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Mathematical physics applies rigorous mathematical ideas to problems inspired by physics.  As such, it is a remarkably broad subject.

Mathematics and Physics are traditionally tightly linked subjects, and many historical figures such as Newton and Gauss were both physicists and mathematicians.  Traditionally mathematical physics has been quite closely associated to ideas in calculus, particularly those of differential equations.  In recent years however, in part due to the rise of superstring theory, many more branches of mathematics have become major contributors to physics.

At Duke, some of the mathematical physics studied is related to geometric ideas.  For example, Hubert Bray uses geometric tools like minimal surfaces and harmonic functions to study gravity, black holes, and the curvature of spacetime. He is also interested in dark matter, which makes up most of the mass of galaxies, and which could just as reasonably be called the "large scale unexplained curvature of the universe."

As well as differential geometry, the subject of algebraic geometry now has many applications in mathematical physics.   Paul Aspinwall is a string theorist who wields algebraic geometry to study the higher-dimensional spaces (and their compactification to the more familiar four dimensional spacetimes) which are string theories candidates for the physical universe.

Mark Stern employs geometry and partial differential equations to study Yang Mills theory, which is both the mathematical model describing the fundamental nuclear forces and a major tool of geometry and topology.

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Since 1960, the Journal of Mathematical Physics (JMP) has published some of the best papers from outstanding mathematicians and physicists. JMP was the first journal in the field of mathematical physics and publishes research that connects the application of mathematics to problems in physics, as well as illustrates the development of mathematical methods for such applications and for the formulation of physical theories.

The Journal of Mathematical Physics  (JMP) features content in all areas of mathematical physics. Specifically, the articles focus on areas of research that illustrate the application of mathematics to problems in physics, the development of mathematical methods for such applications, and for the formulation of physical theories. The mathematics featured in the articles are written so that theoretical physicists can understand them. JMP also publishes review articles on mathematical subjects relevant to physics as well as special issues that combine manuscripts on a topic of current interest to the mathematical physics community.

JMP welcomes original research of the highest quality in all active areas of mathematical physics, including the following:

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Mathematical physics consists of the development of mathematical models to understand physics problems. Given the immense breadth of physics problems that can be addressed mathematically, from quantum theory to mechanics, mathematical physics is a very broad area of research. 

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Mathematical Physics involves using techniques from mathematics to formulate and solve problems in physics as well as applying insights from physics to the solution of problems in mathematics. The Mathematical Physics program offers advanced training for students in the overlapping areas of mathematics, theoretical physics and their applications from a unified point of view and promotes research in the field.

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This program is administered in conjunction with the Department of Mathematics leading to a Ph.D. degree in Mathematical Physics. While no master’s degree is offered, a student may qualify for a master’s degree in Mathematics or Physics during the course of study. Normally a student enters the program at the beginning of the second year of graduate study.

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Because mathematics provides the common reference point for all of physics, the UCSB Department of Physics has had a long-standing goal of building a deeper and stronger connection with the Department of Mathematics here at UCSB.  Towards this end, the Math and Physics Departments bridged the gap between the disciplines of pure mathematics and theoretical physics.

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At Cornell, we use sophisticated mathematics of probability theory, quantum theory, nonlinear dynamics and bifurcation theory, and differential geometry to wide-ranging problems of interest to physics and complex systems. These techniques have been central to our understanding of the synchronization of fireflies, infectious disease dynamics in complex populations, algorithms for 3D X-ray imaging of cellular structures, categorization of topological quantum materials, and the effectiveness of simple models in describing our complex world.

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Mathematical Physics

Mathematical physics is an interdisciplinary subject where theoretical physics and mathematics intersect. The University of Iowa has held an ongoing mathematical physics seminar for over forty years, in which faculty from both the mathematics and physics departments actively participate. Topics of interest include relativistic quantum mechanics, quantum field theory, general relativity, representation theory of groups and quantum groups, theory of dynamical systems, quantum information theory, phase transitions, quantum chaos, lattice gauge theory and C* algebras.

