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Want to solve a complex problem? Applied math can help

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theconversation.com – Alan Veliz-Cuba, Associate Professor of Mathematics, University of Dayton – 2024-10-01 07:25:00

Applied mathematicians use math to model real-world situations.
Ariel Skelley/DigitalVision via Getty Images

Alan Veliz-Cuba, University of Dayton

You can probably think of a time when you’ve used math to solve an everyday problem, such as calculating a tip at a restaurant or determining the square footage of a room. But what role does math play in solving complex problems such as curing a disease?

In my job as an applied mathematician, I use mathematical tools to study and solve complex problems in biology. I have worked on problems involving gene and neural networks such as interactions between cells and decision-making. To do this, I create descriptions of a real-world situation in mathematical language. The act of turning a situation into a mathematical representation is called modeling.

Translating real situations into mathematical terms

If you ever solved an arithmetic problem about the speed of trains or cost of groceries, that’s an example of mathematical modeling. But for more difficult questions, even just writing the real-world scenario as a math problem can be complicated. This process requires a lot of creativity and understanding of the problem at hand and is often the result of applied mathematicians working with scientists in other disciplines.

A group of researchers conversing around a conference table.
Applied mathematicians collaborate with scientists in other fields to answer a wide variety of questions.
Hinterhaus Productions/DigitalVision via Getty Images

As an example, we could represent a game of Sudoku as a mathematical model. In Sudoku, the player fills empty boxes in a puzzle with numbers between 1 and 9 subject to some rules, such as no repeated numbers in any row or column.

The puzzle begins with some prefilled boxes, and the goal is to figure out which numbers go in the rest of the boxes.

Imagine that a variable, say x, represents the number that goes in one of those empty boxes. We can guarantee that x is between 1 and 9 by saying that x solves the equation (x-1)(x-2) … (x-9)=0. This equation is true only when one of the factors on the left side is zero. Each of the factors on the left side is zero only when x is a number between 1 and 9; for example, (x-1)=0 when x=1. This equation encodes a fact about our game of Sudoku, and we can encode the other features of the game similarly. The resulting model of Sudoku will be a set of equations with 81 variables, one for each box in the puzzle.

Another situation we might model is the concentration of a drug, say aspirin, in a person’s bloodstream. In this case, we would be interested in how the concentration changes as we ingest aspirin and the body metabolizes it. Just like with Sudoku, one can create a set of equations that describe how the concentration of aspirin evolves over time and how additional ingestion affects the dynamics of this medication. In contrast to Sudoku, however, the variables that represent concentrations are not static but rather change over time.

A pen resting on a sudoku puzzle in a newspaper.
Sudoku is an example of a situation that can be modeled mathematically.
Peter Dazeley/The Image Bank via Getty Images

But the act of modeling is not always so straightforward. How would we model diseases such as cancer? Is it enough to model the size and shape of a tumor, or do we need to model every single blood vessel inside the tumor? Every single cell? Every single chemical in each cell? There is much that is unknown about cancer, so how can we model such unknown features? Is it even possible?

Applied mathematicians have to find a balance between models that are realistic enough to be useful and simple enough to be implemented. Building these models may take several years, but in collaboration with experimental scientists, the act of trying to find a model often provides novel insight into the real-world problem.

Mathematical models help find real solutions

After writing a mathematical problem to represent a situation, the second step in the modeling process is to solve the problem.

For Sudoku, we need to solve a collection of equations with 81 variables. For the aspirin example, we need to solve an equation that describes the rate of change of concentrations. This is where all the math that has been and is still being invented comes into play. Areas of pure math such as algebra, analysis, combinatorics and many others can be used – in some cases combined – to solve the complex math problems arising from applications of math to the real world.

The third step of the modeling process consists of translating the mathematical solution into the solution to the applied problem. In the case of Sudoku, the solution to the equations tells us which number should go in each box to solve the puzzle. In the case of aspirin, the solution would be a set of curves that tell us the aspirin concentration in the digestive system and bloodstream. This is how applied mathematics works.

