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

某个氚原子会在某个时间前衰变吗？根据我们目前的科学，关于物理现象的这个问题应该通过从概率分布中抽样来回答，这个过程与旋转轮盘或掷骰子没什么不同。然而，Foundations of Physics 上的一篇论文表明，同样的情况也可能适用于关于数学现象的问题，即使是像“2+2 等于多少？”这样平淡无奇的问题。

SFI 教授、物理学家 David Wolpert 和前 SFI Omidyar 研究员、哲学家兼认知科学家 David Kinney 提出了一个统一的概率框架，用于描述数学、物理宇宙，甚至描述人类如何对两者进行推理。

在他们的框架中，数学和科学都被表示为提问和回答问题的过程。给定的数学家或科学家如何回答问题将取决于他们分配给不同答案的概率。在基本形式主义的扩展中，这些相同的概率也决定了数学家或科学家回答这些问题的正确程度。

为了说明这一点，假设你现在回答一个数学问题，而在遥远的未来，一些数学家回答了同样的问题。你答案的正确性取决于你的概率与他们的概率的匹配程度。（在物理学中，你的正确性将涉及物理实验，而不是未来的物理学家。）

从问题选择到给出的答案，随机性渗透到问题-答案过程的每个部分。这导致了所提出的框架的一个重要好处：为科学和数学中的两个常识性捷径提供了一种新颖的、正式的理由——更强烈地相信有多种证据支持的想法，以及更强烈地相信最能解释你已经相信的东西的想法。

要了解更多此类见解，请登录我们的网站 https://international-maths-challenge.com

本文的作者为圣塔菲研究所

Will a certain tritium atom decay by a certain time? According to our current science, this question concerning physical phenomena should be answered by sampling from a probability distribution, a process not unlike spinning a roulette wheel or rolling dice. However, a paper in Foundations of Physics suggests that the same could be true of a question concerning mathematical phenomena, even one as prosaic as “what is 2+2?

SFI Professor David Wolpert, a physicist, and former SFI Omidyar Fellow David Kinney, a philosopher and cognitive scientist, propose a unified, probabilistic framework to describe math, the physical universe, and even describe how humans reason about both.

In their framework, mathematics and science are both represented as a process of asking and answering questions. How a given mathematician or scientist answers a question will depend on the probabilities that they assign to different answers. In an extension of the basic formalism, these same probabilities also determine how correctly that mathematician or scientist answers those questions.

To illustrate, say you answer a mathematics question now, and in the faraway future, some mathematicians answer the same question. The correctness of your answer depends on how your probabilities match theirs. (In physics, your correctness would involve physical experiments, not future physicists.)

Randomness pervades every part of the question-answer process, from question selection to the answer given. This results in an important benefit of the proposed framework: a novel, formal justification for two common-sense shortcuts in science and math—believing more strongly in ideas supported by multiple lines of evidence, and believing more strongly in ideas that best explain something you already believe.

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

Credit of the article given to Santa Fe Institute

Algorithms have become integral to our lives. From social media apps to Netflix, algorithms learn your preferences and prioritise the content you are shown. Google Maps and artificial intelligence are nothing without algorithms.

So, we’ve all heard of them, but where does the word “algorithm” even come from?

Over 1,000 years before the internet and smartphone apps, Persian scientist and polymath Muhammad ibn Mūsā al-Khwārizmī invented the concept of algorithms.

In fact, the word itself comes from the Latinised version of his name, “algorithmi”. And, as you might suspect, it’s also related to algebra.

Largely lost to time

Al-Khwārizmī lived from 780 to 850 CE, during the Islamic Golden Age. He is considered the “father of algebra”, and for some, the “grandfather of computer science”.

Yet, few details are known about his life. Many of his original works in Arabic have been lost to time.

It is believed al-Khwārizmī was born in the Khwarazm regionsouth of the Aral Sea in present-day Uzbekistan. He lived during the Abbasid Caliphate, which was a time of remarkable scientific progress in the Islamic Empire.

Al-Khwārizmī made important contributions to mathematics, geography, astronomy and trigonometry. To help provide a more accurate world map, he corrected Alexandrian polymath Ptolemy’s classic cartography book, Geographia.