Our program in Mathematical Physics is one of the few in the U.S. that is fully interdisciplinary, combining both physicists and mathematicians in a working relationship. Every semester the mathematical physics seminar includes talks given by students, faculty and distinguished visitors, including a fields medalist.  Zoom has allowed us to include many talks by distinguished faculty from external institutions.  The seminar often features topical talks on subjects of interest to the participants.  Quantum information theory and machine learning were recently featured topics.

Students may work on interdisciplinary research topics involving mathematics and theoretical physics.  They can obtain a Ph.D. in the Department of Physics, Department of Mathematics, or through the University's Applied Mathematical and Computational Sciences program, in which a physicist and a mathematician jointly supervise the dissertation.  Students in mathematical physics often work closely with faculty in both the physics and mathematics departments. Several students have recently completed PhD dissertations in related areas of mathematical physics.

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Mathematical physics forms the interplay between physics and mathematics, using pure mathematics to provide a rigorous foundation to open questions from physics. Members of the Math physics group at UCI are working on a wide class of analysis and probability problems, stemming from physics, with areas ranging from electromagnetic theory in linear and nonlinear complex media and statistical mechanics, to solid state physics (random and quasi-periodic Schrödinger operators).

Currently, the group includes three permanent members of faculty, and several excellent graduate students, post-docs, and researchers. In 2012/2013 Charles Newman, a renowned expert in statistical mechanics, currently at the Courant Institute, will be joining UCI faculty. Within UCI Math department, the group has strong interaction with the Ergodic Theory and Dynamical Systems group and the Probability group. Moreover, acclaimed scientists from research institutions world-wide visit on a regular basis, thereby enriching the group's activities and allowing for collaboration with other leading researchers in the field.

Our group is highly regarded in the mathematical physics community. Svetlana Jitomirskaya and Abel Klein gave plenary talks at ICMP, and both have been elected to the executive committee of the International Association of Mathematical Physics (IAMP) . Svetlana Jitomirskaya was elected Vice-President of the IAMP for 2012-14. The work of all faculty members of the group receives steady federal funding.

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• 14 May 2024

Why mathematics is set to be revolutionized by AI

• Thomas Fink 0

Thomas Fink is the director of the London Institute for Mathematical Sciences, UK.

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Giving birth to a conjecture — a proposition that is suspected to be true, but needs definitive proof — can feel to a mathematician like a moment of divine inspiration. Mathematical conjectures are not merely educated guesses. Formulating them requires a combination of genius, intuition and experience. Even a mathematician can struggle to explain their own discovery process. Yet, counter-intuitively, I think that this is the realm in which machine intelligence will initially be most transformative.

In 2017, researchers at the London Institute for Mathematical Sciences, of which I am director, began applying machine learning to mathematical data as a hobby. During the COVID-19 pandemic, they discovered that simple artificial intelligence (AI) classifiers can predict an elliptic curve’s rank 1 — a measure of its complexity. Elliptic curves are fundamental to number theory, and understanding their underlying statistics is a crucial step towards solving one of the seven Millennium Problems, which are selected by the Clay Mathematics Institute in Providence, Rhode Island, and carry a prize of US$1 million each. Few expected AI to make a dent in this high-stakes arena.

AI now beats humans at basic tasks — new benchmarks are needed, says major report

AI has made inroads in other areas, too. A few years ago, a computer program called the Ramanujan Machine produced new formulae for fundamental constants 2 , such as π and e . It did so by exhaustively searching through families of continued fractions — a fraction whose denominator is a number plus a fraction whose denominator is also a number plus a fraction and so on. Some of these conjectures have since been proved, whereas others remain open problems.

Another example pertains to knot theory, a branch of topology in which a hypothetical piece of string is tangled up before the ends are glued together. Researchers at Google DeepMind, based in London, trained a neural network on data for many different knots and discovered an unexpected relationship between their algebraic and geometric structures 3 .

How has AI made a difference in areas of mathematics in which human creativity was thought to be essential?

First, there are no coincidences in maths. In real-world experiments, false negatives and false positives abound. But in maths, a single counterexample leaves a conjecture dead in the water. For example, the Pólya conjecture states that most integers below any given integer have an odd number of prime factors. But in 1960, it was found that the conjecture does not hold for the number 906,180,359. In one fell swoop, the conjecture was falsified.

