reports of mathematical physics

This page uses frames, but your browser doesn't support them.

Reports on Mathematical Physics

Volume 2 • Issue 6

• ISSN: 0034-4877
• 5 Year impact factor: 0.9
• Impact factor: 1
• Journal metrics

Reports on Mathematical Physics publishes papers in theoretical physics which present a rigorous mathematical approach to problems of quantum and classical mechanics and field the… Read more

Subscription options

Institutional subscription on sciencedirect.

Reports on Mathematical Physics publishes papers in theoretical physics which present a rigorous mathematical approach to problems of quantum and classical mechanics and field theories, relativity and gravitation, statistical physics and mathematical foundations of physical theories. Papers using modern methods of functional analysis, probability theory, differential geometry, algebra and mathematical logic are preferred.

Benefits to authors We also provide many author benefits, such as free PDFs, a liberal copyright policy, special discounts on Elsevier publications and much more. Please click here for more information on our author services .

Please see our Guide for Authors for information on article submission. If you require any further information or help, please visit our Support Center

Secondary Menu

• Math Intranet
• Mathematical Physics

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.

• Citable Docs. (3years)
• Total Cites (3years)

-->

	Title	Type
1		journal	4.954 Q1	46	192	492	15899	4995	487	10.21	82.81	13.32
2		journal	3.266 Q1	20	29	54	1432	145	54	2.58	49.38	16.18
3		journal	3.028 Q1	25	23	64	1065	154	64	2.44	46.30	18.18
4		journal	2.641 Q1	76	37	148	1435	307	148	1.84	38.78	20.69
5		journal	1.612 Q1	154	371	963	17839	2532	963	2.51	48.08	13.66
6		journal	1.610 Q1	23	113	260	4346	345	260	1.19	38.46	22.31
7		journal	1.609 Q1	33	160	295	10622	1411	290	4.82	66.39	26.35
8		journal	1.357 Q1	100	232	726	8913	1310	726	1.68	38.42	23.48
9		journal	1.349 Q1	168	1280	3087	59283	21668	3061	5.97	46.31	29.39
10		journal	1.185 Q1	127	169	517	7163	1308	513	2.05	42.38	22.25
11		journal	1.185 Q1	16	16	42	662	56	42	1.14	41.38	26.92
12		journal	1.176 Q1	67	101	347	4930	945	346	2.22	48.81	27.62
13		journal	1.074 Q1	148	286	622	13025	2631	621	3.12	45.54	23.02
14		journal	1.007 Q1	52	145	336	5996	607	336	1.84	41.35	15.03
15		journal	0.998 Q1	39	89	277	3028	327	277	1.13	34.02	28.73
16		journal	0.993 Q1	105	285	615	14560	2978	615	5.07	51.09	20.19
17		journal	0.931 Q1	20	34	154	1176	165	154	1.08	34.59	15.94
18		journal	0.855 Q1	61	123	400	4701	536	400	1.42	38.22	13.39
19		journal	0.844 Q1	11	66	173	2272	219	173	1.29	34.42	26.53
20		journal	0.827 Q1	19	38	90	1329	228	88	2.25	34.97	23.93
21		journal	0.802 Q2	25	43	119	1428	111	119	0.75	33.21	14.81
22		journal	0.798 Q2	117	209	580	7203	897	578	1.34	34.46	18.23
23		journal	0.778 Q2	127	580	1607	28792	4560	1601	2.76	49.64	26.67
24		journal	0.774 Q2	25	94	400	2740	620	399	1.49	29.15	24.87
25		journal	0.769 Q2	167	495	1763	23313	3669	1758	2.08	47.10	17.90
26		journal	0.702 Q2	41	55	152	1113	221	151	1.57	20.24	27.45
27		journal	0.667 Q2	56	59	143	2853	216	142	1.64	48.36	13.51
28		journal	0.645 Q2	35	17	51	833	65	51	1.26	49.00	28.57
29		journal	0.638 Q2	82	114	423	4834	1266	418	2.86	42.40	16.51
30		journal	0.621 Q2	28	28	117	920	140	117	1.06	32.86	10.00
31		journal	0.617 Q2	61	209	622	6657	1079	619	1.86	31.85	20.37
32		journal	0.614 Q2	93	152	593	8422	1131	587	1.82	55.41	22.64
33		journal	0.580 Q2	88	906	2678	19992	3124	2678	1.11	22.07	23.04
34		journal	0.569 Q2	122	446	1314	15494	1705	1311	1.30	34.74	23.46
35		journal	0.541 Q2	101	1663	4973	75993	13291	4890	2.51	45.70	27.11
36		journal	0.521 Q2	65	56	167	1918	264	166	1.55	34.25	19.49
37		journal	0.512 Q2	11	11	34	463	96	34	2.91	42.09	16.00
38		journal	0.493 Q2	59	932	2837	39624	7056	2724	2.10	42.52	31.28
39		journal	0.493 Q2	41	106	358	4007	351	358	0.96	37.80	14.29
40		journal	0.428 Q3	25	23	73	3615	187	73	2.69	157.17	34.38
41		journal	0.415 Q3	97	2375	2859	109655	7450	2854	2.70	46.17	28.72
42		journal	0.410 Q3	41	47	148	1201	165	148	1.04	25.55	18.48
43		journal	0.390 Q3	17	34	83	1185	104	82	1.14	34.85	6.45
44		journal	0.384 Q3	40	50	138	1310	147	138	0.91	26.20	23.08
45		journal	0.370 Q3	8	32	128	702	109	128	0.83	21.94	33.33
46		journal	0.368 Q3	16	8	47	169	30	47	0.64	21.13	30.00
47		journal	0.363 Q3	11	9	22	283	25	21	1.24	31.44	16.67
48		journal	0.343 Q3	33	35	82	845	67	81	0.54	24.14	25.00
49		journal	0.337 Q3	36	25	62	1004	62	58	1.28	40.16	13.46
50		journal	0.330 Q3	25	18	70	399	29	69	0.45	22.17	17.39

Follow us on @ScimagoJR Scimago Lab , Copyright 2007-2024. Data Source: Scopus®

Cookie settings

Cookie Policy

Legal Notice

Privacy Policy

Journal of Mathematical Physics

Journal of Mathematical Physics publishes mathematical developments relevant for the formulation and solution of problems in physics and its applications.

• Comment Comments
• Save Article Read Later Read Later

The Mystery at the Heart of Physics That Only Math Can Solve

June 10, 2021

Olena Shmahalo/Quanta Magazine

Introduction

Over the past century, quantum field theory has proved to be the single most sweeping and successful physical theory ever invented. It is an umbrella term that encompasses many specific quantum field theories — the way “shape” covers specific examples like the square and the circle. The most prominent of these theories is known as the Standard Model, and it is this framework of physics that has been so successful.

