make better decisions later?’”

Essentially, “Money spent now is money that can’t be spent later.” As intuitive as that might seem, “The way we think about money is relative, and it’s not linear.” When you’re already spending thousands of dollars on a car, for instance, an extra five hundred dollars for a feature you may or may not need “feels like a very small amount of money,” but in a different situation, its value can seem higher. How many times, Corbin points out, could you go out to eat with five hundred dollars?

The three games combine financial literacy with behavioral science to explore why people make the decisions they do and how they can start to make better ones.
Source: https://advanced-hindsight.com/wp-content/uploads/2022/03/CAH-NOVA.pdf

There are three games: Shopportunity Cost, Budget Busters, and Exponential Potential. (“One of the people from PBS helped us come up with these cute names,” Corbin says.) They each involve different skills, but they all focus on “financial literacy from a behavioral science perspective.” Players have to contend with both external obstacles and common behavioral biases to make financial decisions for a pet. “I always choose the dog,” Corbin adds, “but I understand other people might choose the cat.” (I chose the cat.)

The first game, Shopportunity Cost, focuses on short-term financial planning. It involves dressing a pet up like a person in order to sneak them into a concert for the night. “You have to make decisions that optimize the pet’s happiness while also being able to make it to the concert and back home,” but you have a limited amount of money to spend. If you spend too much money too soon, you’ll run out, but if you’re too frugal, your pet won’t enjoy the evening. As goofy as the concert scenario is, it introduces players to an important concept known as opportunity cost, which refers to the potential benefits we miss out on when we choose one alternative over another. Say you’re debating between a $50 outfit and a $30 one. The opportunity cost of choosing the more expensive outfit is $20, but shoppers don’t always consider that. “Opportunity cost neglect is the simple idea that when we’re faced with financial decisions, we tend not to consider alternative uses for that money.” Reframing the $30 outfit as “a $30 dress that I’m okay with plus 20 extra dollars” that could be spent elsewhere might lead you to choose the cheaper outfit. Or it might not. “Sometimes you want the $50 outfit, and that’s perfectly fine… but a lot of the time that might not be the right decision.” Like many things, taking opportunity cost into account is a balancing act. “We shouldn’t obsess over every possible opportunity that there is,” Corbin cautions, but “consider[ing] opportunity costs can lead to better financial decisions.”

Budget Busters, meanwhile, involves medium-term planning. Players have to manage checking, credit, and savings accounts while caring for their pet over a six-month period. Along with purchasing essential and non-essential items to attend to their pet’s basic needs and happiness, players have to contend with unforeseen circumstances like medical emergencies. The game introduces people to the 50-30-20 rule, a budgeting concept that involves devoting 50% of income to essentials, 30% to non-essentials, and 20% to savings. Budget Busters also explores the principle of mental accounting, the idea that aside from formal budgets, we have “categories in our head” that change our perception of money. “Let’s say you get birthday money from your relative. That money tends to be a different kind of spending money to you than money you get from your paycheck,” Corbin explains, because “money feels different in different contexts.”

There are parallels in Budget Busters. Sometimes players receive unexpected windfalls like gifts or prizes. (My cat won $40 for being “Best in Show” at the local pet pageant.) Players get to decide whether to use the extra money on a “fun” item for their pet or put it into savings. Corbin says “gift money” is a classic example of a misleading mental account. “We tend to overspend… because it feels like it’s not even our money in a way.” In reality, though, money has “fungibility,” meaning it’s “exchangeable… across any account.” In other words, “money is money,” regardless of where it comes from. A $10 bill, for instance, can be exchanged for two fives without changing its value. (Non-fungible tokens, or NFTs, lack this property. “You can’t exchange the picture of a cat you bought from the internet for Chipotle.”) Like Shopportunity Cost, Budget Busters focuses on both traditional financial concepts and common behavioral tendencies that affect decision-making. “None of these things are necessarily bad,” Corbin emphasizes, “but they’re things that one should be aware of… when that natural proclivity may be swaying them in the wrong way.”

