Author: IMC

French director Justine Triet’s “Anatomy of a Fall,” winner of the 2023 Oscar for best original script, reconstructs a fatal fall in order to dissect the collapse of the romantic relationship between the film’s leading couple, Sandra Voyter and Samuel Maleski.

Far from an exception, breakups of the sort depicted in the film are commonplace: global data shows high levels of marriage failure, with a marked increase towards the end of the last century.

In some Western countries, as many as 50% of marriages do not make it past 25 years, giving rise to the popular maxim “half of all marriages end in divorce.”

According to Triet, “the strange thing is for a relationship to work. The majority are hellish, and the film aims to go deep into that hell.”

Importantly, divorce statistics do not account for the number of relationships that are unhappy. Perhaps the majority are indeed hellish, but some marriages today are long-lasting, and seem stronger and more loving than any that came before. This dichotomy—widespread failure or exceptional success– seems to summarize the current state of marriage in the West. This has been dubbed the “all or nothing” marriage.

Supplying relationship energy

Scientific studies have established that romantic relationships tend to drop off, meaning that, on average, satisfaction levels reduce over time. Successful couples are able to arrest this fall, finding a satisfying level that can last indefinitely. Many others, however, gradually decline to the point where breaking up is only a matter of time.

Relationship psychology shows that love alone is not enough to keep a couple together –it requires effort. Relationship scientist John Gottman likens this to the second law of thermodynamics, whereby a closed system—such as a marriage– degenerates unless energy is supplied. As he puts it, “if you do nothing to make things get better in your marriage but do not do anything wrong, the marriage will still tend to get worse over time.”

The “all or nothing” theory therefore suggests that successful relationships require a significant investment of time and energy. Couples who make this commitment will be rewarded with a high level of satisfaction, while those who fail to do so, like Samuel and Sandra in Triet’s film, are destined to fail.

But why do some couples manage to stop this fall and stay happy? Like Samuel and Sandra, all couples start out in love, and want to be happy together forever. If we assume that they are compatible and willing to make the effort together, they form what some call an “Adam and Eve” relationship—the Biblical archetype of a harmonious, lasting union.

Analysing the ‘Adam and Eve’ relationship

Using dynamic systems to analyse this relationship model confirms the “all or nothing” theory.

Dynamic systems are a mathematical tool for understanding the evolution of a variable over time. In the case of romantic relationships, we are interested in the “feeling” of love in a couple. Because effort is needed to sustain the relationship, it becomes a dynamic system controlled by effort: effort regulates “feeling,” with the objective of making the “feeling” last forever.

By applying this effort control theory, our research has found that a successful relationship requires effort beyond the partners’ preferred level, and that this effort gap is difficult to sustain over time.

The mathematical anatomy of a fall

As Sandra Voyter says in Triet’s film, there are times when a relationship is chaotic, others when you fight alone, sometimes alongside your partner, and sometimes against your partner.

Samuel and Sandra’s relationship has elements in common with any other couple’s relationship. The starting point is very high: “feeling” is at its peak, and there is a shared belief that it will never end. Both are willing to contribute to the happiness of the relationship by making their own individual efforts, and both know that some kind of shock or external event will eventually alter this state.

Generally speaking, couples with the same socioeconomic, cultural, or religious background—known as homogamous couples— are more stable. Many couples, however, are heterogamous, meaning they differ in one or more of these regards.

Heterogamy can extend beyond an individual’s circumstances: on its most elemental level, it can boil down to a mismatch or imbalance in how efficient one member of a couple is in transforming effort into “feeling” or happiness. Such a disparity may lead to asymmetrical levels of effort being dedicated to making the relationship successful, which are already higher than those both members would prefer to make.

This is the case in Samuel and Sandra’s relationship: at one point in the film Samuel highlights this imbalance, and Sandra replies that she does not believe a couple should make equal efforts, saying she finds the idea depressing.

Who contributes more?

Our latest computational models for assessing the dynamics of imbalanced effort levels in couples allow us to simulate the evolution of happiness in a relationship, both in predictable environments and with varying levels of uncertainty. Our simulations suggest that Sandra is right: each partner does not have to make the same level of effort.

One of the film’s scenes—where Sandra and Samuel reproach each other for the efforts made or not made to sustain the relationship—displays typical negative couple dynamics, where each has a bone to pick. The film also implies that Samuel has made or is making more effort than Sandra in their relationship. Our analysis shows, perhaps surprisingly, that the more emotionally efficient partner has to make a greater effort to sustain the relationship. In the film, it appears that this is Samuel.

