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What the towering 20th-century thinkers Simone and André Weil can teach today’s math educators

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theconversation.com – Scott Taylor, Professor of Mathematics, Colby College – 2024-08-20 07:19:08

André Weil (left, in 1956) and Simone Weil (in 1922) were siblings who became prominent in mathematics and philosophy, respectively.
Konrad Jacobs (right) and Anonymous (left) via Wikimedia Commons, CC BY-SA

Scott Taylor, Colby College

Like most mathematicians, I hear confessions from complete strangers: the inevitable “I was always bad at math.” I suppress the response, “You are forgiven, my child.”

Why does it feel like a sin to struggle in math? Why are so many traumatized by their mathematics education? Is learning math worthwhile?

Sometimes agreeing and sometimes disagreeing, André and Simone Weil were the sort of siblings who would argue about such questions. André achieved renown as a mathematician; Simone was a formidable philosopher and mystic. André focused on applying algebra and geometry to deep questions about the structures of whole numbers, while Simone was concerned with how the world can be soul-crushing.

Both wrestled with the best way to teach math. Their insights and contradictions point to the fundamental role that mathematics and mathematics education play in human life and culture.

André Weil’s rigorous mathematics

André Weil, aged about 50, stands on a sunny path. A child is standing next to him shielding their face.
André Weil, pictured here in 1956, was a prominent mathematician in the 20th century.
Konrad Jacobs via Wikimedia Commons, CC BY-SA

Unlike the prominent French mathematicians of previous generations, André, who was born in 1906 and died in 1998, spent little time philosophizing. For him, mathematics was a living subject endowed with a long and substantial history, but as he remarked, he saw “no need to defend (it).”

In his interactions with people, André was an unsparing critic. Although admired by some colleagues, he was feared by and at times disdainful of his students. He co-founded the Bourbaki mathematics collective that used abstraction and logical rigor to restructure mathematics from the ground up.

Nicolas Bourbaki’s commitment to proceeding from first principles, however, did not completely encapsulate his conception of what constituted worthwhile mathematics. André was attuned to how math should be taught differently to different audiences.

Tempering the Bourbaki spirit, he defined rigor as “(not) proving everything, but … endeavoring to assume as little as possible at every stage.”

A black and white photo of one woman and six men standing in front of a doorway.
The Bourbaki congress in 1938. Simone, pictured at front left, accompanies André, obscured at back left.
Unknown author via Wikimedia Commons

In other words, absolute rigor has its place, but teachers must be willing to take their audience into account. He believed that teachers must motivate students by providing them meaningful problems and provocative examples. Excitement for advanced students comes in encountering the unknown; for beginning students, it emerges from solving questions of, as he put it, “theoretical or practical importance.” He insisted that math “must be a source of intellectual excitement.”

André’s own sense of intellectual excitement came from applying insights from one part of mathematics to other parts. In a letter to his sister, André described his work as seeking a metaphorical “Rosetta stone” of analogies between advanced versions of three basic mathematical objects: numbers, polynomials and geometric spaces.

André described mathematics in romantic terms. Initially, the relationship between the different parts of mathematics is that of passionate lovers, exchanging “furtive caresses” and having “inexplicable quarrels.” But as the analogies eventually give way to a single unified theory, the affair grows cold: “Gone is the analogy: gone are the two theories, their conflicts and their delicious reciprocal reflections … alas, all is just one theory, whose majestic beauty can no longer excite us.”

Despite being passionless, this theory that unifies numbers, polynomials and geometry gets to the heart of mathematics; André pursued it intensely. In the words of a colleague, André sought the “real meaning of every basic mathematical phenomenon.” For him, unlike his sister, this real meaning was found in the careful definitions, precisely articulated theorems and rigorous proofs of the most advanced mathematics of his time. Romantic language simply described the emotions of the mathematician encountering the mathematics; it did not point to any deeper significance.

Simone Weil and the philosophy of mathematics

On the other hand, Simone, who was born three years after André and died 55 years before him, used philosophy and religion to investigate the value of mathematics for nongeniuses, in addition to her work on politics, war, science and suffering.

