* . *
  • About
  • Advertise
  • Privacy & Policy
  • Contact
Saturday, May 10, 2025
Earth-News
  • Home
  • Business
  • Entertainment
    Georgia Entertainment CEO says large-scale production is slowing down – Decaturish

    Georgia Entertainment CEO Warns of Slowdown in Large-Scale Productions

    Zugalu Entertainment Welcomes Crimson Herring Studios to Its Family!

    Fall 2025 TV Schedule: Your Guide to the Complete Lineup – Wyoming News Now

    Get Ready for Fall 2025: Your Ultimate Guide to the Exciting TV Lineup!

    Blackstone River Theatre presents music from Scotland with Cantrip – The Valley Breeze

    Experience the Enchanting Sounds of Scotland: Cantrip Takes the Stage at Blackstone River Theatre!

    The viral wireless audio dongle for in-flight entertainment just got a big upgrade – and I love it – ZDNET

    Game-Changer for In-Flight Entertainment: The Upgraded Wireless Audio Dongle You’ll Love!

    Tahoe City Downtown Association opens 4th of July Entertainment Grant Program – Sierra Sun

    Celebrate Independence Day: Tahoe City Launches Exciting 4th of July Entertainment Grant Program!

  • General
  • Health
  • News

    Cracking the Code: Why China’s Economic Challenges Aren’t Shaking Markets, Unlike America’s” – Bloomberg

    Trump’s Narrow Window to Spread the Truth About Harris

    Trump’s Narrow Window to Spread the Truth About Harris

    Israel-Gaza war live updates: Hamas leader Ismail Haniyeh assassinated in Iran, group says

    Israel-Gaza war live updates: Hamas leader Ismail Haniyeh assassinated in Iran, group says

    PAP Boss to Niger Delta Youths, Stay Away from the Protest

    PAP Boss to Niger Delta Youths, Stay Away from the Protest

    Court Restricts Protests In Lagos To Freedom, Peace Park

    Court Restricts Protests In Lagos To Freedom, Peace Park

    Fans React to Jazz Jennings’ Inspiring Weight Loss Journey

    Fans React to Jazz Jennings’ Inspiring Weight Loss Journey

    Trending Tags

    • Trump Inauguration
    • United Stated
    • White House
    • Market Stories
    • Election Results
  • Science
  • Sports
  • Technology
    Artificial intelligence (AI) – The Guardian

    Unlocking the Future: How Artificial Intelligence is Transforming Our World

    Technology Innovation to Take Center Stage at The 2025 National Restaurant Association Show – Restaurant Technology News

    Get Ready for a Tech Revolution: The 2025 National Restaurant Association Show Unveils Cutting-Edge Innovations!

    Newmont signs deal to use Chrysos Corporation technology – Capital Brief

    Newmont Partners with Chrysos Corporation to Revolutionize Mining Technology

    Air Force Invests in Whisper’s Ultraquiet Propulsion Technology – FLYING Magazine

    Air Force Invests in Whisper’s Ultraquiet Propulsion Technology – FLYING Magazine

    Trump administration set to overhaul Biden’s AI chip export regulations – TechHQ

    Trump administration set to overhaul Biden’s AI chip export regulations – TechHQ

    Technology is the key to empathy in insurance – InsuranceNewsNet

    Unlocking Empathy: How Technology is Transforming the Insurance Experience

    Trending Tags

    • Nintendo Switch
    • CES 2017
    • Playstation 4 Pro
    • Mark Zuckerberg
No Result
View All Result
  • Home
  • Business
  • Entertainment
    Georgia Entertainment CEO says large-scale production is slowing down – Decaturish

    Georgia Entertainment CEO Warns of Slowdown in Large-Scale Productions

    Zugalu Entertainment Welcomes Crimson Herring Studios to Its Family!

    Fall 2025 TV Schedule: Your Guide to the Complete Lineup – Wyoming News Now

    Get Ready for Fall 2025: Your Ultimate Guide to the Exciting TV Lineup!