Second, mathematical data — on which AI can be trained — are cheap. Primes, knots and many other types of mathematical object are abundant. The On-Line Encyclopedia of Integer Sequences (OEIS) contains almost 375,000 sequences — from the familiar Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ...) to the formidable Busy Beaver sequence (0, 1, 4, 6, 13, …), which grows faster than any computable function. Scientists are already using machine-learning tools to search the OEIS database to find unanticipated relationships.

DeepMind AI outdoes human mathematicians on unsolved problem

AI can help us to spot patterns and form conjectures. But not all conjectures are created equal. They also need to advance our understanding of mathematics. In his 1940 essay A Mathematician’s Apology , G. H. Hardy explains that a good theorem “should be one which is a constituent in many mathematical constructs, which is used in the proof of theorems of many different kinds”. In other words, the best theorems increase the likelihood of discovering new theorems. Conjectures that help us to reach new mathematical frontiers are better than those that yield fewer insights. But distinguishing between them requires an intuition for how the field itself will evolve. This grasp of the broader context will remain out of AI’s reach for a long time — so the technology will struggle to spot important conjectures.

But despite the caveats, there are many upsides to wider adoption of AI tools in the maths community. AI can provide a decisive edge and open up new avenues for research.

Mainstream mathematics journals should also publish more conjectures. Some of the most significant problems in maths — such as Fermat’s Last Theorem, the Riemann hypothesis, Hilbert’s 23 problems and Ramanujan’s many identities — and countless less-famous conjectures have shaped the course of the field. Conjectures speed up research by pointing us in the right direction. Journal articles about conjectures, backed up by data or heuristic arguments, will accelerate discovery.

Last year, researchers at Google DeepMind predicted 2.2 million new crystal structures 4 . But it remains to be seen how many of these potential new materials are stable, can be synthesized and have practical applications. For now, this is largely a task for human researchers, who have a grasp of the broad context of materials science.

Similarly, the imagination and intuition of mathematicians will be required to make sense of the output of AI tools. Thus, AI will act only as a catalyst of human ingenuity, rather than a substitute for it.

Nature 629 , 505 (2024)

doi: https://doi.org/10.1038/d41586-024-01413-w

He, Y.-H., Lee, K.-H., Oliver, T. & Pozdnyakov, A. Preprint at arXiv https://doi.org/10.48550/arXiv.2204.10140 (2024).

Raayoni, G. et al. Nature 590 , 67–73 (2021).

Article   PubMed   Google Scholar  

Davies, A. et al. Nature 600 , 70–74 (2021).

Merchant, A. et al. Nature 624 , 80–85 (2023).

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A new theory of quantum gravity could explain the biggest puzzle in cosmology, study suggests

A new theory of quantum gravity, which attempts to unite quantum physics with Einstein's relativity, could help solve the puzzle of the universe's expansion, a theoretical paper suggests.

A variation on the theory of quantum gravity — the unification of quantum mechanics and Einstein's general relativity — could help solve one of the biggest puzzles in cosmology, new research suggests.

For nearly a century, scientists have known that the universe is expanding. But in recent decades, physicists have found that different types of measurements of the expansion rate — called the Hubble parameter — produce puzzling inconsistencies.

To resolve this paradox, a new study suggests incorporating quantum effects into one prominent theory used to determine the expansion rate.

"We tried to resolve and explain the mismatch between the values of the Hubble parameter from two different prominent types of observations," study co-author P.K. Suresh , a professor of physics at the University of Hyderabad in India, told Live Science via email.

An expanding problem

The universe's expansion was first identified by Edwin Hubble in 1929. His observations with the largest telescope of that time revealed that galaxies farther from us appear to move away at faster speeds. Although Hubble initially overestimated the expansion rate, subsequent measurements have refined our understanding, establishing the current Hubble parameter as highly reliable.

Later in the 20th century, astrophysicists introduced a novel technique to gauge the expansion rate by examining the cosmic microwave background, the pervasive "afterglow" of the Big Bang .