When creating a model isn’t enough

Or is it? While this three-step process is the ideal process of applied math, reality is more complicated. Once I reach the second step where I want the solution of the math problem, very often, if not most of the time, it turns out that no one knows how to solve the math problem in the model. In some cases, the math to study the problem doesn’t even exist.

For example, it is difficult to analyze models of cancer because the interactions between genes, proteins and chemicals are not as straightforward as the relationships between boxes in a game of Sudoku. The main difficulty is that these interactions are “nonlinear,” meaning that the effect of two inputs is not simply the sum of the individual effects. To address this, I have been working on novel ways to study nonlinear systems, such as Boolean network theory and polynomial algebra. With this and traditional approaches, my colleagues and I have studied questions in areas such as
decision-making,
gene networks,
cellular differentiation and
limb regeneration.

When approaching unsolved applied math problems, the distinction between applied and pure mathematics often vanishes. Areas that were considered at one time too abstract have been exactly what is needed for modern problems. This highlights the importance of math for all of us; current areas of pure mathematics can become the applied mathematics of tomorrow and be the tools needed for complex, real-world problems.The Conversation

Alan Veliz-Cuba, Associate Professor of Mathematics, University of Dayton

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Perfect brownies baked at high altitude are possible thanks to Colorado’s home economics pioneer Inga Allison

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theconversation.com – Tobi Jacobi, Professor of English, Colorado State University – 2025-04-22 07:47:00

Students work in the high-altitude baking laboratory.
Archives and Special Collections, Colorado State University

Tobi Jacobi, Colorado State University and Caitlin Clark, Colorado State University

Many bakers working at high altitudes have carefully followed a standard recipe only to reach into the oven to find a sunken cake, flat cookies or dry muffins.

Experienced mountain bakers know they need a few tricks to achieve the same results as their fellow artisans working at sea level.

These tricks are more than family lore, however. They originated in the early 20th century thanks to research on high-altitude baking done by Inga Allison, then a professor at Colorado State University. It was Allison’s scientific prowess and experimentation that brought us the possibility of perfect high-altitude brownies and other baked goods.

A recipe for brownies at high altitude.
Inga Allison’s high-altitude brownie recipe.
Archives and Special Collections, Colorado State University

We are two current academics at CSU whose work has been touched by Allison’s legacy.

One of us – Caitlin Clark – still relies on Allison’s lessons a century later in her work as a food scientist in Colorado. The other – Tobi Jacobi – is a scholar of women’s rhetoric and community writing, and an enthusiastic home baker in the Rocky Mountains, who learned about Allison while conducting archival research on women’s work and leadership at CSU.

That research developed into “Knowing Her,” an exhibition Jacobi developed with Suzanne Faris, a CSU sculpture professor. The exhibit highlights dozens of women across 100 years of women’s work and leadership at CSU and will be on display through mid-August 2025 in the CSU Fort Collins campus Morgan Library.

A pioneer in home economics

Inga Allison is one of the fascinating and accomplished women who is part of the exhibit.

Allison was born in 1876 in Illinois and attended the University of Chicago, where she completed the prestigious “science course” work that heavily influenced her career trajectory. Her studies and research also set the stage for her belief that women’s education was more than preparation for domestic life.

In 1908, Allison was hired as a faculty member in home economics at Colorado Agricultural College, which is now CSU. She joined a group of faculty who were beginning to study the effects of altitude on baking and crop growth. The department was located inside Guggenheim Hall, a building that was constructed for home economics education but lacked lab equipment or serious research materials.

A sepia-toned photograph of Inga Allison, a white woman in dark clothes with her hair pulled back.
Inga Allison was a professor of home economics at Colorado Agricultural College, where she developed recipes that worked in high altitudes.
Archives and Special Collections, Colorado State University

Allison took both the land grant mission of the university with its focus on teaching, research and extension and her particular charge to prepare women for the future seriously. She urged her students to move beyond simple conceptions of home economics as mere preparation for domestic life. She wanted them to engage with the physical, biological and social sciences to understand the larger context for home economics work.