He produced calculations for tracking the movement of the Sun, Moon and planets. He also wrote about trigonometric functions and produced the first table of tangents.

Al-Khwārizmī was a scholar in the House of Wisdom (Bayt al-Hikmah) in Baghdad. At this intellectual hub, scholars were translating knowledge from around the world into Arabic, synthesising it to make meaningful progress in a range of disciplines. This included mathematics, a field deeply connected to Islam.

There are no images of what al-Khwārizmī looked like, but in 1983 the Soviet Union issued a stamp in honour of his 1,200th birthday. Wikimedia Commons

The ‘father of algebra’

Al-Khwārizmī was a polymath and a religious man. His scientific writings started with dedications to Allah and the Prophet Muhammad. And one of the major projects Islamic mathematicians undertook at the House of Wisdom was to develop algebra.

Around 830 CE, Caliph al-Ma’mun encouraged al-Khwārizmī to write a treatise on algebra, Al-Jabr (or The Compendious Book on Calculation by Completion and Balancing). This became his most important work.

A page from The Compendious Book on Calculation by Completion and Balancing. World Digital Library

At this point, “algebra” had been around for hundreds of years, but al-Khwārizmī was the first to write a definitive book on it. His work was meant to be a practical teaching tool. Its Latin translation was the basis for algebra textbooks in European universities until the 16th century.

In the first part, he introduced the concepts and rules of algebra, and methods for calculating the volumes and areas of shapes. In the second part he provided real-life problems and worked out solutions, such as inheritance cases, the partition of land and calculations for trade.

Al-Khwārizmī didn’t use modern-day mathematical notation with numbers and symbols. Instead, he wrote in simple prose and employed geometric diagrams:

Four roots are equal to twenty, then one root is equal to five, and the square to be formed of it is twenty-five.

In modern-day notation we’d write that like so:

4x = 20, x = 5, x² = 25

Grandfather of computer science

Al-Khwārizmī’s mathematical writings introduced the Hindu-Arabic numerals to Western mathematicians. These are the ten symbols we all use today: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

The Hindu-Arabic numerals are important to the history of computing because they use the number zero and a base-ten decimal system. Importantly, this is the numeral system that underpins modern computing technology.

Al-Khwārizmī’s art of calculating mathematical problems laid the foundation for the concept of algorithms. He provided the first detailed explanations for using decimal notation to perform the four basic operations (addition, subtraction, multiplication, division) and computing fractions.

The contrast between algorithmic computations and abacus computations, as shown in Margarita Philosophica (1517). The Bavarian State Library

This was a more efficient computation method than using the abacus. To solve a mathematical equation, al-Khwārizmī systematically moved through a sequence of steps to find the answer. This is the underlying concept of an algorithm.

Algorism, a Medieval Latin term named after al-Khwārizmī, refers to the rules for performing arithmetic using the Hindu-Arabic numeral system. Translated to Latin, al-Khwārizmī’s book on Hindu numerals was titled Algorithmi de Numero Indorum.

In the early 20th century, the word algorithm came into its current definition and usage: “a procedure for solving a mathematical problem in a finite number of steps; a step-by-step procedure for solving a problem”.

Muhammad ibn Mūsā al-Khwārizmī played a central role in the development of mathematics and computer science as we know them today.

The next time you use any digital technology – from your social media feed to your online bank account to your Spotify app – remember that none of it would be possible without the pioneering work of an ancient Persian polymath.

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

Credit of the article given to The Conversation

Homeschooled children don’t always get a well-rounded curriculum. miniseries via Getty Images

Homeschooling is the fastest-growing education setting in the United States. More than 3 million students were educated at home in the 2021-22 school year, up from 2.5 million in the spring of 2019. Current estimates from the U.S. Census Bureau indicate that there are 3.62 million students homeschooled in the United States. That’s a meteoric increase from about 1 million in 1997.

Some experts, including Harvard law professor Elizabeth Bartholet, find the increase a cause to call for greater regulation. University of Washington education policy professor David Knight agrees, citing a lack of accountability and measures of student progress. Knight also worries about an absence of certain disciplines such as social studies that public schools are required to teach.

For those of us who have researched the homeschool movement and studied its past, these are not new concerns. So, what do we know about homeschooling and preparedness for college?