However, a serious problem arose with these two types of measurements. Specifically, the newer method produced a Hubble parameter value almost 10% lower than the one deduced from the astronomical observations of distant cosmic objects. Such discrepancies between different measurements, called the Hubble tension, signal potential flaws in our understanding of the universe's evolution.

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Related: Newfound 'glitch' in Einstein's relativity could rewrite the rules of the universe, study suggests

In a study published in the journal Classical and Quantum Gravity , Suresh and his colleague from the University of Hyderabad, B. Anupama, proposed a solution to align these disparate results. They underscored that physicists infer the Hubble parameter indirectly, employing our universe's evolutionary model based on Einstein's theory of general relativity.

The team argued for revising this theory to incorporate quantum effects. These effects, intrinsic to fundamental interactions, encompass random field fluctuations and the spontaneous creation of particles from the vacuum of space.

Despite scientists' ability to integrate quantum effects into theories of other fields, quantum gravity remains elusive, making detailed calculations extremely difficult or even impossible. To make matters worse, experimental studies of these effects require reaching temperatures or energies many orders of magnitude higher than those achievable in a lab.

Acknowledging these challenges, Suresh and Anupama focused on broad quantum-gravity effects common to many proposed theories.

"Our equation doesn't need to account for everything, but that does not prevent us from testing quantum gravity or its effects experimentally," Suresh said.

Their theoretical exploration revealed that accounting for quantum effects when describing the gravitational interactions in the earliest stage of the universe's expansion, called cosmic inflation, could indeed alter the theory's predictions regarding the properties of the microwave background at present, making the two types of Hubble parameter measurements consistent.

Of course, final conclusions can be drawn only when a full-fledged theory of quantum gravity is known, but even the preliminary findings are encouraging. Moreover, the link between the cosmic microwave background and quantum gravitational effects opens the way to experimentally studying these effects in the near future, the team said.

"Quantum gravity is supposed to play a role in the dynamics of the early universe; thus its effect can be observed through measurements of the properties of the cosmic microwave background," Suresh said.

— Mysterious 'unparticles' may be pushing the universe apart, new theoretical study suggests

— 'It could be profound': How astronomer Wendy Freedman is trying to fix the universe

— James Webb telescope discovers oldest black hole in the universe  

"Some of the future missions devoted to studying this electromagnetic background are highly probable and promising to test quantum gravity. … It provides a promising suggestion to resolve and validate the inflationary models of cosmology in conjunction with quantum gravity."

Additionally, the authors posit that quantum gravitational phenomena in the early universe might have shaped the properties of gravitational waves emitted during that period. Detecting these waves with future gravitational-wave observatories could further illuminate quantum gravitational characteristics.

"Gravitational waves from various astrophysical sources have only been observed so far, but gravitational waves from the early universe have not yet been detected," Suresh said. "Hopefully, our work will help in identifying the correct inflationary model and detecting the primordial gravitational waves with quantum gravity features."

Andrey got his B.Sc. and M.Sc. degrees in elementary particle physics from Novosibirsk State University in Russia, and a Ph.D. in string theory from the Weizmann Institute of Science in Israel. He works as a science writer, specializing in physics, space, and technology. His articles have been published in  Elements ,  N+1 , and  AdvancedScienceNews .

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A wave of retractions is shaking physics

Grappling with problematic papers and poorly documented data, researchers and journal editors gathered in Pittsburgh to hash out the best way forward.

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Recent highly publicized scandals have gotten the physics community worried about its reputation—and its future. Over the last five years, several claims of major breakthroughs in quantum computing and superconducting research, published in prestigious journals, have disintegrated as other researchers found they could not reproduce the blockbuster results. 

Last week, around 50 physicists, scientific journal editors, and emissaries from the National Science Foundation gathered at the University of Pittsburgh to discuss the best way forward.“To be honest, we’ve let it go a little too long,” says physicist Sergey Frolov of the University of Pittsburgh, one of the conference organizers. 

The attendees gathered in the wake of retractions from two prominent research teams. One team, led by physicist Ranga Dias of the University of Rochester, claimed that it had invented the world’s first room temperature superconductor in a 2023 paper in Nature . After independent researchers reviewed the work, a subsequent investigation from Dias’s university found that he had fabricated and falsified his data. Nature retracted the paper in November 2023. Last year, Physical Review Letters retracted a 2021 publication on unusual properties in manganese sulfide that Dias co-authored. 