Budget Busters, which focuses on monthly budgeting, also encourages players to look closely at discounts when shopping. “Sometimes the discount that looks really good from a percentage-off perspective isn’t actually the better discount” in terms of overall budgeting and total amount of money saved, Corbin warns.

The last game, Exponential Potential, explores concepts like compound interest, debt, and investment. The premise of the game involves traveling back in time to balance debts and investments. The goal is to make your pet a millionaire. By showing players how investment decisions can affect future net worth, the game seeks to increase understanding of processes involving exponential growth. Exponential Potential introduces the concept of exponential growth bias. According to Corbin, “We tend to underestimate things that grow exponentially.” He cites the coronavirus pandemic as an example: “Even the people who were making the graphs of Covid’s growth… it’s really hard for them to figure out how to show that to people.” Log-transformed graphs are one option, but they can be deceptive by making the slope look flatter. Similarly, when dealing with exponential growth in the financial world, “People are going to underestimate how badly they’re going to get burned” by debt, but they may also underestimate how much they’ll benefit by saving for retirement.

With compound interest, for instance, “The interest gets applied both to principle and to interest from the last time, and that’s where exponential growth happens.” In the game, players have the opportunity to adjust how much money to put toward paying off debts, investing, and saving for retirement each month. Then they travel decades into the future to see how their decisions have affected their pet’s net worth. “We’re hoping that that kind of feedback allows you to think through… what you might have done wrong and try to correct,” Corbin says. Once again, though, raw numbers aren’t the only factor at play. “We just want people to understand what the optimal way to do this is, and if there’s a better way for them to do that psychologically, that’s fine.” Debt account aversion, for example, refers to the fact that people want to have fewer debt accounts, meaning they are often eager to pay off accounts in full when they can. Some financial advisers suggest that “because they think it’ll get the ball rolling and you’ll be more likely to pay off the next one.” According to Corbin, there isn’t a lot of evidence for that, and sometimes paying everything off at the outset isn’t ideal. For instance, “It is optimal to start thinking about retirement as soon as you can… but if you’re delaying putting money into retirement because you’re so concerned with your student loan debt,” that can be problematic. Still, Corbin understands the appeal of closing debt accounts. “I am risk-averse, which means if I have a debt I’m probably going to put more money toward that debt that I necessarily should given what the interest rates are and what I could potentially make by investing that money instead.” Financially speaking, “There’s a decent likelihood that I should just pay the minimum on my mortgage… [but] I’ve decided I’m willing to trade off those future gains for the peace of mind that if something goes wrong… I’ll be ahead on my mortgage payment.” Even in Exponential Potential, the right choices aren’t always clear-cut. Corbin describes it as a “sandbox approach” where players are given more opportunity to play around. “This is the trickiest game because there’s no perfect answer for anything,” he says. “Everything has risk.”

Another bias that can affect our financial decisions is known as present bias, the tendency to discount the future in favor of the present. Corbin offers the everyday example of staying up too late. “Nighttime Me wants to stay up and read…. Morning Me is going to be really ticked off at Nighttime Me when they’re exhausted and don’t want to get up.” Research suggests that people can have a harder time identifying with their future selves. That can easily affect our financial decisions, too. “I’m going to let future me worry about that. That guy. Whoever that is.” However, “If you can get people to identify more with that person,” they can sometimes make better decisions. Ultimately, “The game isn’t trying to force people to become investment robots.” We are biased for the present because we live in it, and that’s normal. The purpose of the game is simply “to nudge people… to worry just a little more about the future.”

“Money is basically for safety, security, and happiness,” Corbin says. The ultimate objective is to balance needs, wants, and savings to achieve those three goals both in the present and the future.

