news

Physics and Riemann Hypothesis?

Researchers have recently discovered that solutions to the Riemann zeta function correspond to the solutions of another function that may make it easier to solve the Riemann hypothesis.

Dorje Brody, a mathematical physicist at Brunel University London, says that “to our knowledge, this is the first time that an explicit—and perhaps surprisingly relatively simple—operator has been identified whose eigenvalues [‘solutions’ in matrix terminology] correspond exactly to the nontrivial zeros of the Riemann zeta function“.

Now what remains to be proven is that all of the eigenvalues are real numbers rather than imaginary ones.

This newly discovered operator has close ties with quantum physics. In 1999, Michael Berry and Jonathan Keating made the conjecture (now called the Berry-Keating conjecture) that if such an operator exists, then it should correspond to a theoretical quantum system with particular properties. However, no one has ever found such a system before now.

In general, mathematicians are optimistic that the eigenvalues are actually real, and there is a strong argument for this based on PT symmetry (a concept from quantum physics which says that you can change the signs of all four components of space-time and the result will look the same as the original).

Source

Click here for more on the Riemann hypothesis. M x

NEWS: Abel Prize 2017

The Abel Prize 2017 has been awarded to Yves Meyer of the École normale supérieure Paris-Saclay in France due to his “pivotal role in the development of the mathematical theory of wavelets”, which has applications in data compression, medical imagery and the detection of gravitational waves.

Yves Meyer, en 2010, recevant le prix Gauss.

Meyer, aged 77, will receive 6 million Norwegian krone (around £600,000) for the prize, which aims to recognise outstanding contributions to mathematics. It is often called the ‘Nobel Prize’ of mathematics.

The Abel Prize was previously won by Andrew Wiles in 2016, who solved Fermat’s Last Theorem.

Biography

Yves Meyer, born on the 19th July 1939, grew up in Tunis in the North of Africa. After graduating from École normale supérieure de la rue d’Ulm in Paris and completing a PhD in 1966 at the University of Strasbourg, he became a professor of mathematics at the Université Paris-Sud, then the École Polytechnique and then Université Paris-Dauphine. He then moved to École normale supérieure Paris-Saclay in 1995, until formally retiring in 2008, although he still remains an associate member of the research centre.

To read a full biography of Meyer, click here.

Video of the Ceremony

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NEWS: Vera Rubin

Vera Rubin died on 25th December 2016, aged 88. Rubin was an American astronomer who pioneered work on galaxy rotation rates.

In the 1960s and 70s, Rubin and her collogue Kent Ford noted a discrepancy between the predicted angular motion of galaxies and their observed motion, whilst studying galactic rotation curves.

Galactic Rotation Curve | Source: en.citizendium.org

This led Rubin to conclude that some unseen mass must be influencing the rotation of galaxies. As a result, in an attempt to explain the galaxy rotation problem, the theory of dark matter was created. The existence of this ‘invisible mass’ was first theorised by Fritz Zwicky in the 1930s but until Rubin and Ford’s work it had not been proven to exist.

Although initially it was met with skepticism from the scientific community, Rubin’s results have been confirmed over the subsequent decades.

Emily Levesque from the University of Washington said in an interview with Astronomy magazine:

This discovery “utterly revolutionised our concept of the universe and our entire field.”

It is considered one of the most significant results of the 20th century.

However, Rubin never received the Nobel Prize for Physics, although she was frequently mentioned as a candidate for it. It has been 53 years since a women has won a Nobel Prize in Physics, and now that Vera Rubin has passed away, she is no longer eligible. But, we can take some consolation in the fact that Rubin was indifferent to not being nominated for the Nobel Prize.

“Fame is fleeting,” Rubin said in 1990 to Discover magazine. “My numbers mean more to me than my name. If astronomers are still using my data years from now, that’s my greatest compliment.”

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NEWS: New Twin Primes Found

PrimeGrid is a collaborative website with the aim to search for prime numbers. It is similar to GIMPS, which only searches for Mersenne Primes specifically. It works by allowing anyone to download their software and donate their “unused CPU time” to search for primes. PrimeGrid is responsible for many of the recent prime numbers that have been found, which includes “several in the last few months which rank in the top 160 largest known primes“.

On the 14th of September they announced their most recent discovery made by the user Tom Greer who discovered a new pair of twin primes. (Note that twin primes are prime numbers that differ by two.)

Screen Shot 2016-09-28 at 11.29.24 AM.png

The primes are “388,342 digits long, eclipsing the previous record of 200,700 digits”. These primes have been entered in the database for The Largest Known Primes, which is maintained by Chris Caldwell and are currently ranked 1st for twins and each are ranked 4180th overall.

Source

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abc Conjecture

What is it?

The abc conjecture was first posed by Joseph Oesterlé in 1985 and David Masser in 1988. The conjecture begins by presenting 3 distinct positive integers a, b and that are relatively prime and satisfy the equation a + b = c. It states that, if the number d  is the product of the distinct prime numbers of abc, then d is usually much larger smaller than c. This product is defined mathematically as the ‘radical’ of abc and hence the conjecture can be expressed more formally as:

If a, b, and c are coprime positive integers such that a + b =c, it turns out that “usually” c < rad(abc). 