A headshot of Simone Weil, around age 16. She has short hair and glasses.
Simone Weil, pictured here in 1925, was a prominent philosopher.
Anonymous via Wikimedia Commons

All of her writing – indeed, her life – has a maddening quality to it. In her polished essays, as well as her private letters and journals, she will often make an extreme assertion or enigmatic comment. Such assertions might concern the motivations of scientists, the psychological state of a sufferer, the nature of labor, an analysis of labor unions or an interpretation of Greek philosophers and mathematicians. She is not a systematic thinker but rather circles around and around clusters of ideas and themes. When I read her writing, I am often taken aback. I start to argue with her, bringing up counterexamples and qualifications, but I eventually end up granting the essence of her point. Simone was known for the single-minded pursuit of her ideals.

Despite the discomfort her viewpoints provoke, they are worth engaging. Although her childhood was largely happy, her whole life she felt stupid in comparison with her brother. She channeled her feelings of inadequacy into an exploration of how to experience a meaningful existence in the face of oppression and affliction. Over her life, she developed an interpretation of beauty and suffering intertwined with geometry.

Along with her lifelong mathematical discussions with André, her views were influenced by one of her first jobs as a teacher. In a letter to a colleague, she described her pupils as struggling because they “regarded the various sciences as compilations of cut-and-dried knowledge.” Like André, Simone saw the ability to motivate students as the key to good teaching. She taught mathematics as a subject embedded in culture, emphasizing overarching historical themes. Even those students who were “most ignorant in science” followed her lectures with “passionate interest.”

For Simone, however, the primary purpose of mathematics education was to develop the virtue of attention. Mathematics confronts us with our mistakes, and the contemplation of these inadequacies brings the ability to concentrate on one thing, at the exclusion of all else, to the fore. As a math teacher, I frequently see students grit their teeth and furrow their brow, developing only a headache and resentment. According to Simone, however, true attention arises from joy and desire. We hold our knowledge lightly and wait with detached thought for light to arrive.

For Simone, the “first duty” of teachers is to help students develop, through their studies, the ability to apprehend God, which she conceptualized as a blending of Plato’s description of the ultimate Good with Christian conceptions of the self-abnegating God. A true understanding of God results in love for the afflicted.

Simone might even locate the lingering anxiety and frustration of many former math students in the absence of attention paid to them by their teachers.

Authors grapple with the Weil legacy

Recently, others have wrestled with the Weil legacy.

Sylvie Weil, André’s daughter, was born shortly before Simone’s death. Her family experience was that of being mistaken for her aunt, ignored or demeaned by her father and not being acknowledged and appreciated by those in her orbit.

Similarly, author Karen Olsson uses Simone and André to explore her own conflicted relationship with mathematics. Her forlorn quest to understand André’s mathematics eerily reflects Simone’s desire to understand André’s work and Sylvie’s desire to be seen as her own person, to not be in Simone’s shadow. Olsson studied with exceptional math teachers and students, all the while feeling out of place, overwhelmed and intimidated by her fellow students. Most painfully, in the process of writing her book on the Weil siblings, Olsson asks a mathematician, who had been a student with her, for help in understanding some aspect of André’s mathematics. She was ignored. Both Sylvie Weil and Karen Olsson are living witnesses to Simone’s observation that each of us cries out to be seen.

Christopher Jackson, on the other hand, gives testimony to how mathematics can live up to Simone’s vision. Jackson is incarcerated in a federal prison but found a new life through mathematics. His correspondence with mathematician Francis Su is the backbone of Su’s 2020 book “Mathematics for Human Flourishing,” which uses Simone’s observation that “every being cries out silently to be read differently” as a leitmotif. Su identifies aspects of mathematics that promote human flourishing, such as beauty, truth, freedom and love. In their own ways, both Simone and André would likely agree.The Conversation

Scott Taylor, Professor of Mathematics, Colby College

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As the Taurid meteor shower passes by Earth, pseudoscience rains down – and obscures a potential real threat from space

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theconversation.com – Mark Boslough, Research Associate Professor of Earth and Planetary Sciences, University of New Mexico – 2024-11-15 07:31:00

This image of a Taurid fireball was taken in 2014 by NASA’s All Sky Fireball Network in Tullahoma, Tenn.

NASA

Mark Boslough, University of New Mexico

With the Taurid meteor shower now hitting the night skies worldwide, look for what could be a celestial treat – you might see shooting stars, and maybe even fireballs, the biggest and brightest meteors.