The other high-profile research team consisted of researchers affiliated with Microsoft working to build a quantum computer. In 2021, Nature retracted the team’s 2018 paper that claimed the creation of a pattern of electrons known as a Majorana particle, a long-sought breakthrough in quantum computing. Independent investigations of that research found that the researchers had cherry-picked their data, thus invalidating their findings. Another less-publicized research team pursuing Majorana particles fell to a similar fate, with Science retracting a 2017 article claiming indirect evidence of the particles in 2022.

In today’s scientific enterprise, scientists perform research and submit the work to editors. The editors assign anonymous referees to review the work, and if the paper passes review, the work becomes part of the accepted scientific record. When researchers do publish bad results, it’s not clear who should be held accountable—the referees who approved the work for publication, the journal editors who published it, or the researchers themselves. “Right now everyone’s kind of throwing the hot potato around,” says materials scientist Rachel Kurchin of Carnegie Mellon University, who attended the Pittsburgh meeting.

Much of the three-day meeting, named the International Conference on Reproducibility in Condensed Matter Physics (a field that encompasses research into various states of matter and why they exhibit certain properties), focused on the basic scientific principle that an experiment and its analysis must yield the same results when repeated. “If you think of research as a product that is paid for by the taxpayer, then reproducibility is the quality assurance department,” Frolov told MIT Technology Review . Reproducibility offers scientists a check on their work, and without it, researchers might waste time and money on fruitless projects based on unreliable prior results, he says. 

In addition to presentations and panel discussions, there was a workshop during which participants split into groups and drafted ideas for guidelines that researchers, journals, and funding agencies could follow to prioritize reproducibility in science. The tone of the proceedings stayed civil and even lighthearted at times. Physicist Vincent Mourik of Forschungszentrum Jülich, a German research institution, showed a photo of a toddler eating spaghetti to illustrate his experience investigating another team’s now-retracted experiment. ​​Occasionally the discussion almost sounded like a couples counseling session, with NSF program director Tomasz Durakiewicz asking a panel of journal editors and a researcher to reflect on their “intimate bond based on trust.”

But researchers did not shy from directly criticizing Nature , Science , and the Physical Review family of journals, all of which sent editors to attend the conference. During a panel, physicist Henry Legg of the University of Basel in Switzerland called out the journal Physical Review B for publishing a paper on a quantum computing device by Microsoft researchers that, for intellectual-property reasons, omitted information required for reproducibility. “It does seem like a step backwards,” Legg said. (Sitting in the audience, Physical Review B editor Victor Vakaryuk said that the paper’s authors had agreed to release “the remaining device parameters” by the end of the year.) 

Journals also tend to “focus on story,” said Legg, which can lead editors to be biased toward experimental results that match theoretical predictions. Jessica Thomas, the executive editor of the American Physical Society, which publishes the Physical Review journals, pushed back on Legg’s assertion. “I don’t think that when editors read papers, they’re thinking about a press release or [telling] an amazing story,” Thomas told MIT Technology Review . “I think they’re looking for really good science.” Describing science through narrative is a necessary part of communication, she says. “We feel a responsibility that science serves humanity, and if humanity can’t understand what’s in our journals, then we have a problem.” 

Frolov, whose independent review with Mourik of the Microsoft work spurred its retraction, said he and Mourik have had to repeatedly e-mail the Microsoft researchers and other involved parties to insist on data. “You have to learn how to be an asshole,” he told MIT Technology Review . “It shouldn’t be this hard.” 

At the meeting, editors pointed out that mistakes, misconduct, and retractions have always been a part of science in practice. “I don’t think that things are worse now than they have been in the past,” says Karl Ziemelis, an editor at Nature .

Ziemelis also emphasized that “retractions are not always bad.” While some retractions occur because of research misconduct, “some retractions are of a much more innocent variety—the authors having made or being informed of an honest mistake, and upon reflection, feel they can no longer stand behind the claims of the paper,” he said while speaking on a panel. Indeed, physicist James Hamlin of the University of Florida, one of the presenters and an independent reviewer of Dias’s work, discussed how he had willingly retracted a 2009 experiment published in Physical Review Letters in 2021 after another researcher’s skepticism prompted him to reanalyze the data. 