External events play a big part

Our analysis also shows that when the couple is subjected to a stressful episode, both partners need to increase their effort levels if the relationship is to survive. However, the more efficient partner’s effort level has to increase more. In the film, Sandra and Samuel’s relationship is subjected to a tremendous misfortune, which has a prolonged and pronounced effect on its narrative arc. This is why Samuel feels much more stressed than Sandra.

Mathematics offers an outcome in line with the film’s plot: the continuous overexertion of the most emotionally efficient partner—amplified by a prolonged period of crisis—leads to the relationship falling apart. In the case of the film, this also leads to Samuel’s fall.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to José-Manuel Rey and Jorge Herrera de la Cruz

Math education outcomes in the United States have been unequal for decades. Learners in the top 10% socioeconomically tend to be about four grade levels ahead of learners in the bottom 10%—a statistic that has remained stubbornly persistent for 50 years.

To advance equity, policymakers and educators often focus on boosting test scores and grades and making advanced courses more widely available. Through this lens, equity means all students earn similar grades and progress to similar levels of math.

With more than three decades of experience as a researcher, math teacher and teacher educator, we advocate for expanding what equity means in mathematics education. We believe policymakers and educators should focus less on test scores and grades and more on developing students’ confidence and ability to use math to make smart personal and professional decisions. This is mathematical power—and true equity.

What is ‘equity’ in math?

To understand the limitations of thinking about equity solely in terms of academic achievements, consider a student whom We interviewed during her freshman year of college.

Jasmine took Algebra 1 in ninth grade, followed by a summer online geometry course. This put her on a pathway to study calculus during her senior year in an AP class in which she earned an A. She graduated high school in the top 20% of her class and went to a highly selective liberal arts college. Now in her first year, she plans to study psychology.

Did Jasmine receive an equitable mathematics education? From an equity-as-achievement perspective, yes. But let’s take a closer look.

Jasmine experienced anxiety in her math classes during her junior and senior years in high school. Despite strong grades, she found herself “in a little bit of a panic” when faced with situations that require mathematical analysis. This included deciding the best loan options.

In college, Jasmine’s major required statistics. Her counsellor and family encouraged her to take calculus over statistics in high school because calculus “looked better” for college applications. She wishes now she had studied statistics as a foundation for her major and for its usefulness outside of school. In her psychology classes, knowledge of statistics helps her better understand the landscape of disorders and to ask questions like, “How does gender impact this disorder?”

These outcomes suggest Jasmine did not receive an equitable mathematics education, because she did not develop mathematical power. Mathematical power is the know-how and confidence to use math to inform decisions and navigate the demands of daily life—whether personal, professional or civic. An equitable education would help her develop the confidence to use mathematics to make decisions in her personal life and realize her professional goals. Jasmine deserved more from her mathematics education.

The prevalence of inequitable math education

Experiences like Jasmine’s are unfortunately common. According to one large-scale study, only 37% of U.S. adults have mathematical skills that are useful for making routine financial and medical decisions.

A National Council on Education and the Economy report found that coursework for nine common majors, including nursing, required relatively few of the mainstream math topics taught in most high schools. A recent study found that teachers and parents perceive math education as “unengaging, outdated and disconnected from the real world.”

Looking at student experiences, national survey results show that large proportions of students experience anxiety about math class, low levels of confidence in math, or both. Students from historically marginalized groups experience this anxiety at higher rates than their peers. This can frustrate their postsecondary pursuits and negatively affect their lives.

How to make math education more equitable

In 2023, We collaborated with other educators from Connecticut’s professional math education associations to author an equity position statement. The position statement, which was endorsed by the Connecticut State Board of Education, outlines three commitments to transform mathematics education.

1. Foster positive math identities: The first commitment is to foster positive math identities, which includes students’ confidence levels and their beliefs about math and their ability to learn it. Many students have a very negative relationship with mathematics. This commitment is particularly important for students of colour and language learners to counteract the impact of stereotypes about who can be successful in mathematics.

A growing body of material exists to help teachers and schools promote positive math identities. For example, writing a math autobiography can help students see the role of math in their lives. They can also reflect on their identity as a “math person.” Teachers should also acknowledge students’ strengths and encourage them to share their own ideas as a way to empower them.