“It can explain at a fundamental level literally every single experiment that we’ve ever done,” said David Tong , a physicist at the University of Cambridge.

But quantum field theory, or QFT, is indisputably incomplete. Neither physicists nor mathematicians know exactly what makes a quantum field theory a quantum field theory. They have glimpses of the full picture, but they can’t yet make it out.

“There are various indications that there could be a better way of thinking about QFT,” said Nathan Seiberg , a physicist at the Institute for Advanced Study. “It feels like it’s an animal you can touch from many places, but you don’t quite see the whole animal.”

Mathematics, which requires internal consistency and attention to every last detail, is the language that might make QFT whole. If mathematics can learn how to describe QFT with the same rigor with which it characterizes well-established mathematical objects, a more complete picture of the physical world will likely come along for the ride.

“If you really understood quantum field theory in a proper mathematical way, this would give us answers to many open physics problems, perhaps even including the quantization of gravity,” said Robbert Dijkgraaf , director of the Institute for Advanced Study (and a regular columnist for Quanta ).

Nor is this a one-way street. For millennia, the physical world has been mathematics’ greatest muse. The ancient Greeks invented trigonometry to study the motion of the stars. Mathematics turned it into a discipline with definitions and rules that students now learn without any reference to the topic’s celestial origins. Almost 2,000 years later, Isaac Newton wanted to understand Kepler’s laws of planetary motion and attempted to find a rigorous way of thinking about infinitesimal change. This impulse (along with revelations from Gottfried Leibniz) birthed the field of calculus, which mathematics appropriated and improved — and today could hardly exist without.

Now mathematicians want to do the same for QFT, taking the ideas, objects and techniques that physicists have developed to study fundamental particles and incorporating them into the main body of mathematics. This means defining the basic traits of QFT so that future mathematicians won’t have to think about the physical context in which the theory first arose.

The rewards are likely to be great: Mathematics grows when it finds new objects to explore and new structures that capture some of the most important relationships — between numbers, equations and shapes. QFT offers both.

“Physics itself, as a structure, is extremely deep and often a better way to think about mathematical things we’re already interested in. It’s just a better way to organize them,” said David Ben-Zvi , a mathematician at the University of Texas, Austin.

For 40 years at least, QFT has tempted mathematicians with ideas to pursue. In recent years, they’ve finally begun to understand some of the basic objects in QFT itself — abstracting them from the world of particle physics and turning them into mathematical objects in their own right.

Yet it’s still early days in the effort.

“We won’t know until we get there, but it’s certainly my expectation that we’re just seeing the tip of the iceberg,” said Greg Moore , a physicist at Rutgers University. “If mathematicians really understood [QFT], that would lead to profound advances in mathematics.”

Fields Forever

It’s common to think of the universe as being built from fundamental particles: electrons, quarks, photons and the like. But physics long ago moved beyond this view. Instead of particles, physicists now talk about things called “quantum fields” as the real warp and woof of reality.

These fields stretch across the space-time of the universe. They come in many varieties and fluctuate like a rolling ocean. As the fields ripple and interact with each other, particles emerge out of them and then vanish back into them, like the fleeting crests of a wave.

“Particles are not objects that are there forever,” said Tong. “It’s a dance of fields.”

To understand quantum fields, it’s easiest to start with an ordinary, or classical, field. Imagine, for example, measuring the temperature at every point on Earth’s surface. Combining the infinitely many points at which you can make these measurements forms a geometric object, called a field, that packages together all this temperature information.

In general, fields emerge whenever you have some quantity that can be measured uniquely at infinitely fine resolution across a space. “You’re sort of able to ask independent questions about each point of space-time, like, what’s the electric field here versus over there,” said Davide Gaiotto , a physicist at the Perimeter Institute for Theoretical Physics in Waterloo, Canada.

Quantum fields come about when you’re observing quantum phenomena, like the energy of an electron, at every point in space and time. But quantum fields are fundamentally different from classical ones.

While the temperature at a point on Earth is what it is, regardless of whether you measure it, electrons have no definite position until the moment you observe them. Prior to that, their positions can only be described probabilistically, by assigning values to every point in a quantum field that captures the likelihood you’ll find an electron there versus somewhere else. Prior to observation, electrons essentially exist nowhere — and everywhere.

“Most things in physics aren’t just objects; they’re something that lives in every point in space and time,” said Dijkgraaf.

A quantum field theory comes with a set of rules called correlation functions that explain how measurements at one point in a field relate to — or correlate with — measurements taken at another point.

Each quantum field theory describes physics in a specific number of dimensions. Two-dimensional quantum field theories are often useful for describing the behavior of materials, like insulators; six-dimensional quantum field theories are especially relevant to string theory; and four-dimensional quantum field theories describe physics in our actual four-dimensional universe. The Standard Model is one of these; it’s the single most important quantum field theory because it’s the one that best describes the universe.

There are 12 known fundamental particles that make up the universe. Each has its own unique quantum field. To these 12 particle fields the Standard Model adds four force fields, representing the four fundamental forces: gravity, electromagnetism, the strong nuclear force and the weak nuclear force. It combines these 16 fields in a single equation that describes how they interact with each other. Through these interactions, fundamental particles are understood as fluctuations of their respective quantum fields, and the physical world emerges before our eyes.

It might sound strange, but physicists realized in the 1930s that physics based on fields, rather than particles, resolved some of their most pressing inconsistencies, ranging from issues regarding causality to the fact that particles don’t live forever. It also explained what otherwise appeared to be an improbable consistency in the physical world.

“All particles of the same type everywhere in the universe are the same,” said Tong. “If we go to the Large Hadron Collider and make a freshly minted proton, it’s exactly the same as one that’s been traveling for 10 billion years. That deserves some explanation.” QFT provides it: All protons are just fluctuations in the same underlying proton field (or, if you could look more closely, the underlying quark fields).

But the explanatory power of QFT comes at a high mathematical cost.

“Quantum field theories are by far the most complicated objects in mathematics, to the point where mathematicians have no idea how to make sense of them,” said Tong. “Quantum field theory is mathematics that has not yet been invented by mathematicians.”

Too Much Infinity

What makes it so complicated for mathematicians? In a word, infinity.

When you measure a quantum field at a point, the result isn’t a few numbers like coordinates and temperature. Instead, it’s a matrix, which is an array of numbers. And not just any matrix — a big one, called an operator, with infinitely many columns and rows. This reflects how a quantum field envelops all the possibilities of a particle emerging from the field.