Data shows homeschooled students fare well

In 2020, we reviewed the evidence about how well homeschooling prepares people for college, career and life and published what we learned in a book chapter titled: “Life after Homeschool.”

We found evidence that homeschooled students are just as prepared academically for higher education as traditionally schooled peers. In one study, researchers drew a sample of 825,672 students – including 732 students who had been homeschooled – and found the homeschooled group scored higher on several measures of college preparedness, including the SAT and first-year GPA in college.

Ave Maria University education professor Marc Snyder came to similar results in a 2013 study. Snyder compared homeschooled and traditional students at his Catholic university in Florida to find the average ACT scores for homeschooled students was 26. Public school students averaged 24.22, and students who attended Catholic schools averaged 24.53.

Snyder’s study reinforces data from the ACT itself. The testing outlet reported that from 2001-2019, the average ACT scores for homeschooled students trended up, while public school students’ scores declined slightly. In 2023, the national average on the ACT was 19.9; the average for homeschoolers was 22.8.

Areas of concern abound as homeschool growth accelerates

Still, calls for regulation persist because of a host of challenges homeschooled students present. The Coalition for Responsible Home Education wants states to require minimal qualifications of a high school diploma or GED for the parent providing primary instruction, instruction for students in the same subjects as in public schools, and annual standardized assessments. In some states, they note, parents don’t even have to tell their local school district of their intent to homeschool.

The pro-regulation side points to studies showing homeschooled students feel less prepared for college and are four times less likely to go to college after high school. Homeschool students also take an average of one fewer math and science course than traditional peers.

Homeschooled students also often lack resources and guidance provided in traditional high schools for college prep. And social challenges abound when these students transition; a study of seven homeschooled graduates in Pennsylvania found students struggling to maintain their existing moral beliefs related to drinking, drug use and sexual norms, with the majority admitting they changed some beliefs and practices.

There’s also data that shows homeschooled students find the more structured academic environment on university campuses to be difficult to adjust to after a more lax experience learning at home.

Still, efforts to regulate homeschooling face opposition from parents as well as advocacy groups such as the Home School Legal Defense Association. In March 2024, for example, these forces defeated an attempt in New Hampshire to require homeschool students to take a statewide exam.

3 ways to improve homeschooling

To help homeschooled students transition to college, we recommend parents take three steps to better prepare their kids.

• Prioritize math and scienceto help address the math and science gap. Parents can use online courses offered through virtual high schools or employ tutors.
• Enroll in dual-credit or community college coursesto provide a taste of the structure of college life and to interact with peers from diverse backgrounds.
• Talk to children about the diversity of perspectivesthey will encounter at college. This can help prepare them for how to negotiate and respect the opinions of others.

Homeschooled students can successfully transition to college and compete with their peers. The challenges they face are entirely foreseeable, which means they can be addressed easily.

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

Credit of the article given to The Conversation

A small team of political scientists, statisticians and data scientists from Harvard University, New York University, and Yale University, has found that by switching to a new method to better protect privacy, the U.S. Census Department has introduced factors that reduce accuracy in some cases.

In their paper published in the journal Science Advances, the group describes how they analysed a file provided by Census officials to measure accuracy in publicly available census data and their results.

Prior to the 2020 U.S. census, officials with the U.S. Census Bureau worried about the privacy of the people who provide answers to the census, opted to change the method by which they ensured data security.

The old method was called, “swapping.” It involved swapping data from people living in one block of a city with people in another block, thereby preventing people from being identified based on their data. The new method is called “differential privacy” and it involves adding what the Bureau describes as “noise” to each piece of data that is collected.

In this new effort, the research team could find no instance of an outside entity conducting research to determine if the new method did indeed provide more privacy or if the processed data was more or less accurate than had been the case when swapping was used. So, they began one of their own.

The study began with the research team asking the Census Bureau to give them access to what is called the noisy measurement file (NMF)—the one used for the 2020 census. The Bureau denied the request, which led the team to sue them. Eventually, the lawsuit was dropped when the Bureau agreed to give the team the NMF associated with the much smaller 2010 census—one that was carried out as a way to test the new method, and involved both swapping and differentiating.