2. Modernize math content: The second commitment is to modernize the mathematical content that school districts offer to students. For example, a high school mathematics pathway for students interested in health care professions might include algebra, math for medical professionals and advanced statistics. With these skills, students will be better prepared to calculate drug dosages, communicate results and risk factors to patients, interpret reports and research, and catch potentially life-threatening errors.
3. Align state policies and requirements:The third commitment is to align state policies and school districts in their definition of mathematical proficiency and the requirements for achieving it. In 2018, for instance, eight states had a high school math graduation requirement insufficient for admission to the public universities in the same state. Other states’ requirements exceed the admission requirements. Aligning state and district definitions of math proficiency clears up confusion for students and eliminates unnecessary barriers.

What’s next?

As long as educators and policymakers focus solely on equalizing test scores and enrolment in advanced courses, we believe true equity will remain elusive. Mathematical power—the ability and confidence to use math to make smart personal and professional decisions—needs to be the goal.

No one adjustment to the U.S. math education system will immediately result in students gaining mathematical power. But by focusing on students’ identities and designing math courses that align with their career and life goals, we believe schools, universities and state leaders can create a more expansive and equitable math education system.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Megan Staples, The Conversation

Statisticians from the National University of Singapore (NUS) have introduced a new technique that accurately describes high-dimensional data using lower-dimensional smooth structures. This innovation marks a significant step forward in addressing the challenges of complex nonlinear dimension reduction.

Traditional data analysis methods often rely on Euclidean (linear) dependencies among features. While this approach simplifies data representation, it struggles to capture the underlying complex patterns in high-dimensional data, typically located close to low-dimensional manifolds.

To bridge this gap, manifold-learning techniques have emerged as a promising solution. However, existing methods, such as manifold embedding and denoising, have been limited by a lack of detailed geometric understanding and robust theoretical underpinnings.

The team, led by Associate Professor Zhigang Yao from the Department of Statistics and Data Science, NUS with his Ph.D. student Jiaji Su pioneered a novel method for effectively estimating low-dimensional manifolds hidden within high-dimensional data. This approach not only achieves cutting-edge estimation accuracy and convergence rates but also enhances computational efficiency through the utilization of deep Generative Adversarial Networks (GANs).

This work was conducted in collaboration with Professor Shing-Tung Yau from the Yau Mathematical Sciences Center (YMSC) at Tsinghua University. Part of the work comes from Prof. Yao’s collaboration with Prof. Yau during his sabbatical visit to the Center of Mathematical Sciences and Applications (CMSA) at Harvard University.

Their findings have been published as a methodology paper in the Proceedings of the National Academy of Sciences.

Prof. Yao delivered a 45-minute invited lecture on this research at the recent International Congress of Chinese Mathematicians (ICCM) held in Shanghai, Jan. 2–5, 2024.

Highlighting the significance of the work, Prof. Yao said, “By accurately fitting manifolds, we can reduce data dimensionality while preserving crucial information, including the underlying geometric structure. This represents a major leap in data analysis, enhancing both accuracy and efficiency. By providing a solution that overcomes the limitations of previous methods, our research paves the way for enhanced data analysis and offers valuable insights for diverse applications in the scientific community.”

Looking ahead, Yao’s research team is developing a new framework to process even more complex data, such as single-cell RNA sequence data, while continuing to collaborate with the YMSC team. This ongoing work promises to revolutionize the approach for the reduction and processing of complex datasets, potentially offering new insights into a range of scientific fields.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to National University of Singapore

All a_i lie on the same sheet of a cone with vertex x. The right-hand picture is not true to scale relative to the given numerical example. Credit: Advances in Applied Mathematics (2024). DOI: 10.1016/j.aam.2024.102741

The summer holidays are ending, which for many concludes with a long drive home and reliance on GPS devices to get safely home. But every now and then, GPS devices can suggest strange directions or get briefly confused about your location. But until now, no one knew for sure when the satellites were in a good enough position for the GPS system to give reliable direction.

TU/e’s Mireille Boutin and her co-worker Gregor Kemper at the Technical University of Munich have turned to mathematics to help determine when your GPS system has enough information to determine your location accurately. The research is published in the journal Advances in Applied Mathematics.

“In 200 meters, turn right.” This is a typical instruction that many have heard from their global positioning system (GPS).

Without a doubt, advancements in GPS technologies and mobile navigation apps have helped GPS play a major role in modern car journeys.