“There are infinitely many positions that a particle can have, and this leads to the fact that the matrix that describes the measurement of position, of momentum, also has to be infinite-dimensional,” said Kasia Rejzner of the University of York.

And when theories produce infinities, it calls their physical relevance into question, because infinity exists as a concept, not as anything experiments can ever measure. It also makes the theories hard to work with mathematically.

“We don’t like having a framework that spells out infinity. That’s why you start realizing you need a better mathematical understanding of what’s going on,” said Alejandra Castro , a physicist at the University of Amsterdam.

The problems with infinity get worse when physicists start thinking about how two quantum fields interact, as they might, for instance, when particle collisions are modeled at the Large Hadron Collider outside Geneva. In classical mechanics this type of calculation is easy: To model what happens when two billiard balls collide, just use the numbers specifying the momentum of each ball at the point of collision.

When two quantum fields interact, you’d like to do a similar thing: multiply the infinite-dimensional operator for one field by the infinite-dimensional operator for the other at exactly the point in space-time where they meet. But this calculation — multiplying two infinite-dimensional objects that are infinitely close together — is difficult.

“This is where things go terribly wrong,” said Rejzner.

Smashing Success

Physicists and mathematicians can’t calculate using infinities, but they have developed workarounds — ways of approximating quantities that dodge the problem. These workarounds yield approximate predictions, which are good enough, because experiments aren’t infinitely precise either.

“We can do experiments and measure things to 13 decimal places and they agree to all 13 decimal places. It’s the most astonishing thing in all of science,” said Tong.

One workaround starts by imagining that you have a quantum field in which nothing is happening. In this setting — called a “free” theory because it’s free of interactions — you don’t have to worry about multiplying infinite-dimensional matrices because nothing’s in motion and nothing ever collides. It’s a situation that’s easy to describe in full mathematical detail, though that description isn’t worth a whole lot.

“It’s totally boring, because you’ve described a lonely field with nothing to interact with, so it’s a bit of an academic exercise,” said Rejzner.

But you can make it more interesting. Physicists dial up the interactions, trying to maintain mathematical control of the picture as they make the interactions stronger.

This approach is called perturbative QFT, in the sense that you allow for small changes, or perturbations, in a free field. You can apply the perturbative perspective to quantum field theories that are similar to a free theory. It’s also extremely useful for verifying experiments. “You get amazing accuracy, amazing experimental agreement,” said Rejzner.

But if you keep making the interactions stronger, the perturbative approach eventually overheats. Instead of producing increasingly accurate calculations that approach the real physical universe, it becomes less and less accurate. This suggests that while the perturbation method is a useful guide for experiments, ultimately it’s not the right way to try and describe the universe: It’s practically useful, but theoretically shaky.

“We do not know how to add everything up and get something sensible,” said Gaiotto.

Another approximation scheme tries to sneak up on a full-fledged quantum field theory by other means. In theory, a quantum field contains infinitely fine-grained information. To cook up these fields, physicists start with a grid, or lattice, and restrict measurements to places where the lines of the lattice cross each other. So instead of being able to measure the quantum field everywhere, at first you can only measure it at select places a fixed distance apart.

From there, physicists enhance the resolution of the lattice, drawing the threads closer together to create a finer and finer weave. As it tightens, the number of points at which you can take measurements increases, approaching the idealized notion of a field where you can take measurements everywhere.

“The distance between the points becomes very small, and such a thing becomes a continuous field,” said Seiberg. In mathematical terms, they say the continuum quantum field is the limit of the tightening lattice.

Mathematicians are accustomed to working with limits and know how to establish that certain ones really exist. For example, they’ve proved that the limit of the infinite sequence $latex \frac{1}{2}$ + $latex \frac{1}{4}$ +$latex \frac{1}{8}$ +$latex \frac{1}{16}$ … is 1. Physicists would like to prove that quantum fields are the limit of this lattice procedure. They just don’t know how.

“It’s not so clear how to take that limit and what it means mathematically,” said Moore.

Physicists don’t doubt that the tightening lattice is moving toward the idealized notion of a quantum field. The close fit between the predictions of QFT and experimental results strongly suggests that’s the case.

“There is no question that all these limits really exist, because the success of quantum field theory has been really stunning,” said Seiberg. But having strong evidence that something is correct and proving conclusively that it is are two different things.

It’s a degree of imprecision that’s out of step with the other great physical theories that QFT aspires to supersede. Isaac Newton’s laws of motion, quantum mechanics, Albert Einstein’s theories of special and general relativity — they’re all just pieces of the bigger story QFT wants to tell, but unlike QFT, they can all be written down in exact mathematical terms.

“Quantum field theory emerged as an almost universal language of physical phenomena, but it’s in bad math shape,” said Dijkgraaf. And for some physicists, that’s a reason for pause.

“If the full house is resting on this core concept that itself isn’t understood in a mathematical way, why are you so confident this is describing the world? That sharpens the whole issue,” said Dijkgraaf.

Outside Agitator

Even in this incomplete state, QFT has prompted a number of important mathematical discoveries. The general pattern of interaction has been that physicists using QFT stumble onto surprising calculations that mathematicians then try to explain.

“It’s an idea-generating machine,” said Tong.

At a basic level, physical phenomena have a tight relationship with geometry. To take a simple example, if you set a ball in motion on a smooth surface, its trajectory will illuminate the shortest path between any two points, a property known as a geodesic. In this way, physical phenomena can detect geometric features of a shape.

Now replace the billiard ball with an electron. The electron exists probabilistically everywhere on a surface. By studying the quantum field that captures those probabilities, you can learn something about the overall nature of that surface (or manifold, to use the mathematicians’ term), like how many holes it has. That’s a fundamental question that mathematicians working in geometry, and the related field of topology, want to answer.

“One particle even sitting there, doing nothing, will start to know about the topology of a manifold,” said Tong.

In the late 1970s, physicists and mathematicians began applying this perspective to solve basic questions in geometry. By the early 1990s, Seiberg and his collaborator Edward Witten figured out how to use it to create a new mathematical tool — now called the Seiberg-Witten invariants — that turns quantum phenomena into an index for purely mathematical traits of a shape: Count the number of times quantum particles behave in a certain way, and you’ve effectively counted the number of holes in a shape.

“Witten showed that quantum field theory gives completely unexpected but completely precise insights into geometrical questions, making intractable problems soluble,” said Graeme Segal , a mathematician at the University of Oxford.

Another example of this exchange also occurred in the early 1990s, when physicists were doing calculations related to string theory. They performed them in two different geometric spaces based on fundamentally different mathematical rules and kept producing long sets of numbers that matched each other exactly. Mathematicians picked up the thread and elaborated it into a whole new field of inquiry, called mirror symmetry , that investigates the concurrence — and many others like it.