The researchers then analysed that file as a way to study the impact on accuracy of changing to the new system. In so doing, they found that overall, the two systems provided roughly equal accuracy on a broad scale. But they also found evidence of a reduction in accuracy at the block level of a type that could adversely impact minorities and multiracial populations.

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

Credit of the article given to Bob Yirka , Phys.org

There’s a strange trend in mathematics education in England. Math is the most popular subject at A-level since overtaking English in 2014. It’s taken by around 85,000 and 90,000 students a year.

But many universities—particularly lower-tariff institutions, which accept students with lower A-level grades—are recruiting far fewer students for math degrees. There’s been a 50% drop in numbers of math students at the lowest tariff universities over the five years between 2017 and 2021. As a result, some universities are struggling to keep their mathematics departments open.

The total number of students studying math has remained largely static over the last decade. Prestigious Russell Group universities which require top A-level grades have increased their numbers of math students.

This trend in degree-level mathematics education is worrying. It restricts the accessibility of math degrees, especially to students from poorer backgrounds who are most likely to study at universities close to where they live. It perpetuates the myth that only those people who are unusually gifted at mathematics should study it—and that high-level math skills are not necessary for everyone else.

Research carried out in 2019 by King’s College London and Ipsos found that half of the working age population had the numeracy skills expected of a child at primary school. Just as worrying was that despite this, 43% of those polled said “they would not like to improve their numeracy skills.” Nearly a quarter (23%) stated that “they couldn’t see how it would benefit them.”

Mathematics has been fundamental in recent technological developments such as quantum computing, information security and artificial intelligence. A pipeline of more mathematics graduates from more diverse backgrounds will be essential if the UK is to remain a science and technology powerhouse into the future.

But math is also vital to a huge range of careers, including in business and government. In March 2024, campaign group Protect Pure Math held a summit to bring together experts from industry, academia and government to discuss concerns about poor math skills and the continuing importance of high-quality mathematics education.

Prior to the summit, the London Mathematical Society commissioned a survey of over 500 businesses to gauge their concerns about the potential lack of future graduates with strong mathematical skills.

They found that 72% of businesses agree they would benefit from more math graduates entering the workforce. And 75% would worry if UK universities shrunk or closed their math departments.

A 2023 report on MPs’ staff found that skills in Stem subjects (science, technology, engineering and mathematics) were particularly hard to find among those who worked in Westminster. As many as 90% of those who had taken an undergraduate degree had studied humanities or social sciences. While these subject backgrounds are valuable, the lack of specialized math skills is stark.

Limited options

The mathematics department at Oxford Brookes has closed and other universities have seen recruitment reductions or other cuts. The resulting math deserts will remove the opportunity for students to gain a high-quality mathematics education in their local area. Universities should do their best to keep these departments open.

This might be possible if the way that degrees are set up changes. For many degree courses in countries such as the US and Australia, students are able to take a broad selection of subjects, from science and math subjects through to the humanities. Each are taught in their respective academic departments. This allows students to gain advanced knowledge and see how each field feeds into others.

This is scarcely possible in the UK, where students must choose a specialist and narrow degree program at aged 18.

Another possible solution would be to put core mathematics modules in degree disciplines that rely so heavily on it—such as engineering, economics, chemistry, physics, biology and computer science—and have them taught by specialist mathematicians. This would help keep mathematics departments open, while also ensuring that general mathematical literacy improves in the UK.

The relevance of mathematics and its vast range applications would be abundantly clear, better equipping every student with the necessary mathematical skills the workforce needs.

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

Credit of the article given to Neil Saunders, The Conversation

There’s a strange trend in mathematics education in England. Maths is the most popular subject at A-level since overtaking English in 2014. It’s taken by around 85,000 and 90,000 students a year.

But many universities – particularly lower-tariff institutions, which accept students with lower A-level grades – are recruiting far fewer students for maths degrees. There’s been a 50% drop in numbers of maths students at the lowest tariff universities over the five years between 2017 and 2021. As a result, some universities are struggling to keep their mathematics departments open.

The total number of students studying maths has remained largely static over the last decade. Prestigious Russell Group universities which require top A-level grades have increased their numbers of maths students.