But, strictly adhering to instructions from GPS devices can lead to undesirable situations. Less serious might be turning left instead of right, while more serious could be driving your car into a harbor—just as two tourists did in Hawaii in 2023. The latter incident is very much an exception to the rule, and one might wonder: “How often does this happen and why?”

GPS and your visibility

“The core of the GPS system was developed in the mid-1960s. At the time, the theory behind it did not provide any guarantee that the location given would be correct,” says Boutin, professor at the Department of Mathematics and Computer Science.

It won’t come as a surprise then to learn that calculating an object’s position on Earth relies on some nifty mathematics. And they haven’t changed much since the early days. These are at the core of the GPS system we all use. And it deserved an update.

So, along with her colleague Gregor Kemper at the Technical University of Munich, Boutin turned to mathematics to expand on the theory behind the GPS system, and their finding has recently been published in the journal Advances in Applied Mathematics.

How does GPS work?

Before revealing Boutin and Kemper’s big finding, just how does GPS work?

Global positioning is all about determining the position of a device on Earth using signals sent by satellites. A signal sent by a satellite carries two key pieces of information—the position of the satellite in space and the time at which the position was sent by the satellite. By the way, the time is recorded by a very precise clock on board the satellite, which is usually an atomic clock.

Thanks to the atomic clock, satellites send very accurate times, but the big issue lies with the accuracy of the clock in the user’s device—whether it’s a GPS navigation device, a smartphone, or a running watch.

“In effect, GPS combines precise and imprecise information to figure out where a device is located,” says Boutin. “GPS might be widely used, but we could not find any theoretical basis to guarantee that the position obtained from the satellite signals is unique and accurate.”

Google says ‘four’

If you do a quick Google search for the minimum number of satellites needed for navigation with GPS, multiple sources report that you need at least four satellites.

But the question is not just how many satellites you can see, but also what arrangements can they form? For some arrangements, determining the user position is impossible. But what arrangements exactly? That’s what the researchers wanted to find out.

“We found conjectures in scientific papers that seem to be widely accepted, but we could not find any rigorous argument to support them anywhere. Therefore, we thought that, as mathematicians, we might be able to fill that knowledge gap,” Boutin says.

To solve the problem, Boutin and Kemper simplified the GPS problem to what works best in practice: equations that are linear in terms of the unknown variables.

“A set of linear equations is the simplest form of equations we could hope for. To be honest, we were surprised that this simple set of linear equations for the GPS problem wasn’t already known,” Boutin adds.

The problem of uniqueness

With their linear equations ready, Boutin and Kemper then looked closely at the solutions to the equations, paying special attention as to whether the equations gave a unique solution.

“A unique solution implies that the only solution to the equations is the actual position of the user,” notes Boutin.

If there is more than one solution to the equations, then only one is correct—that is, the true user position—but the GPS system would not know which one to pick and might return the wrong one.

The researchers found that nonunique solutions can emerge when the satellites lie in a special structure known as a “hyperboloid of revolution of two sheets.”

“It doesn’t matter how many satellites send a signal—if they all lie on one of these hyperboloids then it’s possible that the equations can have two solutions, so the one chosen by the GPS could be wrong,” says Boutin.

But what about the claim that you need at least four satellites to determine your position? “Having four satellites can work, but the solution is not always unique,” points out Boutin.

Why mathematics matters

For Boutin, this work demonstrates the power and application of mathematics.

“I personally love the fact that mathematics is a very powerful tool with lots of practical applications,” says Boutin. “I think people who are not mathematicians may not see the connections so easily, and so it is always nice to find clear and compelling examples of everyday problems where mathematics can make a difference.”

Central to Boutin and Kemper’s research is the field of algebraic geometry in which abstract algebraic methods are used to solve geometrical, real-world problems.

“Algebraic geometry is an area of mathematics that is considered very abstract. I find it nice to be reminded that any piece of mathematics, however abstract it might be, may turn out to have practical applications at some point,” says Boutin.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article to be given Eindhoven University of Technology

It is very common in Australian schools to “stream” students for subjects such as English, science and maths. This means students are grouped into different classes based on their previous academic attainment, or in some cases, just a perception of their level of ability.

Students can also be streamed as early as primary school. Yet there are no national or state policies on this. This means school principals are free to decide what will happen in their schools.

Why are students streamed in Australians schools? And is this a good idea? Our research on streaming maths classes shows we need to think much more carefully about this very common practice.

Why do schools stream?