    Blackstone River Theatre presents music from Scotland with Cantrip – The Valley Breeze

    Experience the Enchanting Sounds of Scotland: Cantrip Takes the Stage at Blackstone River Theatre!

    The viral wireless audio dongle for in-flight entertainment just got a big upgrade – and I love it – ZDNET

    Game-Changer for In-Flight Entertainment: The Upgraded Wireless Audio Dongle You’ll Love!

    Tahoe City Downtown Association opens 4th of July Entertainment Grant Program – Sierra Sun

    Celebrate Independence Day: Tahoe City Launches Exciting 4th of July Entertainment Grant Program!

  • General
  • Health
  • News

    Cracking the Code: Why China’s Economic Challenges Aren’t Shaking Markets, Unlike America’s” – Bloomberg

    Trump’s Narrow Window to Spread the Truth About Harris

    Trump’s Narrow Window to Spread the Truth About Harris

    Israel-Gaza war live updates: Hamas leader Ismail Haniyeh assassinated in Iran, group says

    Israel-Gaza war live updates: Hamas leader Ismail Haniyeh assassinated in Iran, group says

    PAP Boss to Niger Delta Youths, Stay Away from the Protest

    PAP Boss to Niger Delta Youths, Stay Away from the Protest

    Court Restricts Protests In Lagos To Freedom, Peace Park

    Court Restricts Protests In Lagos To Freedom, Peace Park

    Fans React to Jazz Jennings’ Inspiring Weight Loss Journey

    Fans React to Jazz Jennings’ Inspiring Weight Loss Journey

    Trending Tags

    • Trump Inauguration
    • United Stated
    • White House
    • Market Stories
    • Election Results
  • Science
  • Sports
  • Technology
    Artificial intelligence (AI) – The Guardian

    Unlocking the Future: How Artificial Intelligence is Transforming Our World

    Technology Innovation to Take Center Stage at The 2025 National Restaurant Association Show – Restaurant Technology News

    Get Ready for a Tech Revolution: The 2025 National Restaurant Association Show Unveils Cutting-Edge Innovations!

    Newmont signs deal to use Chrysos Corporation technology – Capital Brief

    Newmont Partners with Chrysos Corporation to Revolutionize Mining Technology

    Air Force Invests in Whisper’s Ultraquiet Propulsion Technology – FLYING Magazine

    Air Force Invests in Whisper’s Ultraquiet Propulsion Technology – FLYING Magazine

    Trump administration set to overhaul Biden’s AI chip export regulations – TechHQ

    Trump administration set to overhaul Biden’s AI chip export regulations – TechHQ

    Technology is the key to empathy in insurance – InsuranceNewsNet

    Unlocking Empathy: How Technology is Transforming the Insurance Experience

    Trending Tags

    • Nintendo Switch
    • CES 2017
    • Playstation 4 Pro
    • Mark Zuckerberg
No Result
View All Result
Earth-News
No Result
View All Result
Home Science

A Tower of Conjectures That Rests Upon a Needle

September 14, 2023
in Science
A Tower of Conjectures That Rests Upon a Needle
Share on FacebookShare on Twitter

The next step up in that hierarchy is the “restriction” conjecture. If it is true, so is the Kakeya conjecture. (This also means that if the Kakeya conjecture turns out to be false, the restriction conjecture can’t be true.) The restriction conjecture, in turn, is implied by the so-called Bochner-Riesz conjecture. And at the very top sits the local smoothing conjecture.

The first two conjectures deal with the behavior of the Fourier transform, a technique in harmonic analysis for, in effect, calculating how to express almost any function as a sum of sine waves. It is one of the most powerful mathematical tools available to physicists and engineers. The Fourier transform has played a fundamental role in solving differential equations, expressing quantum mechanical ideas like the Heisenberg uncertainty principle, and analyzing and processing signals — making things like modern mobile phones possible.