“Physics would come up with these amazing predictions, and mathematicians would try to prove them by our own means,” said Ben-Zvi. “The predictions were strange and wonderful, and they turned out to be pretty much always correct.”

But while QFT has been successful at generating leads for mathematics to follow, its core ideas still exist almost entirely outside of mathematics. Quantum field theories are not objects that mathematicians understand well enough to use the way they can use polynomials, groups, manifolds and other pillars of the discipline (many of which also originated in physics).

For physicists, this distant relationship with math is a sign that there’s a lot more they need to understand about the theory they birthed. “Every other idea that’s been used in physics over the past centuries had its natural place in mathematics,” said Seiberg. “This is clearly not the case with quantum field theory.”

And for mathematicians, it seems as if the relationship between QFT and math should be deeper than the occasional interaction. That’s because quantum field theories contain many symmetries, or underlying structures, that dictate how points in different parts of a field relate to each other. These symmetries have a physical significance — they embody how quantities like energy are conserved as quantum fields evolve over time. But they’re also mathematically interesting objects in their own right.

“A mathematician might care about a certain symmetry, and we can put it in a physical context. It creates this beautiful bridge between these two fields,” said Castro.

Mathematicians already use symmetries and other aspects of geometry to investigate everything from solutions to different types of equations to the distribution of prime numbers. Often, geometry encodes answers to questions about numbers. QFT offers mathematicians a rich new type of geometric object to play with — if they can get their hands on it directly, there’s no telling what they’ll be able to do.

“We’re to some extent playing with QFT,” said Dan Freed , a mathematician at the University of Texas, Austin. “We’ve been using QFT as an outside stimulus, but it would be nice if it were an inside stimulus.”

Make Way for QFT

Mathematics does not admit new subjects lightly. Many basic concepts went through long trials before they settled into their proper, canonical places in the field.

Take the real numbers — all the infinitely many tick marks on the number line. It took math nearly 2,000 years of practice to agree on a way of defining them. Finally, in the 1850s, mathematicians settled on a precise three-word statement describing the real numbers as a “complete ordered field.” They’re complete because they contain no gaps, they’re ordered because there’s always a way of determining whether one real number is greater or less than another, and they form a “field,” which to mathematicians means they follow the rules of arithmetic.

“Those three words are historically hard fought,” said Freed.

In order to turn QFT into an inside stimulus — a tool they can use for their own purposes — mathematicians would like to give the same treatment to QFT they gave to the real numbers: a sharp list of characteristics that any specific quantum field theory needs to satisfy.

A lot of the work of translating parts of QFT into mathematics has come from a mathematician named Kevin Costello at the Perimeter Institute. In 2016 he coauthored a textbook that puts perturbative QFT on firm mathematical footing, including formalizing how to work with the infinite quantities that crop up as you increase the number of interactions. The work follows an earlier effort from the 2000s called algebraic quantum field theory that sought similar ends, and which Rejzner reviewed in a 2016 book . So now, while perturbative QFT still doesn’t really describe the universe, mathematicians know how to deal with the physically non-sensical infinities it produces.

“His contributions are extremely ingenious and insightful. He put [perturbative] theory in a nice new framework that is suitable for rigorous mathematics,” said Moore.

Costello explains he wrote the book out of a desire to make perturbative quantum field theory more coherent. “I just found certain physicists’ methods unmotivated and ad hoc. I wanted something more self-contained that a mathematician could go work with,” he said.

By specifying exactly how perturbation theory works, Costello has created a basis upon which physicists and mathematicians can construct novel quantum field theories that satisfy the dictates of his perturbation approach. It’s been quickly embraced by others in the field.

“He certainly has a lot of young people working in that framework. [His book] has had its influence,” said Freed.

Costello has also been working on defining just what a quantum field theory is. In stripped-down form, a quantum field theory requires a geometric space in which you can make observations at every point, combined with correlation functions that express how observations at different points relate to each other. Costello’s work describes the properties a collection of correlation functions needs to have in order to serve as a workable basis for a quantum field theory.

The most familiar quantum field theories, like the Standard Model, contain additional features that may not be present in all quantum field theories. Quantum field theories that lack these features likely describe other, still undiscovered properties that could help physicists explain physical phenomena the Standard Model can’t account for. If your idea of a quantum field theory is fixed too closely to the versions we already know about, you’ll have a hard time even envisioning the other, necessary possibilities.

“There is a big lamppost under which you can find theories of fields [like the Standard Model], and around it is a big darkness of [quantum field theories] we don’t know how to define, but we know they’re there,” said Gaiotto.

Costello has illuminated some of that dark space with his definitions of quantum fields. From these definitions, he’s discovered two surprising new quantum field theories. Neither describes our four-dimensional universe, but they do satisfy the core demands of a geometric space equipped with correlation functions. Their discovery through pure thought is similar to how the first shapes you might discover are ones present in the physical world, but once you have a general definition of a shape, you can think your way to examples with no physical relevance at all.

And if mathematics can determine the full space of possibilities for quantum field theories — all the many different possibilities for satisfying a general definition involving correlation functions — physicists can use that to find their way to the specific theories that explain the important physical questions they care most about.

“I want to know the space of all QFTs because I want to know what quantum gravity is,” said Castro.

A Multi-Generational Challenge

There’s a long way to go. So far, all of the quantum field theories that have been described in full mathematical terms rely on various simplifications, which make them easier to work with mathematically.

One way to simplify the problem, going back decades, is to study simpler two-dimensional QFTs rather than four-dimensional ones. A team in France recently nailed down all the mathematical details of a prominent two-dimensional QFT.

Other simplifications assume quantum fields are symmetrical in ways that don’t match physical reality, but that make them more tractable from a mathematical perspective. These include “supersymmetric” and “topological” QFTs.

The next, and much more difficult, step will be to remove the crutches and provide a mathematical description of a quantum field theory that better suits the physical world physicists most want to describe: the four-dimensional, continuous universe in which all interactions are possible at once.

“This is [a] very embarrassing thing that we don’t have a single quantum field theory we can describe in four dimensions, nonperturbatively,” said Rejzner. “It’s a hard problem, and apparently it needs more than one or two generations of mathematicians and physicists to solve it.”

But that doesn’t stop mathematicians and physicists from eyeing it greedily. For mathematicians, QFT is as rich a type of object as they could hope for. Defining the characteristic properties shared by all quantum field theories will almost certainly require merging two of the pillars of mathematics: analysis, which explains how to control infinities, and geometry, which provides a language for talking about symmetry.

“It’s a fascinating problem just in math itself, because it combines two great ideas,” said Dijkgraaf.