At a maths teacher conference in Sydney in late 2023, WEdid a live survey about school approaches to streaming.

This survey was done via interactive software while WEwas giving a presentation. There were 338 responses from head teachers in maths in either high schools or schools that go all the way from Kindergarten to Year 12. Most of the teachers were from public schools.

In a sign of how widespread streaming is, 95% of head teachers said they streamed maths classes in their schools.

Respondents said one of the main reasons is to help high-achieving students and make sure they are appropriately challenged. As one teacher said:

[We stream] to push the better students forward.

But almost half the respondents said they believed all students were benefiting from this system.

We also heard how streaming is seen as a way to cope with the teacher shortage and specific lack of qualified maths teachers. These qualifications include skills in both maths and maths teaching. More than half (65%) of respondents said streaming can “aid differentiation [and] support targeted student learning interventions”. In other words, streaming is a way to cope with different levels of ability in the classrooms and make the job of teaching a class more straightforward. One respondent said:

[we stream because] it’s easier to differentiate with a class of students that have similar perceived ability.

Teachers said they streamed classes to push the best students ‘forward’.

The ‘glass ceiling effect’

But while many schools and teachers assume streaming is good for students, this is not what the research says.

Our 2020 study, on streaming was based on interviews with 85 students and 22 teachers from 11 government schools.

This found streaming creates a “glass ceiling effect” – in other words, students cannot progress out of the stream they are initially assigned to without significant remedial work to catch them up.

As one teacher told us, students in lower-ability classes were then placed at a “massive disadvantage”. This is because they can miss out on segments of the curriculum because the class may progress more slowly or is deliberately not taught certain sections deemed too complex.

Often students in our study were unaware of this missed content until Year 10 and thinking about their options for the final years of school and beyond. They may not be able to do higher-level maths in Year 11 and 12 because they are too far behind. As one teacher explained:

they didn’t have enough of that advanced background for them to be able to study it. It was too difficult for them to begin with.

This comes as fewer students are completing advanced (calculus-based) maths.

If students do not study senior maths, they do not have the background for studying for engineering and other STEM careers, which we know are in very high demand.

On top of this, students may also be stigmatised as “low ability” in maths. While classes are not labelled as such, students are well aware of who is in the top classes and who is not. This can have an impact on students’ confidence about maths.

What does other research say?

International research has also found streaming students is inequitable.

As a 2018 UK study showed, students from disadvantaged backgrounds are more likely to be put in lower streamed classes.

A 2009 review of research studies found that not streaming students was better for low-ability student achievement and had no effect on average and high-ability student achievement.

Streaming is also seen as a way to cope with teachers shortages, and teachers teaching out of their field of expertise.

What should we do instead?

Amid concerns about Australian students’ maths performance in national and international tests, schools need to stop assuming streaming is the best approach for students.

The research indicates it would be better if students were taught in mixed-ability classes – as long as teachers are supported and class sizes are small enough.

This means all students have the opportunity to be taught all of the curriculum, giving them the option of doing senior maths if they want to in Year 11 and Year 12.

It also means students are not stigmatised as “poor at maths” from a young age.

But to do so, teachers and schools must be given more teaching resources and support. And some of this support needs to begin in primary school, rather than waiting until high school to try and catch students up.

Students also need adequate career advice, so they are aware of how maths could help future careers and what they need to do to get there.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Karolina Grabowska/Pexels, CC BY

A pair of mathematicians studied the UK National Lottery and figured out a combination of 27 tickets that guarantees you will always win, but they tell New Scientist they don’t bother to play.

David Cushing and David Stewart calculate a winning solution

Earlier this year, two mathematicians revealed that it is possible to guarantee a win on the UK national lottery by buying just 27 tickets, despite there being 45,057,474 possible draw combinations. The pair were shocked to see their findings make headlines around the world and inspire numerous people to play these 27 tickets – with mixed results – and say they don’t bother to play themselves.

David Cushing and David Stewart at the University of Manchester, UK, used a mathematical field called finite geometry to prove that particular sets of 27 tickets would guarantee a win.

They placed each of the lottery numbers from 1 to 59 in pairs or triplets on a point within one of five geometrical shapes, then used these to generate lottery tickets based on the lines within the shapes. The five shapes offer 27 such lines, meaning that 27 tickets will cover every possible winning combination of two numbers, the minimum needed to win a prize. Each ticket costs £2.