Undergraduate Researchers William He and Annie Wang Dig Deeper into Hypergraphs

By Meghna Datta

On November 22, 2021

In Mathematics, Students

Like most things during the height of the pandemic, research that could be conducted virtually was conducted virtually. And that’s why, although juniors William He and Annie Wang have been working together on a research project since last September, they’ve never actually met in person.

He, a Math major from Houston, and Wang, a Computer Science and Math double-major from Raleigh, both work in the lab of Professor Debmalya Panigrahi, where the focus is on research in theoretical computer science, particularly graph algorithms. Wang and He did work on hypergraphs, and, after I asked them to explain what hypergraphs were in the most elementary terms (I am not a Math major), they went back and forth on how exactly to relay hypergraphs to a lay audience.

Here is what they landed on: hypergraphs are essentially generalizations of normal graphs. In a normal graph, there are edges –each edge connects two points. There are also vertices – each point is a vertex. But in a hypergraph, each edge connects multiple points.

He and Wang were looking at a generalization of graph reliability – if all edges disconnect at a certain probability, what is the probability that the graph itself will break down because crucial edges are disconnecting?

Their research adds to existing research on maximum flow problems, which Wikipedia tells us “entail finding a feasible flow through a flow network to obtain the maximum possible flow rate.” In a landmark paper written by T.E. Harris and F.S. Ross in 1955, the two researchers formulated the maximum flow problem using an example of the Soviet railroad and considering what cuts in the railroad would disconnect the nation entirely – and what cuts could be made with little impact to railway traffic flow.

Maximum flow problems are a core tenet of optimization theory, used widely in disciplines from math to computer science to engineering. You may not know what mathematical optimization is, but you’ve seen it in action before: in electronic circuitry, in economics, or unsurprisingly, used by civil engineers in traffic management.

It’s expected to be incredibly difficult to exactly calculate the target value of He and Wang’s question. They landed on an approximation that they know is far from the exact calculation, but still brings them closer to understanding hypergraph connectivity more fully.

So what draws them to research? For He, it’s like an itch. He describes that “sometimes I’ll be watching a movie, and then thirty minutes in I’m thinking about a possible solution to a math problem and then I can’t focus on the movie anymore.” You can’t get on with things until you scratch the itch, but the best part to him is when things finally start to make sense. For Wang, research is just plain fun. She enjoys learning about algorithms and theorems, and she loves the opportunity to work with professors who are at the forefront of their field.

After Duke, He wants to pursue a PhD, likely in theoretical computer science, while Wang is still weighing her options – whether she wants to go into academia or industry. While He came into Duke as a prospective Economics major, in quarantine especially he realized just how much he enjoyed math for the sake of itself.

Wang, similarly, thought she would want to pursue software engineering, but she’s slowly realizing that she likes “solving the problems within the field – problems that I need a PhD to solve.” The magic of research, for her, is that “you’re solving problems that no one has answers to yet.” And wherever the future takes both of them, she says that in doing research, even at the undergraduate level, “you feel like you’re pushing the boundary a tiny bit, and that’s a cool feeling.”

Post by Meghna Datta, Class of 2023

A Puzzling Saga – Meet Duke Math’s Breakout Star!

By Nidhi Srivaths

On October 21, 2021

In Faculty, Mathematics, Science Communication & Education

When I think of crosswords, I think of young and old alike gathered around a couch, scribbling away on a freshly arrived newspaper on a clear Sunday morning. When I think of math, I think of passionate professors covered in chalk dust from a hard day’s work of etching out complicated Greek symbols and numbers on a huge blackboard. These two visions had always been comfortably separate from each other (disjoint sets, if you will) – until I met Dr. Adam Levine.

Meet Duke’s Adam Simon Levine – who shares his name with a pop star, and is a math and crossword rockstar in his own right.

Levine is an Associate Professor in the Department of Mathematics at Duke who studies low-dimensional topology, surfaces, knots and manifolds, using something called Heegaard Floer homology. But apart, from his incredible passion for his field, numerous grants and tongue-twister-esque research interests, Levine has one accomplishment that sets him apart.