This trend in degree-level mathematics education is worrying. It restricts the accessibility of maths degrees, especially to students from poorer backgrounds who are most likely to study at universities close to where they live. It perpetuates the myth that only those people who are unusually gifted at mathematics should study it – and that high-level maths skills are not necessary for everyone else.

Research carried out in 2019 by King’s College London and Ipsos found that half of the working age population had the numeracy skills expected of a child at primary school. Just as worrying was that despite this, 43% of those polled said “they would not like to improve their numeracy skills”. Nearly a quarter (23%) stated that “they couldn’t see how it would benefit them”.

Mathematics has been fundamental in recent technological developments such as quantum computing, information security and artificial intelligence. A pipeline of more mathematics graduates from more diverse backgrounds will be essential if the UK is to remain a science and technology powerhouse into the future.

But maths is also vital to a huge range of careers, including in business and government. In March 2024, campaign group Protect Pure Maths held a summit to bring together experts from industry, academia and government to discuss concerns about poor maths skills and the continuing importance of high-quality mathematics education.

Prior to the summit, the London Mathematical Society commissioned a survey of over 500 businesses to gauge their concerns about the potential lack of future graduates with strong mathematical skills.

They found that 72% of businesses agree they would benefit from more maths graduates entering the workforce. And 75% would worry if UK universities shrunk or closed their maths departments.

A 2023 report on MPs’ staff found that skills in Stem subjects (science, technology, engineering and mathematics) were particularly hard to find among those who worked in Westminster. As many as 90% of those who had taken an undergraduate degree had studied humanities or social sciences. While these subject backgrounds are valuable, the lack of specialised maths skills is stark.

Limited options

The mathematics department at Oxford Brookes has closed and other universities have seen recruitment reductions or other cuts. The resulting maths deserts will remove the opportunity for students to gain a high-quality mathematics education in their local area. Universities should do their best to keep these departments open.

This might be possible if the way that degrees are set up changes. For many degree courses in countries such as the US and Australia, students are able to take a broad selection of subjects, from science and maths subjects through to the humanities. Each are taught in their respective academic departments. This allows students to gain advanced knowledge and see how each field feeds into others.

This is scarcely possible in the UK, where students must choose a specialist and narrow degree programme at age 18.

Another possible solution would be to put core mathematics modules in degree disciplines that rely so heavily on it – such as engineering, economics, chemistry, physics, biology and computer science – and have them taught by specialist mathematicians. This would help keep mathematics departments open, while also ensuring that general mathematical literacy improves in the UK.

The relevance of mathematics and its vast range of applications would be abundantly clear, better equipping every student with the necessary mathematical skills the workforce needs.

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

Credit of the article given to Neil Saunders, The Conversation

What do the flow of cars on a highway and the movement of bacteria towards a food source have in common? In both cases, annoying traffic jams can form. Especially for cars, we might want to understand how to avoid them, but perhaps we’ve never thought of turning to statistical physics.

Alexandre Solon, a physicist from Sorbonne Université, and Eric Bertin, from the University of Grenoble, both working for the Centre national de la recherche scientifique (CNRS), have done just that. Their research, recently published in the Journal of Statistical Mechanics: Theory and Experiment, has developed a one-dimensional mathematical model that describes the movement of particles in situations similar to cars moving along a road or bacteria attracted to a nutrient source, which they then tested with computer simulations to observe what happened as parameters varied.

“The model is one-dimensional because the elements can only move in one direction, like on a one-lane one-way street,” explains Solon.

It’s an idealized situation, but not so different from what happens on many roads where you can find yourself stuck in rush hour traffic. The models from which this research is derived historically come from studying the behaviour of atoms and molecules: for example, those in a gas being heated or cooled. In the case of Bertin and Solon’s model, however, the behaviour of the individual elements is a bit more sophisticated than that of an atom.

“Among other things, a component of inertia has been inserted, which can be more or less pronounced, replicating for example the reactivity of a driver at the wheel. We can imagine a fresh and reactive driver, who brakes and accelerates at just the right moments, or another one at the end of the day, more tired and struggling to stay in sync with the rhythm of the flow of cars they are in,” Solon explains.