It was an elegant and intuitive solution to a tricky problem, but also an irresistible headline that attracted newspapers, radio stations and television channels from around the world. And it also led many people to chance their luck – despite the researchers always pointing out that it was, statistically speaking, a very good way to lose money, as the winnings were in no way guaranteed to even cover the cost of the tickets.

Cushing says he has received numerous emails since the paper was released from people who cheerily announce that they have won tiny amounts, like two free lucky dips – essentially another free go on the lottery. “They were very happy to tell me how much they’d lost basically,” he says.

The pair did calculate that their method would have won them £1810 if they had played on one night during the writing of their research paper – 21 June. Both Cushing and Stewart had decided not to play the numbers themselves that night, but they have since found that a member of their research group “went rogue” and bought the right tickets – putting himself £1756 in profit.

“He said what convinced him to definitely put them on was that it was summer solstice. He said he had this feeling,” says Cushing, shaking his head as he speaks. “He’s a professional statistician. He is incredibly lucky with it; he claims he once found a lottery ticket in the street and it won £10.”

Cushing and Stewart say that while their winning colleague – who would prefer to remain nameless – has not even bought them lunch as a thank you for their efforts, he has continued to play the 27 lottery tickets. However, he now randomly permutes the tickets to alternative 27-ticket, guaranteed-win sets in case others have also been inspired by the set that was made public. Avoiding that set could avert a situation where a future jackpot win would be shared with dozens or even hundreds of mathematically-inclined players.

Stewart says there is no way to know how many people are doing the same because Camelot, which runs the lottery, doesn’t release that information. “If the jackpot comes up and it happens to match exactly one of the [set of] tickets and it gets split a thousand ways, that will be some indication,” he says.

Nonetheless, Cushing says that he no longer has any interest in playing the 27 tickets. “I came to the conclusion that whenever we were involved, they didn’t make any money, and then they made money when we decided not to put them on. That’s not very mathematical, but it seemed to be what was happening,” he says.

And Stewart is keen to stress that mathematics, no matter how neat a proof, can never make the UK lottery a wise investment. “If every single man, woman and child in the UK bought a separate ticket, we’d only have a quarter chance of someone winning the jackpot,” he says.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Matthew Sparkes*

Avi Wigderson has won the 2023 Turing award for his work on understanding how randomness can shape and improve computer algorithms.

The mathematician Avi Wigderson has won the 2023 Turing award, often referred to as the Nobel prize for computing, for his work on understanding how randomness can shape and improve computer algorithms.

Wigderson, who also won the prestigious Abel prize in 2021 for his mathematical contributions to computer science, was taken aback by the award. “The [Turing] committee fooled me into believing that we were going to have some conversation about collaborating,” he says. “When I zoomed in, the whole committee was there and they told me. I was excited, surprised and happy.”

Computers work in a predictable way at the hardware level, but this can make it difficult for them to model real-world problems, which often have elements of randomness and unpredictability. Wigderson, at the Institute for Advanced Study in Princeton, New Jersey, has shown over a decades-long career that computers can also harness randomness in the algorithms that they run.

In the 1980s, Wigderson and his colleagues discovered that by inserting randomness into some algorithms, they could make them easier and faster to solve, but it was unclear how general this technique was. “We were wondering whether this randomness is essential, or maybe you can always get rid of it somehow if you’re clever enough,” he says.

One of Wigderson’s most important discoveries was making clear the relationship between types of problems, in terms of their difficulty to solve, and randomness. He also showed that certain algorithms that contained randomness and were hard to run could be made deterministic, or non-random, and easier to run.

These findings helped computer scientists better understand one of the most famous unproven conjectures in computer science, called “P ≠ NP”, which proposes that easy and hard problems for a computer to solve are fundamentally different. Using randomness, Wigderson discovered special cases where the two classes of problem were the same.

Wigderson first started exploring the relationship between randomness and computers in the 1980s, before the internet existed, and was attracted to the ideas he worked on by intellectual curiosity, rather than how they might be used. “I’m a very impractical person,” he says. “I’m not really motivated by applications.”

However, his ideas have become important for a wide swath of modern computing applications, from cryptography to cloud computing. “Avi’s impact on the theory of computation in the last 40 years is second to none,” says Oded Goldreich at the Weizmann Institute of Science in Israel. “The diversity of the areas to which he has contributed is stunning.”