As the full moon begins to wane after Nov. 15, the sky will be darker, due to diminishing moonlight, so finding the meteors will get easier. That said, the best visibility for the meteors through the rest of the month will come just before moonrise each night.

Beyond the light show, there is something else that scientists as well as onlookers have long wondered about: the possibility that bigger chunks are in the Taurid meteor streams, chunks the size of boulders, buildings or even mountains.

And if that’s true, could one of those monster-sized Taurid objects collide with Earth? Could they wipe out a city, or worse? Is it possible that’s already happened, sometime in our planet’s past?

This animation simulates the motion of the hypothetical Taurid meteor swarm through space.

As a physicist who researches the risk that comets and asteroids pose to the Earth, I’m aware that this is a subject where pseudoscience often competes with actual science. So let’s try to find the line between fact and fiction.

Pig Pen, glowing tails and shooting stars

Comet Encke is the so-called parent comet of the Taurid meteors. It’s relatively small, just over 3 miles (almost 5 kilometers) in diameter, and crosses inside Earth’s orbit and back out every 3.3 years.

As Encke moves, it sheds dust wherever it goes, like the Peanuts character Pig Pen. A meteor shower occurs when that dust and debris light up while entering Earth’s atmosphere at high speeds. Ultimately, they vanish into an incandescent puff of vapor with a glowing tail, creating the illusion of a “shooting star.”

But dust isn’t all that breaks off the comet. So do bigger chunks, the size of pebbles and stones. When they collide with the air, they create the much brighter fireballs, which sometimes explode.

Against a black and white starscape, a bright spot appears in the center of the photo.

An image of comet Encke, taken by NASA’s MESSENGER spacecraft in November 2013.

NASA/Johns Hopkins University Applied Physics Laboratory/Carnegie Institution of Washington/Southwest Research Institute

Doomsday showers

The “coherent catastrophism” hypothesis suggests that comet Encke was created when an even larger comet broke up into pieces; Encke survived as the largest piece. The hypothesis also suggests that other mountain-sized chunks broke off and coalesced into a large swarm of fragments too. If such a swarm exists, there is a possibility that those large chunks could one day hit Earth as it passes through the swarm.

But just because something might be physically possible doesn’t mean that it exists. Mainstream astronomers have rejected this theory’s most catastrophic predictions. Among other reasons, scientists have never observed high concentrations of these mountain-sized objects.

Despite the lack of evidence, researchers on the fringes of science have embraced the idea. They claim the Earth experienced a global catastrophic swarm 12,900 years ago; the impact, they say, caused continent-wide firestorms, floods and abrupt climate change that led to the mass extinction of large mammals, such as woolly mammoths, and the disappearance of early Americans known as the Clovis people.

The evidence for a catastrophic cause of these events, most of which did not happen, is lacking. Nevertheless, the idea has gained a large following and formed the basis for British author Graham Hancock’s popular TV series, “Ancient Apocalypse.”

A black and white photo of a forest of flattened trees.

This photo shows the flattened trees resulting from the Tunguska event.

Universal History Archives/Universal Images Group via Getty Images

The Tunguska event

But even outlandish ideas can have elements of truth, and there are hints that some objects – more than just dust and debris, but less than doomsday size – indeed exist in the Taurid meteor stream, and that the Earth has already encountered them.

One clue comes from an event on June 30, 1908, when an enormous explosion in the sky blew down millions of trees in Siberia. This was the Tunguska event – an airburst from an object that may have been up to 160 feet (about 50 meters) in diameter.

The collision unleashed several megatons of energy, which is roughly the equivalent of a large thermonuclear bomb. What happens is this: The incoming object penetrates deep into Earth’s atmosphere, and the dense air slows it down and heats it up until it vaporizes and explodes.

Could this object have been a Taurid? After all, the Taurids cross Earth’s orbit twice a year – not just in autumn, but also in June.

A fireball appears in the night sky.

In a 2015 photo, a glowing Taurid fireball descends over Lake Simcoe in Ontario, Canada.

Orchidpost/iStock via Getty Images Plus

Here’s the evidence: First, the descriptions of the trajectory of the Tunguska airburst, as reported by eyewitness observers, is consistent with that of an object coming from the Taurid stream.

