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Sphere packing problem for dimensions 8 and 24 solved

February 27, 2017 by Maths Statistics Science Tutor

Maryna Viazovska, during her time as a postdoctoral researcher at the Berlin Mathematical School and the Humboldt University of Berlin, solved the centuries-old “sphere packing” problem in dimensions 8 and 24 (the latter dimension in collaboration with other researchers).

Interest in dimensions 8 and 24 stem from the fact that in these dimensions, there exist symmetric sphere packings called the E8 (for the 8th dimension) and the Leech lattice (for the 24th dimension) that pack spheres with a higher average density than the best candidates known to mathematicians in other dimensions. The E8 and the Leech lattice also have connections to a wide range of mathematical subjects, including number theory, combinatorics, hyperbolic geometry and even areas of physics such as string theory.

The Ukrainian mathematician proved that the E8 and the Leech lattice are the best sphere packings in their respective dimensions.

The theorem proved by Maryna Viazovska is a higher-dimensional analogue of the original Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler. The Kepler Conjecture, which was settled in 1998 by Thomas Hales of the University of Pittsburgh using computer-assisted “proof by exhaustion”, states that the densest way to pack equal-sized spheres in 3-dimensional Euclidean space is the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is around 74.05%.

Viazovska’s uses modular forms to find a correctly auxiliary function that can calculate the largest allowable sphere density. Her 2013 doctoral dissertation was also on modular forms, but she has additional expertise in discrete optimization, one of the fields that are central to the sphere-packing problem. Thus, a few years ago when Viazovska’s friend Andrii Bondarenko, of the Norwegian University of Science and Technology in Trondheim, suggested that they work together on the 8-dimensional sphere-packing problem, Viazovska agreed. Later this effort become a solo endeavour, and after two years of intense work, Viazovska succeeded in coming up with the right auxiliary function for E8 and proving that it is correct.

A conjecture in mathematics leaves arithmetic geometers puzzled

December 17, 2015 by Maths Statistics Science Tutor

17 December 2015

A collective effort to solve a recently discovered problem in mathematics has yielded few clues and no definitive answers.

The problem is entitled the abc conjecture (also known as the Oesterlé–Masser conjecture). It was first proposed by Joseph Oesterlé (1988) and David Masser (1985), and can be stated as follows: For every ε > 0, there are only finitely many triples of coprime positive integers a + b = c such that c > d^1+ε, where d denotes the product of the distinct prime factors of the product abc.

Approximately three years ago, an alleged proof for the abc conjecture appeared online. Written by Shinichi Mochizuki, the proof is deep and seemingly impenetrable, according to a team of leading specialists who had gathered at the University of Oxford on 7–11 December earlier this year to discuss the matter.

The quest to understand Mochizuki’s proof dates back to August 2012, when he quietly posted four papers on his website in which he claimed to have solved the abc conjecture.

Gödel’s Incompleteness Theorem may render a famous physics problem unanswerable

December 13, 2015 by Maths Statistics Science Tutor

13 December 2015

A well-known thereom at the heart of mathematical logic, Gödel’s Incompleteness Theorem, has deep implications for physics because it makes a fundamental question about matter literally unanswerable.

In 1931, Austrian-born mathematician Kurt Gödel proved that some statements are ‘undecidable’, meaning that it is impossible to prove them either true or false within the confines of a set of axioms adopted by mathematics.

Three researchers have now found that the same principle makes it impossible to calculate the gaps between the lowest energy levels of its electrons from an idealized model of its atoms.

Toby Cubitt, a quantum-information theorist at University College London, and his collaborators have focused on calculating the ‘spectral gap’: the gap between the lowest energy level that electrons can occupy in a material, and the next one up.

The quantum states of the atoms in an atomic lattice of a material contain the information needed to find the material’s spectral gap. Cubitt and his colleagues showed that for an infinite lattice, the question of whether the gap exists is undecidable.

50-Year-Old Math Problem solved

December 7, 2015 by Maths Statistics Science Tutor

07 December 2015

Daniel Spielman of Yale University, together with postdoc Adam Marcus, now at Princeton University, and his graduate student Nikhil Srivastava, now at the University of California, Berkeley, have succeeded in solving the famous Kadison-Singer problem. This problem is one of the paramount problems in C*-algebras and quantum physics that had remained unsolved for almost 50 years.

Over the past two years, the experts in the Kadison-Singer problem have had to work hard to assimilate the ideas of the proof, which was first released in 2013. But whilst mathematicians are still grappling with its intricacies, computer scientists have been quick to exploit the new techniques. Last year, for instance, two researchers made use of these new mathematical tools to enhance their understanding of the famously difficult traveling salesman problem.

The original question Richard Kadison and Isadore Singer posed in 1959 asks how much it is possible to learn about a “state” of a quantum system if you have complete information about that state in a special subsystem. Their question builds on Werner Heisenberg’s uncertainty principle, which says that certain pairs of attributes, like the position and the momentum of a particle, cannot simultaneously be measured to arbitrary precision.

A Calculating Web Site Could Ignite a New Campus ‘Math War’

June 13, 2009 by Maths Science Tutor

Source: Chronicle of Higher Education, 12 June 2009

By Jeffrey R. Young

The long-running debate over whether students should be allowed to wield calculators during mathematics examinations may soon seem quaint.

The latest dilemma facing professors is whether to let students turn to a Web site called WolframAlpha, which not only solves complex math problems, but also can spell out the steps leading to those solutions. In other words, it can instantly do most of the homework and test questions found in many calculus textbooks.

The new tool will be a bane to teaching, some professors say—but others see a blessing.

Iraq-born teen cracks maths puzzle

June 12, 2009 by Maths Science Tutor

Source: Yahoo News, 28 May 2009

STOCKHOLM (AFP) – A 16-year-old Iraqi immigrant living in Sweden has cracked a maths puzzle that has stumped experts for more than 300 years, Swedish media reported on Thursday.

In just four months, Mohamed Altoumaimi has found a formula to explain and simplify the so-called Bernoulli numbers, a sequence of calculations named after the 17th century Swiss mathematician Jacob Bernoulli, the Dagens Nyheter daily said.

Altoumaimi, who came to Sweden six years ago, said teachers at his high school in Falun, central Sweden were not convinced about his work at first.

Maths and markets

March 31, 2009 by Maths Science Tutor

Financial Times, 20 March 2009
FT link

Markets + maths = mayhem. That equation sums up an erroneous view of the role played by mathematics in the banking crisis, which is gaining currency in financial and regulatory circles. For example, this week’s report by Lord Turner, chairman of the UK Financial Services Authority, blamed “misplaced reliance on sophisticated maths” for lulling banks’ top managers into a false sense of security about the risks they were taking. Terms such as quant, geek and rocket scientist, once used in affectionate respect, now have darker connotations.

Mathematicians tend to be shy and retiring, compared with other professional groups, and they have not leapt up to defend themselves in public. In private, however, they are seething – understandably so, since the problem was not the maths itself but the way banks used it.

Contrary to Lord Turner’s assertion, the banks’ sums were not sophisticated enough. They over-simplified, and assumed away the limitations and caveats of their models. They did this to convey an illusion of accuracy and precision, and so convince the market that they had everything under control.