One of the unexpected ways in which Wigderson’s ideas are now widely used was his work, with Goldreich and others, on zero-knowledge proofs, which detail ways of verifying information without revealing the information itself. These methods are fundamental for cryptocurrencies and blockchains today as a way to establish trust between different users.

Although great strides in the theory of computation have been made over Wigderson’s career, he says that the field is still full of interesting and unsolved problems. “You can’t imagine how happy I am that I am where I am, in the field that I’m in,” he says. “It’s bursting with intellectual questions.”

Wigderson will receive a $1 million prize as part of the Turing award.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Alex Wilkins*

Math experts have developed new ways to provide further evidence for human-caused global heating and predict how close Earth is to reaching dangerous climate tipping points.

Tipping points occur when lots of small changes in climate build up to create a sudden, large change. Passing these thresholds leads to accelerated warming and extreme weather events.

For example, in years to come, the Atlantic Meridional Overturning Circulation (AMOC), is at risk of collapsing. The AMOC is an ocean current transport system that brings warm waterto the North Atlantic and helps to regulate temperatures in Europe and North America. If it collapses, the regional and global climate may become more erratic.

In July 2023, a study suggested the AMOC could collapse any time between 2025 and 2095. Another recent study, from October 2023, indicates that we are in dangerous proximity of the collapse of the Greenland ice sheet, yet with some hope of avoiding it if proper action is taken.

Now, a study published in the journal Nature Reviews Physics combines mathematics and physics to study climate change across time scales. It will help scientists refine their predictions for AMOC’s collapse as well as help them understand the proximity to various other climate tipping points.

Lead author Professor Valerio Lucarini, Professor of Statistical Mechanics at the University of Reading, said, “Our study provides academics and policymakers with rigorous mathematical tools needed to understand and predict climate change, and, specifically to detect and avoid climate tipping points resulting from human activities. It gives us a unified perspective linking climate variability and climate change. Our approach allows one to seamlessly study gradual climate changes and tipping points.

“We now have a robust framework for capturing the richness of climate dynamics across timescales. The findings provide tools for building up further evidence for human-caused global heating.”

Innovations and collaborations

The work will significantly advance the development of climate models and theories to generate more accurate predictions of climate change and evaluation of tipping point proximity, helping to inform mitigation policies and climate adaptation strategies.

The research critically investigated 2021 Nobel Prize in Physics winner Klaus Hasselmann’s stochastic modeling approach, which revolutionized climate change analysis. The new study refined Hasselmann’s techniques using recent tools developed within the mathematical and physical scientific literature.

The scientists noted how their methodology allows to better understand the nearing of the collapse of the AMOC. Observations show the AMOC is weakening. In addition to AMOC, the new method could be used to detect proximity to tipping points for:

• The collapse of ecosystems
• The melting of Greenland ice sheets
• The dieback of the Amazon rainforest

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Credit of the article given to University of Reading

Chinese ice-ray lattice, or “binglie” as it is called in Chinese, is an intricate pattern that looks like cracked ice and is a common decorative element used in traditional Chinese window designs.

Originally inspired by fragmented patterns on ice or crackle-glazed ceramic surfaces, the design represents the melting of the ice and the beginning of a thriving spring.

When Dr. Iasef Md Rian, now an Associate Professor at Xi’an Jiaotong-Liverpool University’s Department of Architecture, arrived in China for the first time in 2019, he was immediately captivated by the latticed window designs in the classical gardens of Suzhou.

“Classical gardens in China strike me as very different from the Western ones, which are more symmetrical and organized,” he says. “Chinese gardens, however, have a more natural formation in their layout and design. The ice-ray window design is one of the manifestations.”

Having focused on fractal geometry in architectural design for many years, Dr. Rian felt an urge to explore the beauty in the patterns.

“My mind is always looking for this kind of inspiration source, so I was motivated right away to study the underlying geometric principles of the ice-ray patterns, he says.”

Revealing the underlying rule

Dr. Rian finds that the rule of creating ice-ray patterns is actually very simple.

He explains, “Take Type 1 as an example; a square is first divided into two quadrilaterals, and then each quadrilateral is further divided into two quadrilaterals. In each step, the proportions of the subdivided quadrilaterals are different, and this is how the random pattern is created using a simple rule.

“Through this configuration, Chinese craftsmen might have intended to increase its firmness so it can function as a window fence to provide protection. The random configuration of ice-ray lattices provides multi-angled connections, which transform the window into a collection of resultant forces and uniform stress distribution, in turn achieving a unique degree of stiffness.