By conducting simulations with different values of certain parameters (the density of the elements, inertia, speed), Solon and Bertin were able to determine both situations in which traffic flowed smoothly, or on the contrary, became congested, as well as the type of jams that formed: large and centralized, or smaller and distributed along the route, akin to a “stop-and-go” pattern.

Borrowing language from statistical mechanics, Solon speaks of phase transitions: “Just as when the temperature changes water becomes ice, when the values of some parameters change, a smooth flow of cars becomes a congestion, a knot where no movement is possible.”

When the system reaches a critical density or when movement conditions favour accumulation rather than dispersion, the particles begin to form dense clusters, similar to traffic jams, while other areas may remain relatively empty. Traffic jams, therefore, can be seen as the dense phase in a system that has undergone a phase transition, characterized by low mobility and high localization of particles.

Solon and Bertin have thus identified conditions that can favour this congestion. Continuing with the metaphor of cars, contributing to the formation of traffic jams is the high density of vehicles, which reduces the space between one vehicle and another and increases the likelihood of interaction (and thus slowdown). Another condition is the frequent entries and exits from the flow: The addition of vehicles from the access ramp or attempts to change lanes in dense areas increase the risk of slowdowns, especially if vehicles try to merge without leaving sufficient space.

A third factor is the already-mentioned inertia in the behaviour of drivers, who—when they react with some delay to changes in the speed of the vehicles ahead of them—create a chain reaction of braking that can lead to the formation of a traffic jam. In contrast, the aggregation observed in bacterial colony happens in absence of any inertia, and bacteria can move in any direction contrary to cars that need to follow the direction of traffic.

As Bertin says, “It is thus interesting and surprising to find that both types of behaviours are connected and can be continuously transformed into one another.”

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Credit of the article given to International School of Advanced Studies (SISSA)

A mathematical link between two key equations—one that deals with the very big and the other, the very small—has been developed by a young mathematician in China.

The mathematical discipline known as differential geometry is concerned with the geometry of smooth shapes and spaces. With roots going back to antiquity, the field flourished in the early 20th century, enabling Einstein to develop his general theory of relativity and other physicists to develop quantum field theory and the Standard Model of particle physics.

Gao Chen, a 29-year-old mathematician at the University of Science and Technology of China in Hefei, specializes in a branch known as complex differential geometry. Its complexity is not in dealing with complicated structures, but rather because it is based on complex numbers—a system of numbers that extends everyday numbers by including the square root of -1.

This area appeals to Chen because of its connections with other fields. “Complex differential geometry lies at the intersection of analysis, algebra, and mathematical physics,” he says. “Many tools can be used to study this area.”

Chen has now found a new link between two important equations in the field: the Kähler–Einstein equation, which describes how mass causes curvature in space–time in general relativity, and the Hermitian–Yang–Mills equation, which underpins the Standard Model of particle physics.

Chen was inspired by his Ph.D. supervisor Xiuxiong Chen of New York’s Stony Brook University, to take on the problem. “Finding solutions to the Hermitian–Yang–Mills and the Kähler–Einstein equations are considered the most important advances in complex differential geometry in previous decades,” says Gao Chen. “My results provide a connection between these two key results.”

“The Kähler –Einstein equation describes very large things, as large as the universe, whereas the Hermitian–Yang–Mills equation describes tiny things, as small as quantum phenomena,” explains Gao Chen. “I’ve built a bridge between these two equations.” Gao Chen notes that other bridges existed previously, but that he has found a new one.

“This bridge provides a new key, a new tool for theoretical research in this field,” Gao Chen adds. His paper describing this bridge was published in the journal Inventiones mathematicae in 2021.

In particular, the finding could find use in string theory—the leading contender of theories that researchers are developing in their quest to unite quantum physics and relativity. “The deformed Hermitian–Yang–Mills equation that I studied plays an important role in the study of string theory,” notes Gao Chen.

Gao Chen now has his eyes set on other important problems, including one of the seven Millennium Prize Problems. These are considered the most challenging in the field by mathematicians and carry a $1 million prize for a correct solution. “In the future, I hope to tackle a generalization of the Kähler–Einstein equation,” he says. “I also hope to work on other Millennium Prize problems, including the Hodge conjecture.”

Provided by University of Science and Technology of China

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Credit of the article given to University of Science and Technology of China