Since each statement in the hierarchy implies the one below it, if the Kakeya conjecture is false, none of the other conjectures are true. The entire tower will come crashing down. “You can create a super monster counterexample that would break a lot of conjectures,” Hickman said.

On the other hand, proving the Kakeya conjecture true wouldn’t automatically imply the truth of those other conjectures — but it would give mathematicians important insights into how to proceed.

And so, “nearly half of the community of harmonic analysis that I know of is working on this and related problems, or has worked on them at some point,” said Shaoming Guo of the University of Wisconsin, Madison.

More recently, mathematicians have discovered, to their surprise, that the techniques they’ve developed to tackle these problems can also be used to prove major results in the seemingly unrelated field of number theory. “It’s a much more general phenomenon than people thought,” Guo said.

Layer Cake

The story starts with the Fourier transform. “You want to decompose [functions] into small pieces, analyze their interactions, and add them back together,” said Yumeng Ou of the University of Pennsylvania. For one-dimensional functions — curves that you can plot on a piece of paper — mathematicians have a good understanding of how to do this, even when they need to reverse the Fourier transform using only some of the pieces.

But in two or more dimensions, things can get messy.

In 1971, Charlie Fefferman, a mathematician at Princeton University, figured out how to use Kakeya sets to demonstrate that reversing the Fourier transform can lead to strange and surprising outcomes in multiple dimensions.

Mathematicians found a fix in the form of the Bochner-Riesz conjecture, which essentially states that there are more sophisticated ways to recover the original function that don’t break down like Fefferman’s example. But that fix depended on the truth of the Kakeya conjecture.

If it’s true, “truncating frequencies will only lead to small errors,” said Betsy Stovall of the University of Wisconsin, Madison. “It means the small errors don’t blow up.”

So began the hierarchy. Later, mathematicians discovered another important connection: If true, the Bochner-Riesz conjecture also implied a statement called the restriction conjecture. This conjecture states that if you start with a limited version of the Fourier transform — “restricting” the values you look at to only those that live on particular surfaces — this can still give you important information about the original function. And it turned out that if the restriction conjecture was true, so was the Kakeya conjecture. (This placed the restriction conjecture between Kakeya and Bochner-Riesz in the tower.)

The crowning problem in the hierarchy, called the local smoothing conjecture, doesn’t deal with the Fourier transform directly, but rather puts bounds on the size of solutions to equations describing the behavior of waves.

You can think of this, too, in terms of the geometry of lines in a Kakeya set. You can break up a general solution to the wave equation into a bunch of pieces that move in different directions and interact with each other in different ways over time. Each of those pieces mathematically resembles a needle in a Kakeya set. The Kakeya conjecture asserts that such a configuration can’t have too much overlap. In this physical context, overlaps would correspond to the persistence of irregular and unexpected behaviors in the solution. For instance, a sound wave could amplify in lots of regions at lots of different times.

The local smoothing conjecture states that such irregularities should average out. “It’s like taking the average of the financial market,” said Ciprian Demeter of Indiana University Bloomington. “There could be crashes here and there, but if you invest your money and retire in 40 years, there’s a good chance you’ll get some good investments.”

But as with all the conjectures in the hierarchy, that depends on the truth of the Kakeya conjecture. “The idea is that if you rule out a lot of intersection in Kakeya sets, that means you can rule out these situations where parts of your solution conspire together to create some sort of blowup,” Stovall said.

This conjecture is the most difficult of the bunch: While the two-dimensional cases of the Kakeya, restriction and Bochner-Riesz problems were solved decades ago, the two-dimensional local smoothing conjecture was only proved a few years ago. (In higher dimensions, all these problems remain open.)

But despite the slow progress on proving the local smoothing conjecture, work on it has led to tremendous progress elsewhere. In 1999, while trying to tackle the conjecture, the mathematician Thomas Wolff introduced a method known as decoupling. Since then, that technique has taken on a life of its own: It’s been used to make major breakthroughs not just in harmonic analysis, but in number theory, geometry and other areas. “Using decoupling results, you now have world records in very famous, important problems,” said Christopher Sogge of Johns Hopkins University, who first formulated the local smoothing conjecture in the 1990s. For instance, decoupling has been used to help count how many ways an integer can be represented as the sum of squares, cubes or some other power.