He published a crossword puzzle in the New York Times on Sept. 25!

In our Zoom conversation, Levine was bright, chirpy and incredibly excited to talk about his unique love for crosswords and how he’s kept his hobby alive even under the pressures of academic life. As a teenager he loved working on puzzles like Scrabble, playing around with words and etymology, and of course, crosswords. As he grew older, he continued to be fascinated by words.

As an undergrad taking a History of Life class at Harvard, Levine learned about the the impact site of the asteroid that wiped out the dinosaurs, the Chicxulub crater. His first thought was, “That’s a fantastic word– that would make the perfect crossword entry!” His fascination with the word would stay alive for years, until it became the seed entry (the word that anchors the puzzle) for his first crossword to be published in the Times. His love for wordplay through the years is clear, from a self-composed song on Heegaard Floer homology, to a sonnet summary of his PhD thesis.

Levine continued to solve puzzles as he earned his multiple graduate degrees in mathematics, and began publishing his own puzzles a few years on his aptly named blog – Knotty Grids. Crosswords have been a fun and necessary hobby for him – a way to de-stress or take his mind off his academic work. He thinks it is an important example to set for undergraduate students hoping to pursue a career in math – to show them that it is possible to be “full human beings” and have diverse and unique hobbies, interests and a life outside of your research.

Levine often intertwines his puzzles with his primary interest. Some of his puzzles are “themed” – with “Tying Up Loose Ends” built around topological concepts, and the math-y “Indivisible” which was published in The Mathematical Intelligencer. His personal favorites, however, delve into the meta-puzzle genre with “The Queen’s Gambit” and “A Series of Unfortunate Events.”

Levine’s crossword grids are very literally inspired by his daily work with knots.

When I talked to him shortly after the NYT puzzle appeared, Levine was still visibly excited. He compared the puzzle-publishing journey to that of submitting his mathematical research to academic journals – tedious, long and not very fast-moving. Having submitted the crossword to the newspaper more than a year ago, he heard back from them early this year, at which point they began editing and working on the final publication.

As Levine spoke about the editing process, he brought up an issue that has been important to him – diversity in the puzzle world. Because most crosswords have been written by white men, clues and answers are often tilted towards what that demographic considers common knowledge, and minority and female creators have remained a very small voice. Erasure of their identities is common, and Levine related a firsthand experience. His clue for “ROBIN” was “the first Black woman to host Jeopardy.” But it was initially edited to be a reference to the bird.

Levine says he pushed back on the parts of the puzzle that were important to him, and was glad with the final published result.

He says he was amazed by the outpouring of support and congratulations he received after the crossword came out. His parents were “really pumped,” family excited, and colleagues incredibly surprised to see his name in print (on a non-academic paper). Unfortunately, the weekend that the crossword was published, his undergraduate students were cramming for a midterm scheduled that Monday, which Levine figures might be the reason some of his crossword-fanatic students seem to have missed his big moment of glory.

In the following week, I spoke to one of Levine’s academic advisees and other Duke students passionate about puzzles, who shared an initial shock and an immediate subsequent joy in the Blue Devil puzzle representation that this publication debuted.

While Levine has no imminent plans of writing another NYT puzzle, he continues to write for his personal blog. Hopefully, his unique journey inspires students and faculty from different fields and of varied backgrounds to contribute, create and participate, helping make his dream of a diverse puzzle world a reality.

Post by Nidhi Srivaths, Class of 2024

New Blogger Shariar Vaez-Ghaemi: Arts and Artificial Intelligence

By Shariar Vaez-Ghaemi

On October 13, 2021

In Art, Artificial Intelligence, Data, Mathematics, Statistics, Students

Hi! My name is Shariar. My friends usually pronounce that as Shaw-Ree-Awr, and my parents pronounce it as a Share-Ee-Awr, but feel free to mentally process my name as “Sher-Rye-Eer,” “Shor-yor-ior-ior-ior-ior,” or whatever phonetic concoction your heart desires. I always tell people that there’s no right way to interpret language, especially if you’re an AI (which you might be).