What’s new is that “the ease of sharing data has enabled scrutiny to a larger extent than existed before,” says Jelena Stajic, an editor at Science . Journals and researchers need a “more standardized approach to how papers should be written and what needs to be shared in peer review and publication,” she says.

Focusing on the scandals “can be distracting” from systemic problems in reproducibility, says attendee Frank Marsiglio, a physicist at the University of Alberta in Canada. Researchers aren’t required to make unprocessed data readily available for outside scrutiny. When Marsiglio has revisited his own published work from a few years ago, sometimes he’s had trouble recalling how his former self drew those conclusions because he didn’t leave enough documentation. “How is somebody who didn’t write the paper going to be able to understand it?” he says.

Problems can arise when researchers get too excited about their own ideas. “What gets the most attention are cases of fraud or data manipulation, like someone copying and pasting data or editing it by hand,” says conference organizer Brian Skinner, a physicist at Ohio State University. “But I think the much more subtle issue is there are cool ideas that the community wants to confirm, and then we find ways to confirm those things.”

But some researchers may publish bad data for a more straightforward reason. The academic culture, popularly described as “publish or perish,” creates an intense pressure on researchers to deliver results. “It’s not a mystery or pathology why somebody who’s under pressure in their work might misstate things to their supervisor,” said Eugenie Reich, a lawyer who represents scientific whistleblowers, during her talk.

Notably, the conference lacked perspectives from researchers based outside the US, Canada, and Europe, and from researchers at companies. In recent years, academics have flocked to companies such as Google, Microsoft, and smaller startups to do quantum computing research, and they have published their work in Nature , Science , and the Physical Review journals. Frolov says he reached out to researchers from a couple of companies, but “that didn’t work out just because of timing,” he says. He aims to include researchers from that arena in future conversations.

After discussing the problems in the field, conference participants proposed feasible solutions for sharing data to improve reproducibility. They discussed how to persuade the community to view data sharing positively, rather than seeing the demand for it as a sign of distrust. They also brought up the practical challenges of asking graduate students to do even more work by preparing their data for outside scrutiny when it may already take them over five years to complete their degree. Meeting participants aim to publicly release a paper with their suggestions. “I think trust in science will ultimately go up if we establish a robust culture of shareable, reproducible, replicable results,” says Frolov. 

Deepfakes of your dead loved ones are a booming Chinese business

People are seeking help from AI-generated avatars to process their grief after a family member passes away.

• Zeyi Yang archive page

Technology is probably changing us for the worse—or so we always think

For nearly a hundred years in this publication (and long before that elsewhere) people have worried that new technologies could alter what it means to be human.

• Timothy Maher archive page

Why Threads is suddenly popular in Taiwan

During Taiwan’s presidential election, Meta’s social network emerged as a surprise hit.

Threads is giving Taiwanese users a safe space to talk about politics

But Meta's discomfort with political content could end up pushing them away.

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May 21, 2024

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The case for 'math-ish' thinking

by Stanford University

For everyone whose relationship with mathematics is distant or broken, Jo Boaler, a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start—to approach the subject with playfulness and curiosity, not anxiety or dread.

"Most people have only ever experienced what I call narrow mathematics—a set of procedures they need to follow, at speed," Boaler says. "Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience."

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed, a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced How to Learn Math, the first massive open online course (MOOC) on mathematics education . She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users.

In her new book, " Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics ," Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls "ishing" a math problem can help students make better sense of the answer.

What do you mean by 'math-ish' thinking?

It's a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport—these are generally answered with what I call "ish" numbers, and that's very different from the way we use and learn numbers in school.

In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They're given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2—but the most common answer 13-year-olds gave was 19. The second most common was 21.

I'm not surprised, because when students learn fractions, they often don't learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don't you also risk sending the message that mathematical precision isn't important?