Such thinking, according to CSU historian James E. Hansen, pushed women college students in the early 20th century to expand the reach of home economics to include “extension and welfare work, dietetics, institutional management, laboratory research work, child development and teaching.”

News articles from the early 1900s track Allison giving lectures like “The Economic Side of Natural Living” to the Colorado Health Club and talks on domestic science to ladies clubs and at schools across Colorado. One of her talks in 1910 focused on the art of dishwashing.

Allison became the home economics department chair in 1910 and eventually dean. In this leadership role, she urged then-CSU President Charles Lory to fund lab materials for the home economics department. It took 19 years for this dream to come to fruition.

In the meantime, Allison collaborated with Lory, who gave her access to lab equipment in the physics department. She pieced together equipment to conduct research on the relationship between cooking foods in water and atmospheric pressure, but systematic control of heat, temperature and pressure was difficult to achieve.

She sought other ways to conduct high-altitude experiments and traveled across Colorado where she worked with students to test baking recipes in varied conditions, including at 11,797 feet in a shelter house on Fall River Road near Estes Park.

Early 1900s car traveling in the Rocky Mountains.
Inga Allison tested her high-altitude baking recipes at 11,797 feet at the shelter house on Fall River Road, near Estes Park, Colorado.
Archives and Special Collections, Colorado State University

But Allison realized that recipes baked at 5,000 feet in Fort Collins and Denver simply didn’t work in higher altitudes. Little advancement in baking methods occurred until 1927, when the first altitude baking lab in the nation was constructed at CSU thanks to Allison’s research. The results were tangible — and tasty — as public dissemination of altitude-specific baking practices began.

A 1932 bulletin on baking at altitude offers hundreds of formulas for success at heights ranging from 4,000 feet to over 11,000 feet. Its author, Marjorie Peterson, a home economics staff person at the Colorado Experiment Station, credits Allison for her constructive suggestions and support in the development of the booklet.

Science of high-altitude baking

As a senior food scientist in a mountain state, one of us – Caitlin Clark – advises bakers on how to adjust their recipes to compensate for altitude. Thanks to Allison’s research, bakers at high altitude today can anticipate how the lower air pressure will affect their recipes and compensate by making small adjustments.

The first thing you have to understand before heading into the kitchen is that the higher the altitude, the lower the air pressure. This lower pressure has chemical and physical effects on baking.

Air pressure is a force that pushes back on all of the molecules in a system and prevents them from venturing off into the environment. Heat plays the opposite role – it adds energy and pushes molecules to escape.

When water is boiled, molecules escape by turning into steam. The less air pressure is pushing back, the less energy is required to make this happen. That’s why water boils at lower temperatures at higher altitudes – around 200 degrees Fahrenheit in Denver compared with 212 F at sea level.

So, when baking is done at high altitude, steam is produced at a lower temperature and earlier in the baking time. Carbon dioxide produced by leavening agents also expands more rapidly in the thinner air. This causes high-altitude baked goods to rise too early, before their structure has fully set, leading to collapsed cakes and flat muffins. Finally, the rapid evaporation of water leads to over-concentration of sugars and fats in the recipe, which can cause pastries to have a gummy, undesirable texture.

Allison learned that high-altitude bakers could adjust to their environment by reducing the amount of sugar or increasing liquids to prevent over-concentration, and using less of leavening agents like baking soda or baking powder to prevent dough from rising too quickly.

Allison was one of many groundbreaking women in the early 20th century who actively supported higher education for women and advanced research in science, politics, humanities and education in Colorado.

Others included Grace Espy-Patton, a professor of English and sociology at CSU from 1885 to 1896 who founded an early feminist journal and was the first woman to register to vote in Fort Collins. Miriam Palmer was an aphid specialist and master illustrator whose work crafting hyper-realistic wax apples in the early 1900s allowed farmers to confirm rediscovery of the lost Colorado Orange apple, a fruit that has been successfully propagated in recent years.