What’s more, the pattern of blast damage on the ground beneath an airburst depends on the trajectory of the exploding object. Supercomputer simulations show that the shape of the surface blast that would be caused by an exploding Taurid object matches the pattern of fallen trees at Tunguska.

Finally, during the Taurid meteor shower in 1975, people observed large fireballs – and seismometers, previously placed on the Moon by Apollo astronauts, detected seismic events on the lunar surface. Scientists interpreted those events as impacts, presumably made by the Taurid meteors.

In 2032 and 2036, the Taurid swarm – assuming it exists – is predicted to be closer to the Earth than any time since 1975. That might mean the Moon, and perhaps the Earth, could be pelted again in those years.

There is time to figure this out. Scientists can expand their astronomical surveys to look for Tunguska-sized objects at the locations where they are predicted to be the next time they are in our vicinity.

Most scientists remain skeptical that such a swarm exists, but it’s the job of planetary defenders to investigate possible threats, even if the risk is small. After all, a Tunguska-sized object could conceivably demolish a major city and kill millions; an accurate count of objects on a potential collision course is essential.

Put doomsday scenarios and ancient apocalypses aside. The real question, and still an open one, is whether a Taurid swarm could deliver more Tunguska-sized objects than would otherwise be expected. This would mean we have underestimated the risk from future airbursts.The Conversation

Mark Boslough, Research Associate Professor of Earth and Planetary Sciences, University of New Mexico

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Knee problems tend to flare up as you age – an orthopedic specialist explains available treatment options

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theconversation.com – Angie Brown, Clinical Associate Professor of Physical Therapy, Quinnipiac University – 2024-11-15 07:32:00

Knee problems can hinder mobility and erode your quality of life.

Witthaya Prasongsin/Moment via Getty Images

Angie Brown, Quinnipiac University

Knee injuries are common in athletes, accounting for 41% of all athletic injuries. But knee injuries aren’t limited to competitive athletes. In our everyday lives, an accident or a quick movement in the wrong direction can injure the knee and require medical treatment. A quarter of the adult population worldwide experiences knee pain each year

As a physical therapist and board-certified orthopedic specialist, I help patients of all ages with knee injuries and degenerative conditions.

Your knees have a huge impact on your mobility and overall quality of life, so it’s important to prevent knee problems whenever possible and address pain in these joints with appropriate treatments.

Healthy knees

The knee joint bones consist of the femur, tibia and patella. As in all healthy joints, smooth cartilage covers the surfaces of the bones, forming the joints and allowing for controlled movement.

diagram of a healthy knee

A healthy knee.

Inna Kharlamova/iStock/Getty Images Plus via Getty Images

Muscles, ligaments and tendons further support the knee joint. The anterior cruciate ligament, commonly known as the ACL, and posterior cruciate ligament, or PCL, provide internal stability to the knee. In addition, two tough pieces of fibrocartilage, called menisci, lie inside the joint, providing further stability and shock absorption.

All these structures work together to enable the knee to move smoothly and painlessly throughout everyday movement, whether bending to pick up the family cat or going for a run.

Causes of knee pain

Two major causes of knee pain are acute injury and osteoarthritis.

Ligaments such as the ACL and PCL can be stressed and torn when a shear force occurs between the femur and tibia. ACL injuries often occur when athletes land awkwardly on the knee or quickly pivot on a planted foot. Depending on the severity of the injury, these patients may undergo physical therapy, or they may require surgery for repair or replacement.

PCL injuries are less common. They occur when the tibia experiences a posterior or backward force. This type of injury is common in car accidents when the knee hits the dashboard, or when patients fall forward when walking up stairs.

The menisci can also experience degeneration and tearing from shear and rotary forces, especially during weight-bearing activities. These types of injuries often require rehabilitation through physical therapy or surgery.

Knee pain can also result from injury or overuse of the muscles and tendons surrounding the knee, including the quadriceps, hamstrings and patella tendon.

Both injuries to and overuse of the knee can lead to degenerative changes in the joint surfaces, known as osteoarthritis. Osteoarthritis is a progressive disease that can lead to pain, swelling and stiffness. This disease affects the knees of over 300 million people worldwide, most often those 50 years of age and up. American adults have a 40% chance of developing osteoarthritis that affects their daily lives, with the knee being the most commonly affected joint.