“The microstructure of trabecular bone tissue in our own bodies serves as an excellent natural example of the potential of random lattices. It balances high stiffness, which contributes to strength, with a surprisingly lightweight structure.”

Dr. Rian recently published a paper in Frontiers of Architectural Research that explores the geometric qualities of ice-ray patterns and expands the possibilities of integrating random patterns into structural designs, especially the lattice shell design, which is often used in spherical domes and curved structures.

“In my research, I developed an algorithm to model the ice-ray patterns for lattice shell designs and assessed their feasibility and effectiveness compared to conventional gridshells. These gridshells, made from regular grids, contrast with continuous shells.

“While regular gridshells perform well under uniform loads, the ice-ray lattice offers strength under asymmetric loads. Some ice-ray patterns, resulting from optimization, surprisingly provide better strength than regular gridshells under self-weight. There is also an additional aesthetic advantage when applying the ice-ray pattern to a lattice shell design.

“I extend the application of this pattern to curved surfaces, which helps to unlock its potential in the geometric, structural, and constructional aspects of lattice shell design,” he says.

Dr. Rian has also integrated ice-ray patterns and complex geometries into his teaching. In 2022, he organized a workshop for students to design ice-ray lattice roofs.

He explains that learning the concept of fractal geometry can really push the students’ ideas toward a unique design.

“This is very different from what they’ve learned in high school. In learning to create this geometry system, they will also learn computational modeling and simulations. In the end, they’ll get comprehensive knowledge of advanced architectural and digital design,” he says.

Rediscovering traditional designs

To extend the research in this field, Dr. Rian is investigating the effectiveness of complex geometry in various aspects like micro-scale material design and structural design.

He says, “For instance, in facade design, we usually use conventional or parametric geometry to design regular shapes. However, the random shapes designed with complex geometry can offer a more natural impression and daylight penetration.”

He encourages design students and researchers to learn from the past.

“Any traditional design has a hidden rule in it. We can now use digital technologies and advanced tools to extend and expand the knowledge of traditional craftsmanship for contemporary design.

“There are many inspirations behind the traditional designs, and those principles can really inspire us designers to make innovative designs for the future,” he says.

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Credit of the article given to Yi Qian, Xi’an jiaotong-Liverpool University

Putting off college math could improve the likelihood that students remain in college. But that may only be true as long as students don’t procrastinate more than one year. This is what colleagues and I found in a study published in 2023 of 1,119 students at a public university for whom no remedial coursework was required during their first year.

Enrolling in a math course during the first semester of college resulted in students being four times more likely to drop out. Although delayed enrollment in a math course had benefits in the first year, its advantages vanished by the end of the second year. In our study, almost 40% of students who postponed the course beyond a year did not attempt it at all and failed to obtain a degree within six years.

Why it matters

Nearly 1.7 million students who recently graduated from high school will immediately enroll in college. Math is a requirement for most degrees, but students aren’t always ready to do college-level math. By putting off college math for a year, it gives students time to adjust to college and prepare for more challenging coursework.

Approximately 40% of four-year college students must first take a remedial math course. This can extend the time it takes to graduate and increase the likelihood of dropping out. Our study did not apply to students who need remedial math.

For students who do not require remedial courses, some delay can be beneficial, but students’ past experiences in math can lead to avoidance of math courses. Many students experience math anxiety. Procrastination can be an avoidance strategy for managing fears about math. The fear of math for students may be a more significant barrier than their performance.

It is estimated that at least 17% of the population will likely experience high levels of math anxiety. Math anxiety can lead to a drop in math performance. It can also lead to avoiding majors and career paths involving math.

Our study fills the void in research on the effects of how soon students take college-level math courses. It also supports prior evidence that students benefit from a mix of coursework that is challenging yet not overwhelming as they transition to college.

What still isn’t known

We believe colleges need to better promote student confidence in math by examining how student success courses can reduce math anxiety. Student success courses provide students with study skills, note-taking skills, goal setting, time management and stress management, as well as career and financial decision making to support the transition to college. Although student success courses are a proven practice that help students stick with college, rarely do these courses address students’ fear of math.

Students are at the greatest risk of dropping out of college during their first year. Advisors play a crucial role in providing students with resources for success. This includes recommendations on what courses to take and when to take them. More research is also needed about how advisors can effectively communicate the impact of when math is taken by students.

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Credit of the article given to Forrest Lane, The Conversation