In 1945, Allison retired as both an emerita professor and emerita dean at CSU. She immediately stepped into the role of student and took classes in Russian and biochemistry.

In the fall of 1958, CSU opened a new dormitory for women that was named Allison Hall in her honor.

“I had supposed that such a thing happened only to the very rich or the very dead,” Allison told reporters at the dedication ceremony.

Read more of our stories about Colorado.The Conversation

Tobi Jacobi, Professor of English, Colorado State University and Caitlin Clark, Senior Food Scientist at the CSU Spur Food Innovation Center, Colorado State University

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Why don’t humans have hair all over their bodies? A biologist explains our lack of fur

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theconversation.com – Maria Chikina, Assistant Professor of Computational and Systems Biology, University of Pittsburgh – 2025-04-21 07:33:00

Some mammals are super hairy, some are not.
Ed Jones/AFP via Getty Images

Maria Chikina, University of Pittsburgh

Curious Kids is a series for children of all ages. If you have a question you’d like an expert to answer, send it to CuriousKidsUS@theconversation.com.


Why don’t humans have hair all over their bodies like other animals? – Murilo, age 5, Brazil


Have you ever wondered why you don’t have thick hair covering your whole body like a dog, cat or gorilla does?

Humans aren’t the only mammals with sparse hair. Elephants, rhinos and naked mole rats also have very little hair. It’s true for some marine mammals, such as whales and dolphins, too.

Scientists think the earliest mammals, which lived at the time of the dinosaurs, were quite hairy. But over hundreds of millions of years, a small handful of mammals, including humans, evolved to have less hair. What’s the advantage of not growing your own fur coat?

I’m a biologist who studies the genes that control hairiness in mammals. Why humans and a small number of other mammals are relatively hairless is an interesting question. It all comes down to whether certain genes are turned on or off.

Hair benefits

Hair and fur have many important jobs. They keep animals warm, protect their skin from the sun and injuries and help them blend into their surroundings.

They even assist animals in sensing their environment. Ever felt a tickle when something almost touches you? That’s your hair helping you detect things nearby.

Humans do have hair all over their bodies, but it is generally sparser and finer than that of our hairier relatives. A notable exception is the hair on our heads, which likely serves to protect the scalp from the sun. In human adults, the thicker hair that develops under the arms and between the legs likely reduces skin friction and aids in cooling by dispersing sweat.

So hair can be pretty beneficial. There must have been a strong evolutionary reason for people to lose so much of it.

Why humans lost their hair

The story begins about 7 million years ago, when humans and chimpanzees took different evolutionary paths. Although scientists can’t be sure why humans became less hairy, we have some strong theories that involve sweat.

Humans have far more sweat glands than chimps and other mammals do. Sweating keeps you cool. As sweat evaporates from your skin, heat energy is carried away from your body. This cooling system was likely crucial for early human ancestors, who lived in the hot African savanna.

Of course, there are plenty of mammals living in hot climates right now that are covered with fur. Early humans were able to hunt those kinds of animals by tiring them out over long chases in the heat – a strategy known as persistence hunting.

Humans didn’t need to be faster than the animals they hunted. They just needed to keep going until their prey got too hot and tired to flee. Being able to sweat a lot, without a thick coat of hair, made this endurance possible.

Genes that control hairiness

To better understand hairiness in mammals, my research team compared the genetic information of 62 different mammals, from humans to armadillos to dogs and squirrels. By lining up the DNA of all these different species, we were able to zero in on the genes linked to keeping or losing body hair.

Among the many discoveries we made, we learned humans still carry all the genes needed for a full coat of hair – they are just muted or switched off.

In the story of “Beauty and the Beast,” the Beast is covered in thick fur, which might seem like pure fantasy. But in real life some rare conditions can cause people to grow a lot of hair all over their bodies. This condition, called hypertrichosis, is very unusual and has been called “werewolf syndrome” because of how people who have it look.