Age is also a factor in knee pain. The structure and function of your joints change as you age. Cartilage starts to break down, your body produces less synovial fluid to lubricate your joints, and muscle strength and flexibility decrease. This can lead to painful, restricted movement in the joint.

Risk factors

There are some risk factors for knee osteoarthritis that you cannot control, such as genetics, age, sex and your history of prior injuries.

Fortunately, there are several risk factors you can control that can predispose you to knee pain and osteoarthritis specifically. The first is excessive weight. Based on studies between 2017 and 2020, nearly 42% of all adult Americans are obese. This obesity is a significant risk factor for diabetes and osteoarthritis and can also play a role in other knee injuries.

A lack of physical activity is another risk, with 1 in 5 U.S. adults reporting that they’re inactive outside of work duties. This can result in less muscular support for the knee and more pressure on the joint itself.

An inflammatory diet also adds to the risk of knee pain from osteoarthritis. Research shows that the average American diet, often high in sugar and fat and low in fiber, can lead to changes to the gut microbiome that contribute to osteoarthritis pain and inflammation.

Preventing knee pain

Increasing physical activity is one of the key elements to preventing knee pain. Often physical therapy intervention for patients with knee osteoarthritis focuses on strengthening the knee to decrease pain and support the joint during movement.

The U.S. Department of Health and Human Services recommends that adults spend at least 150 to 300 minutes per week on moderate-intensity, or 75 to 150 minutes per week on vigorous-intensity aerobic physical activity. These guidelines do not change for adults who already have osteoarthritis, although their exercise may require less weight-bearing activities, such as swimming, biking or walking.

The agency also recommends that all adults do some form of resistance training at least two or more days a week. Adults with knee osteoarthritis particularly benefit from quadriceps-strengthening exercises, such as straight leg raises.

Treatments for knee pain

Conservative treatment of knee pain includes anti-inflammatory and pain medications and physical therapy.

Medical treatment for knee osteoarthritis may include cortisone injections to decrease inflammation or hyaluronic acid injections, which help lubricate the joint. The relief from these interventions is often temporary, as they do not stop the progression of the disease. But they can delay the need for surgery by one to three years on average, depending on the number of injections.

Physical therapy is generally a longer-lasting treatment option for knee pain. Physical therapy treatment leads to more sustained pain reduction and functional improvements when compared with cortisone injections treatment and some meniscal repairs.

Patients with osteoarthritis often benefit from total knee replacement, a surgery with a high success rate and lasting results.

Surgical interventions for knee pain include the repair, replacement or removal of the ACL, PCL, menisci or cartilage. When more conservative approaches fail, patients with osteoarthritis may benefit from a partial or total knee replacement to allow more pain-free movement. In these procedures, one or both sides of the knee joint are replaced by either plastic or metal components. Afterward, patients attend physical therapy to aid in the return of range of motion.

Although there are risks with any surgery, most patients who undergo knee replacement benefit from decreased pain and increased function, with 90% of all replacements lasting more than 15 years. But not all patients are candidates for such surgeries, as a successful outcome depends on the patient’s overall health and well-being.

New treatments on the horizon

New developments for knee osteoarthritis are focused on less invasive therapies. Recently, the U.S. Food and Drug Administration approved a new implant that acts as a shock absorber. This requires a much simpler procedure than a total knee replacement.

Other promising interventions include knee embolization, a procedure in which tiny particles are injected into the arteries near the knee to decrease blood flow to the area and reduce inflammation near the joint. Researchers are also looking into injectable solutions derived from human bodies, such as plasma-rich protein and fat cells, to decrease inflammation and pain from osteoarthritis. Human stem cells and their growth factors also show potential in treating knee osteoarthritis by potentially improving muscle atrophy and repairing cartilage.

Further research is needed on these novel interventions. However, any intervention that holds promise to stop or delay osteoarthritis is certainly encouraging for the millions of people afflicted with this disease.The Conversation

Angie Brown, Clinical Associate Professor of Physical Therapy, Quinnipiac University

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Get chronic UTIs? Future treatments may add more bacteria to your bladder to beat back harmful microbes

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theconversation.com – Sarguru Subash, Assistant Professor of Veterinary Pathobiology, Texas A&M University – 2024-11-15 07:31:00

Certain strains of E. coli can outcompete disease-causing microbes for resources.