As Demeter put it, these results are possible because “we can look at numbers as waves.” That all these problems link back to Kakeya needle sets “is fascinating,” he added. “You don’t think that so much beauty, difficulty and importance can be hidden in something that can be formulated using line segments.”

>>> Read full article>>>
Copyright for syndicated content belongs to the linked Source : Quanta Magazine – https://www.quantamagazine.org/a-tower-of-conjectures-that-rests-upon-a-needle-20230912/

Tags: ConjecturesscienceTower
Previous Post

New public protector ‘must restore dignity’ after Mkhwebane disaster: Outa’

Next Post

‘Species Repulsion’ Enables High Biodiversity in Tropical Trees

New committee launches for female health and safety – IAFF

New committee launches for female health and safety – IAFF

May 9, 2025
How smart people fall for conspiracy theories – CNN

Unraveling the Paradox: Why Intelligent Minds Are Drawn to Conspiracy Theories

May 9, 2025
Artificial intelligence (AI) – The Guardian

Unlocking the Future: How Artificial Intelligence is Transforming Our World

May 9, 2025
Donovan Mitchell won’t let Cavs go quietly after ‘one of the best seasons of my life’ – FOX Sports

Donovan Mitchell won’t let Cavs go quietly after ‘one of the best seasons of my life’ – FOX Sports

May 9, 2025
Cash Flows: Industry, Ecology, and the Future of the Great Lakes Blue Economy – Michigan Public

Cash Flows: Industry, Ecology, and the Future of the Great Lakes Blue Economy – Michigan Public

May 9, 2025
Under Trump, job prospects look ‘bleak’ for Mass. grads in environmental science – WBUR

Massachusetts Environmental Science Graduates Face Gloomy Job Prospects Under Trump Administration

May 9, 2025
A Leaked Memo Reveals Details About Trump’s War on Science at the NIH – Mother Jones

Inside the Leaked Memo: Uncovering Trump’s Battle Against Science at the NIH

May 9, 2025
Hyatt Lifestyle Group opens inaugural property in Florida – Hotel Dive

Hyatt Lifestyle Group Unveils Its First Florida Gem!

May 9, 2025
2025 IIHF World Championships Begin for Danault, Chromiak, Slukynsky – LA Kings Insider

Exciting Kickoff: Danault, Chromiak, and Slukynsky Take Center Stage at the 2025 IIHF World Championships!

May 9, 2025
US Farm Economy Is Starting to See First Hits From Trump Tariffs – Bloomberg.com

US Farm Economy Faces Initial Impact from Trump Tariffs

May 9, 2025

Categories

Archives

May 2025
MTWTFSS
 1234
567891011
12131415161718
19202122232425
262728293031 
« Apr    
Earth-News.info

The Earth News is an independent English-language daily published Website from all around the World News

Browse by Category

  • Business (20,132)
  • Ecology (595)
  • Economy (606)
  • Entertainment (21,520)
  • General (15,210)
  • Health (9,649)
  • Lifestyle (611)
  • News (22,149)
  • People (609)
  • Politics (614)
  • Science (15,828)
  • Sports (21,117)
  • Technology (15,597)
  • World (596)

Recent News

New committee launches for female health and safety – IAFF

New committee launches for female health and safety – IAFF

May 9, 2025
How smart people fall for conspiracy theories – CNN

Unraveling the Paradox: Why Intelligent Minds Are Drawn to Conspiracy Theories

May 9, 2025
  • About
  • Advertise
  • Privacy & Policy
  • Contact

© 2023 earth-news.info

No Result
View All Result

© 2023 earth-news.info

No Result
View All Result

© 2023 earth-news.info

Go to mobile version