A detailed painting of a man and a woman standing next to one another in historical looking clothes. The man's face is covered in hair, while the woman's is not.
Petrus Gonsalvus and his wife, Catherine, painted by Joris Hoefnagel, circa 1575.
National Gallery of Art

In the 1500s, a Spanish man named Petrus Gonsalvus was born with hypertrichosis. As a child he was sent in an iron cage like an animal to Henry II of France as a gift. It wasn’t long before the king realized Petrus was like any other person and could be educated. In time, he married a lady, forming the inspiration for the “Beauty and the Beast” story.

While you will probably never meet someone with this rare trait, it shows how genes can lead to unique and surprising changes in hair growth.


Hello, curious kids! Do you have a question you’d like an expert to answer? Ask an adult to send your question to CuriousKidsUS@theconversation.com. Please tell us your name, age and the city where you live.

And since curiosity has no age limit – adults, let us know what you’re wondering, too. We won’t be able to answer every question, but we will do our best.The Conversation

Maria Chikina, Assistant Professor of Computational and Systems Biology, University of Pittsburgh

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Scientists found a potential sign of life on a distant planet – an astronomer explains why many are still skeptical

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theconversation.com – Daniel Apai, Associate Dean for Research and Professor of Astronomy and Planetary Sciences, University of Arizona – 2025-04-18 17:44:00

An illustration of the exoplanet K2-18b, which some research suggests may be covered by deep oceans.
NASA, ESA, CSA, Joseph Olmsted (STScI)

Daniel Apai, University of Arizona

A team of astronomers announced on April 16, 2025, that in the process of studying a planet around another star, they had found evidence for an unexpected atmospheric gas. On Earth, that gas – called dimethyl sulfide – is mostly produced by living organisms.

In April 2024, the James Webb Space Telescope stared at the host star of the planet K2-18b for nearly six hours. During that time, the orbiting planet passed in front of the star. Starlight filtered through its atmosphere, carrying the fingerprints of atmospheric molecules to the telescope.

A diagram showing planets and stars emitting light, which goes through JWST detectors, where it's split into different wavelengths to make a spectrum. Each spectrum suggests the presence of a different element.
JWST’s cameras can detect molecules in the atmosphere of a planet by looking at light that passed through that atmosphere.
European Space Agency

By comparing those fingerprints to 20 different molecules that they would potentially expect to observe in the atmosphere, the astronomers concluded that the most probable match was a gas that, on Earth, is a good indicator of life.

I am an astronomer and astrobiologist who studies planets around other stars and their atmospheres. In my work, I try to understand which nearby planets may be suitable for life.

K2-18b, a mysterious world

To understand what this discovery means, let’s start with the bizarre world it was found in. The planet’s name is K2-18b, meaning it is the first planet in the 18th planetary system found by the extended NASA Kepler mission, K2. Astronomers assign the “b” label to the first planet in the system, not “a,” to avoid possible confusion with the star.

K2-18b is a little over 120 light-years from Earth – on a galactic scale, this world is practically in our backyard.

Although astronomers know very little about K2-18b, we do know that it is very unlike Earth. To start, it is about eight times more massive than Earth, and it has a volume that’s about 18 times larger. This means that it’s only about half as dense as Earth. In other words, it must have a lot of water, which isn’t very dense, or a very big atmosphere, which is even less dense.

Astronomers think that this world could either be a smaller version of our solar system’s ice giant Neptune, called a mini-Neptune, or perhaps a rocky planet with no water but a massive hydrogen atmosphere, called a gas dwarf.

Another option, as University of Cambridge astronomer Nikku Madhusudhan recently proposed, is that the planet is a “hycean world”.

That term means hydrogen-over-ocean, since astronomers predict that hycean worlds are planets with global oceans many times deeper than Earth’s oceans, and without any continents. These oceans are covered by massive hydrogen atmospheres that are thousands of miles high.

Astronomers do not know yet for certain that hycean worlds exist, but models for what those would look like match the limited data JWST and other telescopes have collected on K2-18b.