Speaking of AI, I’m excited to study statistics and mathematics at Duke! This dream was born out of my high school research internship with New York Times bestselling author Jonah Berger, through which I immersed myself in the applications of machine learning to the social sciences. Since Dr. Berger and I completed our ML-guided study of the social psychology of communicative language, I’ve injected statistical learning techniques into my investigations of political science, finance, and even fantasy football.

*Unwinding in the orchestra room after a performance*

When I’m not cramped behind a Jupyter Notebook or re-reading a particularly long research abstract for the fourth time, I’m often pursuing a completely different interest: the creative arts. I’m an orchestral clarinetist and quasi-jazz pianist by training, but my proudest artistic endeavours have involved cinema. During high school, I wrote and directed three short films, including a post-apocalyptic dystopian comedy and a silent rendition of the epic poem “Epopeya de la Gitana.”

I often get asked whether there’s any bridge between machine learning and the creative arts*, to which the answer is yes! In fact, as part of my entry project for Duke-based developer team Apollo Endeavours, I created a statistical language model that writes original poetry. Wandering
Mind, as I call the system, is just one example of the many ways that artificial intelligence can do what we once considered exclusively-human tasks. The program isn’t quite as talented as Frost or Dickinson, but it’s much better at writing poetry than I am.

*In a movie production (I’m the one wearing a Totoro onesie)*

I look forward to presenting invigorating research topics to blog readers for the next year or more. Though machine learning is my scientific expertise, my investigations could transcend all boundaries of discipline, so you may see me passionately explaining biology experiments, environmental studies, or even macroeconomic forecasts. Go Blue Devils!

(* In truth, I almost never get asked this question by real people unless I say, “You know, there’s actually a connection between machine learning and arts.”)

Introducing: The Duke Space Initiative

By Zella Hanson

On September 17, 2021

In Behavior/Psychology, Biology, Biomedical Engineering, Chemistry, Climate/Global Change, Engineering, Environment/Sustainability, Mathematics, Medicine, Physics, Science Communication & Education

Engineers, medical students, ecologists, political scientists, ethicists, policymakers — come one, come all to the Duke Space Initiative (DSI), “the interdisciplinary home for all things space at Duke.”

At Duke Polis’ “Perspectives on Space: Introducing the Duke Space Initiative” on Sept. 9, DSI co-founder and undergraduate student Ritika Saligram introduced the initiative and moderated a discussion on the current landscape of space studies both at Duke and beyond.

William R. & Thomas L. Perkins Professor of Law Jonathan Wiener began by expressing his excitement in the amount of interest he’s observed in space at Duke.

One of these interested students was Spencer Kaplan. Kaplan, an undergraduate student studying public policy, couldn’t attend Wiener’s Science & Society Dinner Dialogue about policy and risk in the settlement of Mars. Unwilling to miss the learning opportunity, Kaplan set up a one-on-one conversation with Wiener. One thing led to another: the two created a readings course on space law — Wiener hired Kaplan as a research assistant and they worked together to compile materials for the syllabus — then thought, “Why stop there?”

Wiener and Kaplan, together with Chase Hamilton, Jory Weintraub, Tyler Felgenhauer, Dan Buckland, and Somia Youssef, created the Bass Connections project “Going to Mars: Science, Society, and Sustainability,” through which a highly interdisciplinary team of faculty and students discussed problems ranging from the science and technology of getting to Mars, to the social and political reality of living on another planet.