If mathematicians can understand QFT, there’s no telling what mathematical discoveries await in its unlocking. Mathematicians defined the characteristic properties of other objects, like manifolds and groups, long ago, and those objects now permeate virtually every corner of mathematics. When they were first defined, it would have been impossible to anticipate all their mathematical ramifications. QFT holds at least as much promise for math.

“I like to say the physicists don’t necessarily know everything, but the physics does,” said Ben-Zvi. “If you ask it the right questions, it already has the phenomena mathematicians are looking for.”

And for physicists, a complete mathematical description of QFT is the flip side of their field’s overriding goal: a complete description of physical reality.

“I feel there is one intellectual structure that covers all of it, and maybe it will encompass all of physics,” said Seiberg.

Now mathematicians just have to uncover it.

Get highlights of the most important news delivered to your email inbox

Comment on this article

Quanta Magazine moderates comments to facilitate an informed, substantive, civil conversation. Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected. Moderators are staffed during regular business hours (New York time) and can only accept comments written in English. 

Next article

Help | Advanced Search

Mathematical Physics

Title: localization for lipschitz monotone quasi-periodic schrödinger operators on $\mathbb{z}^d$ via rellich functions analysis.

Abstract: We establish the Anderson localization and exponential dynamical localization for a class of quasi-periodic Schrödinger operators on $\mathbb{Z}^d$ with bounded or unbounded Lipschitz monotone potentials via multi-scale analysis based on Rellich function analysis in the perturbative regime. We show that at each scale, the resonant Rellich function uniformly inherits the Lipschitz monotonicity property of the potential via a novel Schur complement argument.

Comments:	Comments welcome; 62 pages
Subjects:	Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Dynamical Systems (math.DS); Spectral Theory (math.SP)
Cite as:	[math-ph]
	(or [math-ph] for this version)
	Focus to learn more arXiv-issued DOI via DataCite

Submission history

Access paper:.

• Other Formats

References & Citations

• Google Scholar
• Semantic Scholar

BibTeX formatted citation

Bibliographic and Citation Tools

Code, data and media associated with this article, recommenders and search tools.

• Institution

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs .

Identifiers

Linking ISSN (ISSN-L): 0034-4877

URL http://www.sciencedirect.com/science/journal/00344877

KEEPERS link https://archive.org/details/pub_reports-on-mathematical-physics

Google https://www.google.com/search?q=ISSN+%221879-0674%22

Bing https://www.bing.com/search?q=ISSN+%221879-0674%22

Yahoo https://search.yahoo.com/search?p=ISSN%20%221879-0674%22

Koninklijke Bibliotheek https://webggc.oclc.org/cbs/DB=2.37/SET=4/TTL=1/CMD?ACT=SRCHA&IKT=1007&SRT=RLV&TRM=1879-0674

Resource information

Archival status.

Title proper: Reports on mathematical physics.

Country: Netherlands

Medium: Online

Status	Publisher	Keeper	From	To	Updated	Extent of archive
Preserved	Elsevier	CLOCKSS Archive	1970	2024	01/07/2024
Preserved	ELSEVIER LTD.	Internet Archive	1977	1993	26/04/2024
Preserved	ELSEVIER LTD.	Internet Archive	1986	1993	26/04/2024
Preserved	Elsevier	National Library of the Netherlands	1970	2019	20/10/2023
Preserved	Elsevier	Portico	1970	2024	01/07/2024
Preserved	Elsevier	Scholars Portal	1970	2024	02/06/2024

Record information

Last modification date: 06/02/2021

Type of record: Confirmed

ISSN Center responsible of the record: ISSN National Centre for The Netherlands Please contact this ISSN Centre by clicking on it for any request or query concerning the publication

downloads requested

Discover all the features of the complete ISSN records

Display mode x.

Labelled view

MARC21 view

UNIMARC view

Mathematical Physics, Analysis and Geometry

An International Journal devoted to the Theory and Applications of Analysis and Geometry to Physics

The journal provides a reputable forum for research articles in the areas of Probability Theory and Statistical Physics, Quantum Theory, Integrable Systems and Geometry. 

• The Probability Theory and Statistical Physics section addresses probabilistic models and spatial stochastic processes in statistical physics.
• The Quantum Theory section features research on geometry, probability, and analysis relevant to quantum theory.
• The Integrable Systems section explores connections between integrable systems, both discrete and continuous, and classical/modern mathematics.
• The section on Geometry considers research in differential geometry, algebraic geometry and topology which is inspired by, or of relevance to, physics.
• Cédric Bernardin,
• Benoit Charbonneau,
• Margherita Disertori,
• Cristian Giardinà,
• Alexander V. Mikhailov,
• Frank Nijhoff,
• Martin Speight,
• Walter van Suijlekom

Latest articles

On the resolvent of h+a \(^{*}\) +a.

• Andrea Posilicano

Fluctuation Moments for Regular Functions of Wigner Matrices

Generating Function of q - and Elliptic Multiple Polylogarithms of Hurwitz Type

• Masaki Kato

Quasi-free Isomorphisms of Second Quantization Algebras and Modular Theory

• Roberto Conti
• Gerardo Morsella

Space-Time Fluctuations in a Quasi-static Limit

• Cédric Bernardin
• Patricia Gonçalves
• Stefano Olla

Journal updates

Mpag launches a new section devoted to geometry, the probability and statistical physics section celebrates the new composition of its editorial board, mpag welcomes the new co-eic of the probability theory and statistical physics section, cédric bernardin, take a look at the list of the 10 most downloaded articles of 2021.

Mathematical Physics, Analysis and Geometry would like to congratulate and express gratitude to all the authors who have published their work in our journal. 

Journal information

• Astrophysics Data System (ADS)
• Google Scholar
• INIS Atomindex
• Japanese Science and Technology Agency (JST)
• Mathematical Reviews
• OCLC WorldCat Discovery Service
• Science Citation Index Expanded (SCIE)
• TD Net Discovery Service
• UGC-CARE List (India)

Rights and permissions

Editorial policies

© Springer Nature B.V.

• Find a journal
• Publish with us
• Track your research

Login to your account

Change password, your password must have 8 characters or more and contain 3 of the following:.

• a lower case character, 
• an upper case character, 
• a special character 

Password Changed Successfully

Your password has been changed

Create a new account

Can't sign in? Forgot your password?

Enter your email address below and we will send you the reset instructions

If the address matches an existing account you will receive an email with instructions to reset your password

Request Username

Can't sign in? Forgot your username?

Enter your email address below and we will send you your username

If the address matches an existing account you will receive an email with instructions to retrieve your username

• This Journal
•   
• Institutional Access

Cookies Notification

Our site uses javascript to enchance its usability. you can disable your ad blocker or whitelist our website www.worldscientific.com to view the full content., select your blocker:, adblock plus instructions.