There are a few exceptions to this. To deal with this, the abc conjecture specifically states that:

For every ε > 0, there exist only finitely many triples (a, b, c) of coprime positive integers, with a +b = c, such that:

c>\operatorname {rad} (abc)^{1+\varepsilon }.

 

Consequences if it is TRUE

The abc conjecture is linked to many other questions in number theory; if it were to be true, then it would imply that many other conjectures are also true, including:

and many more.

Proof

 

In 2012, the Japanese mathematician Shinchi Mochizuki published a 500 page proof of the abc conjecture. However, it was so complex that no mathematicians could understand it due to the fact that it used a new mathematical framework known as inter-universal Teichmüller Theory. In a verification report published on December 2014, Mochizuki stated that

“With the exception of the handful of researchers already involved in the verification activities concerning IUTeich (inter-universal Teichmüller Theory) discussed in the present report, every researcher in arithmetic geometry throughout the world is a complete novice with respect to the mathematics surrounding IUTeich, and hence, in particular, is simply not qualified to issue a definitive (i.e., mathematically meaningful) judgment concerning the validity of IUTeich on the basis of a ‘deep understanding’ arising from his/her previous research achievements.”

However, a few weeks ago, four dozen mathematicians converged to hear Mochizuki present his own work at a conference in Kyoto University. This was a ‘breakthrough’ as described by Ivan Fesenko: “It was the key part of the meeting… He was climbing the summit of his theory, and pulling other participants with him, holding their hands.”

Now, around 10 mathematicians have a solid understanding of this new theory, giving a glimmer of hope for our future understanding of this proof.

Video

Sources: 1 | 2 | 3 | 4 | 5 | 6

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NEWS: World’s Largest Proof

Recently, a trio of mathematicians – Marijn Heule from the University of Texas, Victor Marek from the University of Kentucky, and Oliver Kullmann from Swansea University – have solved a problem in mathematics and the solution takes up 200 terabytes of basic text (just consider the fact that 1 terabyte can hold 337,920 copies of War and Peace)! This breaks the previous recorded of a 13-gigabyte proof, which was published in 2014.

The mathematics problem is named the ‘Boolean Pythagorean Triples problem’, and was posed by Ronald Graham in the 1980s, who offered a $100 prize for the solution.

The problem is part of Ramsey theory and asks:

“Is it possible to colour all the integers either red or blue so that no Pythagorean triple of integers a, b, c, satisfying a^2+b^2=c^2 are all the same colour. For example if you would colour a and b red, and c blue, this would successfully not satisfy the tested triple, but all triples would have to be tested.”

Andrew Moseman from Popular Mechanics details how:

“What makes it so hard is that one integer can be part of multiple Pythagorean triples. Take 5. So 3, 4, and 5 are a Pythagorean triple. But so are 5, 12, and 13. If 5 is blue in the first example, then it has to be blue in the second, meaning either 12 or 13 has to be red.

The proof found that it is only possible to colour the numbers in such a way up to the number 7,824 and that 102,300 such colourings exist. Hence, there is no solution to the problem question. The proof took a supercomputer two days to solve, and generated 200TB of data!

The paper describing the proof was published on arXiv on the 3rd of May 2016.

Although the computer has proved that the colouring is impossible, it has not provided the underlying reason why this is, or explored why the number 7,824 is important. This highlights the objection to the value of computer-assisted proofs; yes, they may be correct, but to what extent are they mathematics?

Sources: 1 | 2

Let me know what you think of computer assisted proofs below! M x

NEWS: Steinburg Conjecture is False!

The Steinburg conjecture, proposed by Richard Steinberg in 1976, is a problem in Graph Theory which states that:

“Every planar graph without 4-cycles and 5-cycles is 3-colourable.”

The property of a graph being 3-colourable means that it is possible to assign colours to its vertices so that no pair of adjacent vertices have the same colour, using only three different colours.

Robert Steinburg

Robert Steinberg was born in 1922 and was a mathematician at UCLA. He made many important contributions to mathematics, including the Steinberg representation, the Lang-Steinberg theorem, the Steinberg group in algebraic K-theory and Steinberg’s formula in representation theory.

Notable Achievements:

  • 1966: he was an invited speaker at the International Congress of Mathematics;
  • 1985: won a Steele Prize, which is presented for distinguished research work and writing in mathematics;
  • 1985: he was elected to the National Academy of Sciences;
  • 1990: won the Jeffery-Williams Prize.

“I have had a good life.” – Robert Steinberg

Counterexample

A paper, recently uploaded to the arXiv, by Vincent Cohn-Addad, Michael Hebdige, Daniel Kral, Zhentao Li and Esteban Salgado, shows the construction of a graph with no cycles of length 4 or 5 that isn’t 3 colourable. This is a direct counterexample to the conjecture, hence proving it to be false.

Sources: 1 | 2

I had another post planned for today but I recently read this news and decided to make a post about it! Hope you enjoy. M x