The team produced a website, research papers, policy memos and recommendations, and a policy report for stakeholders including NASA and some prestigious actors in the private sector. According to Saligram, through their work, the team realized the need for a concerted “space for space” at Duke, and the DSI was born. The Initiative seeks to serve more immediately as a resource center for higher education on space, and eventually as the home of a space studies certificate program for undergraduates at Duke.

Wiener sees space as an “opportunity to reflect on what we’ve learned from being on Earth” — to consider how we could avoid mistakes made here and “try to do better if we settle another planet.” He listed a few of the many problems that the Bass Connections examined.

The economics of space exploration have changed: once, national governments funded space exploration; now, private companies like SpaceX, Blue Origin, and Virgin Galactic seek to run the show. Space debris, satellite and launch junk that could impair future launches, is the tragedy of the commons at work — in space. How would we resolve international disputes on other planets and avoid conflict, especially when settlements have different missions? Can we develop technology to ward off asteroids? What if we unintentionally brought microorganisms from one planet to another? How will we make the rules for the settlement of other planets?

These questions are vast — thereby reflecting the vastness of space, commented Saligram — and weren’t answerable within the hour. However, cutting edge research and thinking around them can be found on the Bass Connections’ website.

Earth and Climate Sciences Senior Lecturer Alexander Glass added to Wiener’s list of problems: “terraforming” — or creating a human habitat — on Mars. According to Glass, oxygen “isn’t a huge issue”: MOXIE can buzz Co2 with electricity to produce it. A greater concern is radiation. Without Earth’s magnetosphere, shielding of some sort will be necessary; it takes sixteen feet of rock to produce the same protection. Humans on Mars might have to live underground.

Glass noted that although “we have the science to solve a lot of these problems, the science we’re lagging in is the human aspects of it: the psychological, of humanity living in conditions like isolation.” The engineering could be rock solid. But the mission “will fail because there will be a sociopath we couldn’t predict beforehand.”

Bass Connections project leader and PhD candidate in political science Somia Youssef discussed the need to examine deeply our laws, systems, and culture. Youssef emphasized that we humans have been on Earth for six million years. Like Wiener, she asked how we will “apply what we’ve learned to space” and what changes we should make. How, she mused, do prevailing ideas about humanity “transform in the confines, the harsh environment of space?” Youssef urged the balancing of unity with protection of the things that make us different, as well as consideration for voices that aren’t being represented.

Material Science Professor, Assistant Professor of Surgery, and NASA Human System Risk Manager Dr. Dan Buckland explained that automation has exciting potential in improving medical care in space. If robots can do the “most dangerous aspects” of mission medical care, humans won’t have to. Offloading onto “repeatable devices” will reduce the amount of accidents and medical capabilities needed in space.

Multiple panelists also discussed the “false dichotomy” between spending resources on space and back home on Earth. Youssef pointed out that many innovations which have benefited (or will benefit) earthly humanity have come from the excitement and passion that comes from investing in space. Saligram stated that space is an “extension of the same social and policy issues as the ones we face on Earth, just in a different context.” This means that solutions we find in our attempt to settle Mars and explore the universe can be “reverse engineered” to help Earth-dwelling humans everywhere.

Saligram opened up the panel for discussion, and one guest asked Buckland how he ended up working for NASA. Buckland said his advice was to “be in rooms you’re not really supposed to be in, and eventually people will start thinking you’re supposed to be there.”

Youssef echoed this view, expressing the need for diverse perspectives in space exploration. She’s most excited by all the people “who are interested in space, but don’t know if there’s enough space for them.”

If this sounds like you, check out the Duke Space Initiative. They’ve got space.

Post by Zella Hanson

A New Algorithm for “In-Betweening” images applied to Covid, Aging and Continental Drift

By Guest Post

On April 14, 2021

In Computers/Technology, Data, Faculty, Guest Post, Mathematics, Physics, Visualization

Collaborating with a colleague in Shanghai, we recently published an article that explains the mathematical concept of ‘in-betweening,’in images – calculating intermediate stages of changes in appearance from one image to the next.