• Click the AdBlock Plus icon in the extension bar
• Click the blue power button
• Click refresh

Adblock Instructions

• Click the AdBlock icon
• Click "Don't run on pages on this site"

uBlock Origin Instructions

• Click on the uBlock Origin icon in the extension bar
• Click on the big, blue power button
• Refresh the web page

uBlock Instructions

• Click on the uBlock icon in the extension bar

Adguard Instructions

• Click on the Adguard icon in the extension bar
• Click on the toggle next to the "Protection on this website" text

Brave Instructions

• Click on the orange lion icon to the right of the address bar
• Click the toggle on the top right, shifting from "Up" to "Down

Adremover Instructions

• Click on the AdRemover icon in the extension bar
• Click the "Don’t run on pages on this domain" button
• Click "Exclude"

Adblock Genesis Instructions

• Click on the Adblock Genesis icon in the extension bar
• Click on the button that says "Whitelist Website"

Super Adblocker Instructions

• Click on the Super Adblocker icon in the extension bar
• Click on the "Don’t run on pages on this domain" button
• Click the "Exclude" button on the pop-up

Ultrablock Instructions

• Click on the UltraBlock icon in the extension bar
• Click on the "Disable UltraBlock for ‘domain name here’" button

Ad Aware Instructions

• Click on the AdAware icon in the extension bar
• Click on the large orange power button

Ghostery Instructions

• Click on the Ghostery icon in the extension bar
• Click on the "Trust Site" button

Firefox Tracking Protection Instructions

• Click on the shield icon on the left side of the address bar
• Click on the toggle that says "Enhanced Tracking protection is ON for this site"

Duck Duck Go Instructions

• Click on the DuckDuckGo icon in the extension bar
• Click on the toggle next to the words "Site Privacy Protection"

Privacy Badger Instructions

• Click on the Privacy Badger icon in the extension bar
• Click on the button that says "Disable Privacy Badger for this site"

Disconnect Instructions

• Click on the Disconnect icon in the extension bar
• Click the button that says "Whitelist Site"

Opera Instructions

• Click on the blue shield icon on the right side of the address bar
• Click the toggle next to "Ads are blocked on this site"

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

Reviews in Mathematical Physics

• Recommend to library
• Alert me on new issues
• Aims & Scope
• Editorial Board
• Sample Issue
• Abstracted & Indexed in
• Price Information
• Contact RMP
• Submission Guidelines
• Open Access
• Peer Review Policy
• Ethical Policy
• Vol. 36 No. 06 CURRENT ISSUE
• Vol. 36 No. 05
• Vol. 36 No. 04
• Vol. 36 No. 03
• Vol. 36 No. 02
• Vol. 36 No. 01

Top Downloaded Articles 2023

Latest articles, lattice green functions for pedestrians: exponential decay.

• Wojciech Dybalski , 
• Alexander Stottmeister , and 
• Yoh Tanimoto

Kolmogorov extension theorem for non-probability measures on Cayley trees

• F. H. Haydarov

On spectral and scattering theory for one-body quantum systems in crossed constant electric and magnetic fields

• Tadayoshi Adachi  and 
• Yuta Tsujii

On uniform decay of the Maxwell fields on black hole space-times

• Sari Ghanem

Perturbation theory and canonical coordinates in celestial mechanics

• Gabriella Pinzari

Spectral asymptotics of elliptic operators on manifolds

• Ivan G. Avramidi

MOST READ ARTICLES The most read articles in the last 3 years

Twistorial cohomotopy implies green–schwarz anomaly cancellation.

• Domenico Fiorenza , 
• Hisham Sati , and 
• Urs Schreiber

Yang–Mills theories on geometric spaces

Coulomb scattering in the massless nelson model iv. atom–electron scattering.

• Wojciech Dybalski  and 
• Alessandro Pizzo

T-duality, vertical holonomy line bundles and loop Hori formulae

• Fei Han  and 
• Varghese Mathai

Determination of black holes by boundary measurements

Continuum limit for lattice schrödinger operators.

• Hiroshi Isozaki  and 
• Arne Jensen

MOST CITED ARTICLES The most cited articles in the last 3 years. Source: Crossref

Solitary waves of the nonlinear klein-gordon equation coupled with the maxwell equations.

• VIERI BENCI  and 
• DONATO FORTUNATO FORTUNATO

Entanglement Breaking Channels

• Michael Horodecki , 
• Peter W. Shor , and 
• Mary Beth Ruskai

Celestial operator products of gluons and gravitons

• A.-M. Raclariu , 
• A. Strominger , and 

Solutions to the modified Korteweg–de Vries equation

• Da-Jun Zhang , 
• Song-Lin Zhao , 
• Ying-Ying Sun , and 

CONTINUOUS SLICE FUNCTIONAL CALCULUS IN QUATERNIONIC HILBERT SPACES

• RICCARDO GHILONI , 
• VALTER MORETTI , and 
• ALESSANDRO PEROTTI

QUANTIZATION METHODS: A GUIDE FOR PHYSICISTS AND ANALYSTS

• S. TWAREQUE ALI  and 
• MIROSLAV ENGLIŠ

TWO-POINT FUNCTIONS AND QUANTUM FIELDS IN DE SITTER UNIVERSE

• JACQUES BROS  and 
• UGO MOSCHELLA

MONOTONICITY OF QUANTUM RELATIVE ENTROPY REVISITED

Causality theory for closed cone structures with applications.

• Ettore Minguzzi

Home → Science → Mathematics

Scientists find a faster way to express pi by accident

It's a major breakthrough in both math and physics.

Researchers from the Indian Institute of Science (IISc) and the University of Calgary have come across a new way to represent the mathematical constant π. The new formula for expressing the irrational number was found unintentionally while the researchers were working on applying quantum field theory (QFT) principles to string theory amplitudes.

A happy accident

The mathematical constant π has been calculated in numerous ways over the centuries. For instance, π is defined as the ratio of the circumference of a circle to its diameter. However, the constant can also be expressed as integrals, continued fractions, and — perhaps most elegantly — using infinite series.

One of the most important π infinite series that math students learn early on is the Gregory-Leibniz and Nilakantha series. This new approach, however, is rooted in the intersection of QFT and string theory, two fundamental frameworks in modern physics.

Researchers Arnab Priya Saha and Aninda Sinha stumbled upon this novel mathematical series by accident, while exploring string theory’s applications in high-energy physics. Even more remarkably, the researchers found that the series closely mirrors the representation of pi proposed by 15th-century mathematician and astronomer Sangamagrama Madhava .