I'm not saying precision isn't important. What I'm suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they'll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, "Isn't 'ishing' just estimating?" It is, but when we ask students to estimate, they often groan, thinking it's yet another mathematical method. But when we ask them to "ish" a number, they're more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mindset. I think we all need a little more ish in our lives.

You also argue that mathematics should be taught in more visual ways. What do you mean by that?

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things—all of that contributes to our understanding of how it works.

There's an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik's Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking.

Some years back we were interviewing students a year after they'd done that activity in our summer camp and asked what had stayed with them. One student said, "I'm in geometry class now, and I still remember that sugar cube, what it looked like and felt like." His class had been asked to estimate the volume of their shoes, and he said he'd imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don't get to see and manipulate them. When we learn about square roots, we don't take squares and look at their diagonals. We just manipulate numbers.

I wonder if people consider the physical representations more appropriate for younger kids.

That's the thing—elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it's all symbolic. There's a myth that there's a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.

There's an example in the book where you've asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn't it be easier to teach students one standard method?

That narrow, rigid version of mathematics where there's only one right approach is what most students experience, and it's a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they're not developing number sense.

They don't learn how to use numbers flexibly in different situations. It also makes students who think differently believe there's something wrong with them.

When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics.

When we bring those forms of diversity together, it's powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.

Provided by Stanford University

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A Groundbreaking Scientific Discovery Just Created the Instruction Manual for Light-Speed Travel

In a first for warp drives, this research actually obeys the laws of physics.

If a superluminal—meaning faster than the speed of light—warp drive like Alcubierre’s worked, it would revolutionize humanity’s endeavors across the universe , allowing us, perhaps, to reach Alpha Centauri, our closest star system, in days or weeks even though it’s four light years away.

The clip above from the 2016 film Star Trek Beyond showcases the effect of a starship zipping through space inside a faster-than-light warp bubble. You can see the imagined but hypothetically accurate warping of spacetime.

However, the Alcubierre drive has a glaring problem: the force behind its operation, called “negative energy,” involves exotic particles—hypothetical matter that, as far as we know, doesn’t exist in our universe. Described only in mathematical terms, exotic particles act in unexpected ways, like having negative mass and working in opposition to gravity (in fact, it has “anti-gravity”). For the past 30 years, scientists have been publishing research that chips away at the inherent hurdles to light speed revealed in Alcubierre’s foundational 1994 article published in the peer-reviewed journal Classical and Quantum Gravity .

Now, researchers at the New York City-based think tank Applied Physics believe they’ve found a creative new approach to solving the warp drive’s fundamental roadblock. Along with colleagues from other institutions, the team envisioned a “positive energy” system that doesn’t violate the known laws of physics . It’s a game-changer, say two of the study’s authors: Gianni Martire, CEO of Applied Physics, and Jared Fuchs, Ph.D., a senior scientist there. Their work, also published in Classical and Quantum Gravity in late April, could be the first chapter in the manual for interstellar spaceflight.

Positive energy makes all the difference. Imagine you are an astronaut in space, pushing a tennis ball away from you. Instead of moving away, the ball pushes back, to the point that it would “take your hand off” if you applied enough pushing force, Martire tells Popular Mechanics . That’s a sign of negative energy, and, though the Alcubierre drive design requires it, there’s no way to harness it.

Instead, regular old positive energy is more feasible for constructing the “ warp bubble .” As its name suggests, it’s a spherical structure that surrounds and encloses space for a passenger ship using a shell of regular—but incredibly dense—matter. The bubble propels the spaceship using the powerful gravity of the shell, but without causing the passengers to feel any acceleration. “An elevator ride would be more eventful,” Martire says.

That’s because the density of the shell, as well as the pressure it exerts on the interior, is controlled carefully, Fuchs tells Popular Mechanics . Nothing can travel faster than the speed of light, according to the gravity-bound principles of Albert Einstein’s theory of general relativity . So the bubble is designed such that observers within their local spacetime environment—inside the bubble—experience normal movement in time. Simultaneously, the bubble itself compresses the spacetime in front of the ship and expands it behind the ship, ferrying itself and the contained craft incredibly fast. The walls of the bubble generate the necessary momentum, akin to the momentum of balls rolling, Fuchs explains. “It’s the movement of the matter in the walls that actually creates the effect for passengers on the inside.”