Our equilibrium-driven deformation algorithm (EDDA) was used to demonstrate three difficult tasks of ‘in-betweening’ images: Facial aging, coronavirus spread in the lungs, and continental drift.

Part I. Understanding Pneumonia Invasion and Retreat in COVID-19

The pandemic has influenced the entire world and taken away nearly 3 million lives to date. If a person were unlucky enough to contract the virus and COVID-19, one way to diagnose them is to carry out CT scans of their lungs to visualize the damage caused by pneumonia.

However, it is impossible to monitor the patient all the time using CT scans. Thus, the invading process is usually invisible for doctors and researchers.

To solve this difficulty, we developed a mathematical algorithm which relies on only two CT scans to simulate the pneumonia invasion process caused by COVID-19.

We compared a series of CT scans of a Chinese patient taken at different times. This patient had severe pneumonia caused by COVID-19 but recovered after a successful treatment. Our simulation clearly revealed the pneumonia invasion process in the patient’s lungs and the fading away process after the treatment.

Our simulation results also identify several significant areas in which the patient’s lungs are more vulnerable to the virus and other areas in which the lungs have better response to the treatment. Those areas were perfectly consistent with the medical analysis based on this patient’s actual, real-time CT scan images. The consistency of our results indicates the value of the method.

The COVID-19 pneumonia invading (upper panel) and fading away (lower panel) process from the data-driven simulations. Red circles indicate four significant areas in which the patient’s lungs were more vulnerable to the pneumonia and blue circles indicate two significant areas in which the patient’s lungs had better response to the treatment. (Image credit: Gao et al., 2021)

We also applied this algorithm to simulate human facial changes over time, in which the aging processes for different parts of a woman’s face were automatically created by the algorithm with high resolution. (Image credit: Gao et al., 2021. Video)

Part II. Solving the Puzzle of Continental Drift

It has always been mysterious how the continents we know evolved and formed from the ancient single supercontinent, Pangaea. But then German polar researcher Alfred Wegener proposed the continental drift hypothesis in the early 20th century. Although many geologists argued about his hypothesis initially, more sound evidence such as continental structures, fossils and the magnetic polarity of rocks has supported Wegener’s proposition.

Our data-driven algorithm has been applied to simulate the possible evolution process of continents from Pangaea period.

The underlying forces driving continental drift were determined by the equilibrium status of the continents on the current planet. In order to describe the edges that divide the land to create oceans, we proposed a delicate thresholding scheme.

The formation and deformation for different continents is clearly revealed in our simulation. For example, the ‘drift’ of the Antarctic continent from Africa can be seen happening. This exciting simulation presents a quick and obvious way for geologists to establish more possible lines of inquiry about how continents can drift from one status to another, just based on the initial and equilibrium continental status. Combined with other technological advances, this data-driven method may provide a path to solve Wegener’s puzzle of continental drift.

The theory of continental drift reconciled similar fossil plants and animals now found on widely separated continents. The southern part after Pangaea breaks (Gondwana) is shown here evidence of Wegener’s theory. (Image credit: United States Geological Survey)

The continental drift process of the data-driven simulations. Black arrow indicates the formation of the Antarctic. (Image credit: Gao et al., 2021)

The study was supported by the Department of Mathematics and Physics, Duke University.

CITATION: “Inbetweening auto-animation via Fokker-Planck dynamics and thresholding,” Yuan Gao, Guangzhen Jin & Jian-Guo Liu. Inverse Problems and Imaging, February, 2021, DOI: 10.3934/ipi.2021016. Online: http://www.aimsciences.org/article/doi/10.3934/ipi.2021016

Yuan Gao is the William W. Elliot Assistant Research Professor in the department of mathematics, Trinity College of Arts & Sciences.

Jian-Guo Liu is a Professor in the departments of mathematics and physics, Trinity College of Arts & Sciences.