“Our efforts, initially, were never to find a way to look at pi. All we were doing was studying high-energy physics in quantum theory and trying to develop a model with fewer and more accurate parameters to understand how particles interact. We were excited when we got a new way to look at pi,” Sinha says.

A new and fast ‘recipe’ for pi

Traditional series representations of π can require millions of terms to achieve high precision. The new representation developed by Saha and Sinha converges much more rapidly. For instance, setting a parameter to 41.5 allows convergence to 15 decimal places with only 40 terms. Meanwhile, the traditional series may need 50 million terms for similar precision.

The new representation is derived from the Euler-Beta function and tree-level string theory amplitudes. By tweaking parameters and utilizing the crossing symmetric dispersion relation, the researchers obtained a series that converges quickly to π.

In mathematics, a series breaks down a complex parameter like pi into simpler components. This new series offers a rapid approach to approximate pi, which is super important for calculations in high-energy particle physics. The formula they presented is as follows:

“Physicists (and mathematicians) have missed this so far since they did not have the right tools, which were only found through work we have been doing with collaborators over the last three years or so,” Sinha explains. “In the early 1970s, scientists briefly examined this line of research but quickly abandoned it since it was too complicated.”

The rapid convergence of the new series can significantly reduce the computational effort required to calculate π to high precision. This can be particularly useful in fields where π is used extensively, such as numerical simulations, cryptographic algorithms, and computational geometry.

The study showcases how ideas from high-energy physics can lead to practical advancements in pure mathematics. It opens up new avenues for exploring other mathematical constants and functions using similar techniques. Researchers may apply these principles to develop new representations for constants like e or the natural logarithm, further enhancing our computational toolkit.

The findings appeared in the Physical Review Letters.

Was this helpful?

Related posts.

• Americans don’t talk about the environmental impact of their food — and that’s a problem
• Dogs successfully diagnose malaria in children
• How to get rid of hiccups, according to science
• How much you’d pay for something depends on what prices you’ve seen recently

Recent news

Researchers find traces of 12,000-year-old Aboriginal ritual carried out for millennia

Big Banks Break Their Climate Promises by Propping Up Big Meat

One Question About Climate, and Barely an Answer at Biden-Trump Debate

• Editorial Policy
• Privacy Policy and Terms of Use
• How we review products

© 2007-2023 ZME Science - Not exactly rocket science. All Rights Reserved.

• Science News
• Environment
• Natural Sciences
• Matter and Energy
• Quantum Mechanics
• Thermodynamics
• Periodic Table
• Applied Chemistry
• Physical Chemistry
• Biochemistry
• Microbiology
• Plants and Fungi
• Planet Earth
• Earth Dynamics
• Rocks and Minerals
• Invertebrates
• Conservation
• Animal facts
• Climate change
• Weather and atmosphere
• Diseases and Conditions
• Mind and Brain
• Food and Nutrition
• Anthropology
• Archaeology
• The Solar System
• Asteroids, meteors & comets
• Astrophysics
• Exoplanets & Alien Life
• Spaceflight and Exploration
• Computer Science & IT
• Engineering
• Sustainability
• Renewable Energy
• Green Living
• Editorial policy
• Privacy Policy

How do shells get their shapes? These are the forces behind their twists and coils

Math and physics, along with a little evolutionary luck, combine to help form some of the world’s most fascinating creatures.

You can see it in the   logarithmic whorl of a nautilus. The milky iridescence of an abalone. The spiked turret of a queen conch. Precision, elegance, strength. The seashell is a consonance of form and function that both rivals and guides human creativity. From Botticelli’s “Birth of Venus” to Frank Lloyd Wright’s design for the Guggenheim Museum in New York, shells have inspired artists and architects throughout history. Our admiration for seashells even predates writing. Archaeologists in Israel recently unearthed the remains of a clamshell necklace made about 120,000 years ago.

As enduring as our fascination with seashells are the mysteries they hold. How did such complex and beautiful structures come into existence? Evolution by natural selection can help explain why shells became widespread among mollusks, but it cannot fully elucidate how an individual creature builds a skeleton as exquisite as, say, the exceptionally spiny shell of a Venus comb murex. To truly understand the seashell, we must peer beneath its biology to the underlying math and physics. In recent years, scientists undertaking such work have developed a considerably more sophisticated understanding of the physical forces that guide shell formation. Innovative studies by Derek Moulton, a professor of applied mathematics at the University of Oxford, and his colleagues have revealed that much of the astonishing variety of seashells can be explained by a few simple mathematical principles.

( Try designing your own shell in our interactive. )

Every mollusk’s body is surrounded by a mantle, a cloaklike organ that secretes calcium carbonate and mixes it with a scaffolding of protein to form a shell one layer at a time. As a mollusk expands its shell, it adds new material to only one area: the opening, or aperture. Envision this as a circle. If, with each new layer, a mollusk deposits a ring of material the same size as the aperture, its shell will lengthen into a cylinder. If, instead, the mollusk increases the circumference of each new ring, its shell will become a cone. By depositing more material on one side of the opening than the other, a mollusk can curve a cylindrical shell into a doughnut. Imagine gluing a stack of lopsided rings, one atop another—each thicker on its left side than its right. Instead of forming a neat cylindrical tower as uniform rings would, they’d arc to one side—like the wedges in a stone archway—and eventually make a loop. By rotating the points where the mantle secretes more or less material, a mollusk can twist such a loop into a spiraling tube.

While these straightforward mathematical rules account for many of the fundamental forms of seashells, a more convoluted set of interactions generates their most ostentatious features. Spines, knobs, and other ornaments arise from a mismatch between the growth rates of the soft-bodied mollusk’s mantle and its rigid shell. This mismatch creates bulges that are exaggerated with each new layer. The mantle’s relative growth rate and stiffness determine the length, thickness, and curvature of these projections. Such embellishments may initially form because of a combination of chance mutations and inescapable outcomes of the mechanical forces involved in shell growth. If they prove advantageous to a creature’s survival and reproduction, however, they may be perpetuated by natural selection, eventually becoming a prevalent trait.

Although we primarily prize shells for their aesthetic attributes, the animals they house value them for a more pragmatic reason: protection. About 540 million years ago, during a burst of evolutionary innovation known as the Cambrian explosion, predatory animals became much more numerous and proficient. Soft-bodied, slow-moving organisms found themselves in constant jeopardy. Around this time, marine mollusks evolved simple but robust shells to protect their vulnerable exposed tissues. In response, crustaceans, fish, and other predators developed more sophisticated weaponry and hunting techniques.