Building on its 2021 paper published in Classical and Quantum Gravity —which details the same researchers’ earlier work on physical warp drives—the team was able to model the complexity of the system using its own computational program, Warp Factory. This toolkit for modeling warp drive spacetimes allows researchers to evaluate Einstein’s field equations and compute the energy conditions required for various warp drive geometries. Anyone can download and use it for free . These experiments led to what Fuchs calls a mini model, the first general model of a positive-energy warp drive. Their past work also demonstrated that the amount of energy a warp bubble requires depends on the shape of the bubble; for example, the flatter the bubble in the direction of travel, the less energy it needs.

☄️ DID YOU KNOW? People have been imagining traveling as fast as light for nearly a century, if not longer. The 1931 novel Islands of Space by John W. Campbell mentions a “warp” method in the context of superluminal space travel.

This latest advancement suggests fresh possibilities for studying warp travel design, Erik Lentz, Ph.D., tells Popular Mechanics . In his current position as a staff physicist at Pacific Northwest National Laboratory in Richland, Washington, Lentz contributes to research on dark matter detection and quantum information science research. His independent research in warp drive theory also aims to be grounded in conventional physics while reimagining the shape of warped space. The topic needs to overcome many practical hurdles, he says.

Controlling warp bubbles requires a great deal of coordination because they involve enormous amounts of matter and energy to keep the passengers safe and with a similar passage of time as the destination. “We could just as well engineer spacetime where time passes much differently inside [the passenger compartment] than outside. We could miss our appointment at Proxima Centauri if we aren’t careful,” Lentz says. “That is still a risk if we are traveling less than the speed of light.” Communication between people inside the bubble and outside could also become distorted as it passes through the curvature of warped space, he adds.

While Applied Physics’ current solution requires a warp drive that travels below the speed of light, the model still needs to plug in a mass equivalent to about two Jupiters. Otherwise, it will never achieve the gravitational force and momentum high enough to cause a meaningful warp effect. But no one knows what the source of this mass could be—not yet, at least. Some research suggests that if we could somehow harness dark matter , we could use it for light-speed travel, but Fuchs and Martire are doubtful, since it’s currently a big mystery (and an exotic particle).

Despite the many problems scientists still need to solve to build a working warp drive, the Applied Physics team claims its model should eventually get closer to light speed. And even if a feasible model remains below the speed of light, it’s a vast improvement over today’s technology. For example, traveling at even half the speed of light to Alpha Centauri would take nine years. In stark contrast, our fastest spacecraft, Voyager 1—currently traveling at 38,000 miles per hour—would take 75,000 years to reach our closest neighboring star system.

Of course, as you approach the actual speed of light, things get truly weird, according to the principles of Einstein’s special relativity . The mass of an object moving faster and faster would increase infinitely, eventually requiring an infinite amount of energy to maintain its speed.

“That’s the chief limitation and key challenge we have to overcome—how can we have all this matter in our [bubble], but not at such a scale that we can never even put it together?” Martire says. It’s possible the answer lies in condensed matter physics, he adds. This branch of physics deals particularly with the forces between atoms and electrons in matter. It has already proven fundamental to several of our current technologies, such as transistors, solid-state lasers, and magnetic storage media.

The other big issue is that current models allow a stable warp bubble, but only for a constant velocity. Scientists still need to figure out how to design an initial acceleration. On the other end of the journey, how will the ship slow down and stop? “It’s like trying to grasp the automobile for the first time,” Martire says. “We don’t have an engine just yet, but we see the light at the end of the tunnel.” Warp drive technology is at the stage of 1882 car technology, he says: when automobile travel was possible, but it still looked like a hard, hard problem.

The Applied Physics team believes future innovations in warp travel are inevitable. The general positive energy model is a first step. Besides, you don’t need to zoom at light speed to achieve distances that today are just a dream, Martire says. “Humanity is officially, mathematically, on an interstellar track.”

Before joining Popular Mechanics , Manasee Wagh worked as a newspaper reporter, a science journalist, a tech writer, and a computer engineer. She’s always looking for ways to combine the three greatest joys in her life: science, travel, and food.

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