This is where the story becomes exciting. Mini-Neptunes and gas dwarfs are unlikely to be hospitable for life, because they probably don’t have liquid water, and their interior surfaces have enormous pressures. But a hycean planet would have a large and likely temperate ocean. So could the oceans of hycean worlds be habitable – or even inhabited?

Detecting DMS

In 2023, Madhusudhan and his colleagues used the James Webb Space Telescope’s short-wavelength infrared camera to inspect starlight that filtered through K2-18b’s atmosphere for the first time.

They found evidence for the presence of two simple carbon-bearing molecules – carbon monoxide and methane – and showed that the planet’s upper atmosphere lacked water vapor. This atmospheric composition supported, but did not prove, the idea that K2-18b could be a hycean world. In a hycean world, water would be trapped in the deeper and warmer atmosphere, closer to the oceans than the upper atmosphere probed by JWST observations.

Intriguingly, the data also showed an additional, very weak signal. The team found that this weak signal matched a gas called dimethyl sulfide, or DMS. On Earth, DMS is produced in large quantities by marine algae. It has very few, if any, nonbiological sources.

This signal made the initial detection exciting: on a planet that may have a massive ocean, there is likely a gas that is, on Earth, emitted by biological organisms.

An illustration of what scientists imagine K2-18b to look like, which looks a little like Earth, with clouds and a translucent surface.
K2-18b could have a deep ocean spanning the planet, and a hydrogen atmosphere.
Amanda Smith, Nikku Madhusudhan (University of Cambridge), CC BY-SA

Scientists had a mixed response to this initial announcement. While the findings were exciting, some astronomers pointed out that the DMS signal seen was weak and that the hycean nature of K2-18b is very uncertain.

To address these concerns, Mashusudhan’s team turned JWST back to K2-18b a year later. This time, they used another camera on JWST that looks for another range of wavelengths of light. The new results – announced on April 16, 2025 – supported their initial findings.

These new data show a stronger – but still relatively weak – signal that the team attributes to DMS or a very similar molecule. The fact that the DMS signal showed up on another camera during another set of observations made the interpretation of DMS in the atmosphere stronger.

Madhusudhan’s team also presented a very detailed analysis of the uncertainties in the data and interpretation. In real-life measurements, there are always some uncertainties. They found that these uncertainties are unlikely to account for the signal in the data, further supporting the DMS interpretation. As an astronomer, I find that analysis exciting.

Is life out there?

Does this mean that scientists have found life on another world? Perhaps – but we still cannot be sure.

First, does K2-18b really have an ocean deep beneath its thick atmosphere? Astronomers should test this.

Second, is the signal seen in two cameras two years apart really from dimethyl sulfide? Scientists will need more sensitive measurements and more observations of the planet’s atmosphere to be sure.

Third, if it is indeed DMS, does this mean that there is life? This may be the most difficult question to answer. Life itself is not detectable with existing technology. Astronomers will need to evaluate and exclude all other potential options to build their confidence in this possibility.

The new measurements may lead researchers toward a historic discovery. However, important uncertainties remain. Astrobiologists will need a much deeper understanding of K2-18b and similar worlds before they can be confident in the presence of DMS and its interpretation as a signature of life.

Scientists around the world are already scrutinizing the published study and will work on new tests of the findings, since independent verification is at the heart of science.

Moving forward, K2-18b is going to be an important target for JWST, the world’s most sensitive telescope. JWST may soon observe other potential hycean worlds to see if the signal appears in the atmospheres of those planets, too.

With more data, these tentative conclusions may not stand the test of time. But for now, just the prospect that astronomers may have detected gasses emitted by an alien ecosystem that bubbled up in a dark, blue-hued alien ocean is an incredibly fascinating possibility.

Regardless of the true nature of K2-18b, the new results show how using the JWST to survey other worlds for clues of alien life will guarantee that the next years will be thrilling for astrobiologists.The Conversation

Daniel Apai, Associate Dean for Research and Professor of Astronomy and Planetary Sciences, University of Arizona

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