Eventually, the coevolution of predator and prey contributed to a dramatic increase in the diversity of mollusk shells. Some evolved particularly large and thick shells that only the strongest or craftiest animals could crack. Narrow openings, sometimes reinforced with doors called opercula, made it more difficult for predators to reach their prey. Tall spires allowed some mollusks to retreat even further from danger. Spikes, spines, and knobs thwarted pincers and jaws, as did smooth and slippery surfaces.

Seashells, then, are not purely constructs of either biology or physics, nor can they be adequately described by either mathematical modeling or Darwinian theory alone. The exacting, sculptural beauty of shells that we find so enchanting emerges from a confluence of geometry, mechanics, ecology, evolution, and luck. Every shell you’ve ever scooped out of the sand or marveled at in a museum is a palimpsest layered with secrets that science is still untangling—a physical manifestation of our planet’s complexity and splendor.

Related Topics

You may also like.

From fish pie to beef tartare, 5 things to do with oysters

Home sweet … garbage? The Great Pacific Garbage Patch is teeming with life

Snail mucus is a skin care phenomenon—but does it really work?

One of the world’s biggest sea snails at risk of extinction

Why seashells are getting harder to find on the seashore

• Environment

History & Culture

• History & Culture
• Mind, Body, Wonder
• Interactive Graphic
• Paid Content
• Terms of Use
• Privacy Policy
• Your US State Privacy Rights
• Children's Online Privacy Policy
• Interest-Based Ads
• About Nielsen Measurement
• Do Not Sell or Share My Personal Information
• Nat Geo Home
• Attend a Live Event
• Book a Trip
• Inspire Your Kids
• Shop Nat Geo
• Visit the D.C. Museum
• Learn About Our Impact
• Support Our Mission
• Advertise With Us
• Customer Service
• Renew Subscription
• Manage Your Subscription
• Work at Nat Geo
• Sign Up for Our Newsletters
• Contribute to Protect the Planet

Copyright © 1996-2015 National Geographic Society Copyright © 2015-2024 National Geographic Partners, LLC. All rights reserved

More From Forbes

The most rigorous math program you've never heard of.

• Share to Facebook
• Share to Twitter
• Share to Linkedin

Math-M-Addicts students eagerly dive into complex math problems during class.

In the building of the Speyer Legacy School in New York City, a revolutionary math program is quietly producing some of the city's most gifted young problem solvers and logical thinkers. Founded in 2005 by two former math prodigies, Math-M-Addicts has grown into an elite academy developing the skills and mindset that traditional schooling often lacks.

"We wanted to establish the most advanced math program in New York," explains Ruvim Breydo, co-founder of Math-M-Addicts. "The curriculum focuses not just on mathematical knowledge, but on developing a mastery of problem-solving through a proof-based approach aligned with prestigious competitions like the International Mathematical Olympiad."

From its inception, Math-M-Addicts took an unconventional path. What began as an attempt to attract only the highest caliber high school students soon expanded to offer multiple curriculum levels. "We realized we couldn't find enough kids at the most advanced levels," says Breydo. "So we decided to develop that talent from an earlier age."

The program's approach centers on rigor. At each of the 7 levels, the coursework comprises just a handful of fiendishly difficult proof-based math problems every week. "On average, we expect them to get about 50% of the solutions right," explains instructor Natalia Lukina. "The problems take hours and require grappling with sophisticated mathematical concepts."

But it's about more than just the content. Class sizes are small, with two instructors for every 15-20 students. One instructor leads the session, while the other teacher coordinates the presentation of the homework solutions by students. The teachers also provide customized feedback by meticulously reviewing each student's solutions. "I spend as much time analyzing their thought processes as I do teaching new material," admits instructor Bobby Lee.

Best High-Yield Savings Accounts Of 2024

Best 5% interest savings accounts of 2024.

Lee and the Math-M-Addicts faculty embrace an unconventional pedagogy focused on developing logic, creativity, and a tenacious problem-solving mindset over procedures. "We don't dumb it down for them," says Breydo. "We use technical math language and allow students to struggle through the challenges because that's where real learning happens."

Impressive results of Math-M-addicts students in selective math competitions highlight their ... [+] preparation and dedication.

For the Math-M-Addicts team, finding the right teachers is as essential as shaping brilliant students. Prospective instructors go through a rigorous multi-stage vetting process. "We seek passionate mathematical problem solvers first," says program director Sonali Jasuja. "Teaching experience is great, but first and foremost, we need people who deeply understand and enjoy the reasoning behind mathematics."

Even exceptional instructors undergo extensive training by co-teaching for at least a year alongside veteran Math-M-Addicts faculty before taking the lead role. "Our approach is different from how most US teachers learned mathematics," explains instructor Tanya Gross, the director of Girls Adventures in Math (GAIM) competition. "We immerse them in our unique math culture, which focuses on the 'why' instead of the 'how,' empowering a paradigm shift."

That culture extends to the students as well. In addition to the tools and strategies imparted in class, Math-M-Addicts alumni speak of an unshakable confidence and camaraderie that comes from up to several thousands of hours grappling with mathematics at the highest levels alongside peers facing the same challenges.

As Math-M-Addicts ramps up efforts to expand access through online classes and global partnerships, the founders remain devoted to their core mission. "Math education should not obsess with speed and memorization of math concepts," argues Breydo. "This is not what mathematics is about. To unlock human potential, we must refocus on cognitive reasoning and problem-solving skills. We are seeking to raise young people unafraid to tackle any complex challenge they face"

• Editorial Standards
• Reprints & Permissions

Join The Conversation

One Community. Many Voices. Create a free account to share your thoughts. 

Forbes Community Guidelines

Our community is about connecting people through open and thoughtful conversations. We want our readers to share their views and exchange ideas and facts in a safe space.

In order to do so, please follow the posting rules in our site's  Terms of Service.   We've summarized some of those key rules below. Simply put, keep it civil.

Your post will be rejected if we notice that it seems to contain:

• False or intentionally out-of-context or misleading information
• Insults, profanity, incoherent, obscene or inflammatory language or threats of any kind
• Attacks on the identity of other commenters or the article's author
• Content that otherwise violates our site's  terms.

User accounts will be blocked if we notice or believe that users are engaged in:

• Continuous attempts to re-post comments that have been previously moderated/rejected
• Racist, sexist, homophobic or other discriminatory comments
• Attempts or tactics that put the site security at risk
• Actions that otherwise violate our site's  terms.

So, how can you be a power user?

• Stay on topic and share your insights
• Feel free to be clear and thoughtful to get your point across
• ‘Like’ or ‘Dislike’ to show your point of view.
• Protect your community.
• Use the report tool to alert us when someone breaks the rules.

Thanks for reading our community guidelines. Please read the full list of posting rules found in our site's  Terms of Service.