NIAID/Flickr, CC BY-SA

Sarguru Subash, Texas A&M University

Millions of people in the U.S. and around the world suffer from urinary tract infections every year. Some groups are especially prone to chronic UTIs, including women, older adults and some veterans.

These infections are typically treated with antibiotics, but overusing these drugs can make the microbes they target become resistant and reduce the medicines’ effectiveness.

To solve this problem of chronic UTIs and antibiotic resistance, we combined our expertise in microbiology and engineering to create a living material that houses a specific strain of beneficial E. coli. Our research shows that the “good” bacteria released from this biomaterial can compete with “bad” bacteria for nutrients and win, dramatically reducing the number of disease-causing microbes.

With further development, we believe this technique could help manage recurring UTIs that do not respond to antibiotics.

Bringing bacteria to the bladder

For the microbes living in people, nutrients are limited their presence varies between different parts of the body. Bacteria have to compete with other microbes and the host to acquire essential nutrients. By taking up available nutrients, beneficial bacteria can stop or slow the growth of harmful bacteria. When harmful bacteria are starved of important nutrients, they aren’t able to reach high enough numbers to cause disease.

Delivering beneficial bacteria to the bladder to prevent UTIs in challenging, though. For one, these helpful bacteria can naturally colonize only in people who are unable to fully empty their bladder, a condition called urinary retention. Even among these patients, how long these bacteria can colonize their bladders varies widely.

Current methods to deliver bacteria to the bladder are invasive and require repeated catheter insertion. Even when bacteria are successfully released into the bladder, urine will flush out these microbes because they cannot stick to the bladder wall.

Micrograph of clusters of pink rods scattered across a blue textured wall, while yellow blobs extend thin tendrils across the clumps

This microscopy image shows the bladder of a mouse (blue) covered with E. coli (pink) and the white blood cells (yellow) attacking them.

Valerie O’Brien, Matthew Joens, Scott J. Hultgren, James A.J. Fitzpatrick, Washington University, St. Louis/NIH via Flickr, CC BY-NC

Biomaterials to treat UTIs

Since beneficial bacteria cannot attach to and survive in the bladder for long, we developed a biomaterial that could slowly release bacteria in the bladder over time.

Our biomaterial is composed of living E. coli embedded in a matrix structure made of gel. It resembles a piece of jelly about 500 times smaller than a drop of water and can release bacteria for up to two weeks in the bladder. By delivering the bacteria via biomaterial, we overcome the need for the bacteria to attach to the bladder to persist in the organ.

We tested our biomaterial by placing it in human urine in petri dishes and exposing it to bacterial pathogens that cause UTIs. Our results showed that when mixed in a 50:50 ratio, the E. coli outcompeted the UTI-causing bacteria by increasing to around 85% of the total population. When we added more E. coli than UTI-causing bacteria, which is what we envision for future development and testing, the proportion of E. coli increased to over 99% of the population, essentially wiping out the UTI-causing bacteria. Moreoever, the biomaterial continued releasing E. coli for up to two weeks in human urine.

Our findings suggest that E.coli could stick around and survive in the bladder for extended periods of time and successfully decrease the growth of many types of bacteria that cause UTIs.

Person pressing hands against stomach

UTIs can be painful.

Images we create and what actually happens are always beautiful when we have imagination/iStock via Getty Images Plus

Improving biomaterials

Our findings show that E. coli can not only control harmful bacteria it’s closely related to but also a broad range of disease-causing bacteria in humans and animals. This means scientists might not need to identify different types of beneficial bacteria to control each pathogen – and there are many – that can cause a UTI.

Our team is currently evaluating how effectively our biomaterial can cure UTIs in mice. We are also working to identify the specific nutrients that beneficial and harmful bacteria compete over and what factors may help beneficial bacteria win. We could add these nutrients to our biomaterial to be released or withheld.

This research is still at an early stage, and clinical uses are not in development yet, so if it does reach patients it will be well in the future. We hope that our technology could be refined and applied to control other bacterial infections and some cancers caused by bacteria.The Conversation

Sarguru Subash, Assistant Professor of Veterinary Pathobiology, Texas A&M University

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