What's Happening in Mathematics?
page 5 (newest)
An occasionally updated series of links to mathematical news — new applications,
new discoveries, problems, personalities, prizes...
new discoveries, problems, personalities, prizes...
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Take any integer n > 0. If n is even, divide by 2; if odd, replace by 3n+1. Repeat indefinitely. The Collatz (or 3n+1) Conjecture states that the process always reaches n = 1, and then cycles 1, 4, 2, 1, 4, 2, ... . This conjecture has many names, and has long resisted efforts at a proof or disproof. Now Terry Tao has made significant progress using techniques from probability theory. Roughly, he proves that almost all n (in the sense of logarithmic density) eventually attain a value much smaller than n (less than f(n) for any f such that f(n) tends to infinity with n, such as log log log log n). A complete proof remains out of reach by these methods.
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42 is interesting after all...
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Which numbers are sums of three (positive or negative) cubes? A team led by the University of Bristol and MIT has solved a famous maths puzzle with an answer for the most elusive number of all: 42. The problem, set in 1954, looked for Solutions of the Diophantine Equation x^3+y^3+z^3=k, with k being all the numbers from 1 to 100. Until recently only 33 and 42 held out. Then 33 was solved, and finally 42. The numbers are x = -80538738812075974, y = 80435758145817515, z = 12602123297335631.
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Paul Krapivsky and Sidney Redner take a new look at an old optimisation problem to compare three strategies for parking your car. The "meek" strategy picks the first available. The "optimistic" strategy looks for a space next to the entrance, then backtracks to the closest vacancy. "Prudent" drivers drive past the first available space, betting that another space exists further in; then backtrack to the space a meek driver would have claimed initially if there isn't one. Which is best? There's also an unexpected connection to the microtubule "scaffolding" of living cells, which made it possible to find a solution.
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A method for finding primes first devised by the ancient Greek mathematician Eratosthenes in 240 BC has been upgraded. Peruvian mathematician Harald Helfgott has invented an improved version of the sieve of Eratosthenes. It reduces the amount of computer memory required, so the calculation can be carried out much faster. To find all primes up to a trillion, the new sieve uses a few million bits instead of a billion .
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Leif Ristroph, a mathematician at New York University’s Courant Institute, has evolved efficient birds' wings mathematically. The method is to 3D-print a selection of wing shapes, test them to see which is fastest, feed the data into an algorithm that simulates evolution, and repeat. The result suggest that an asymmetric teardrop-shaped wing with a very thin trailing edge suppresses the formation of vortices, and is fastest for both flight and (when applied to fish fins) swimming. Real bird's wings often have this shape.
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First woman to win this prestigious mathematics prize
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The Norwegian Academy of Science and Letters has awarded the 2019 Abel Prize to Karen Uhlenbeck “for her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.” King Harald V will present the prize in Oslo on 21 May.
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When the signal is a change in tunnel length one thousandth of the diameter of a proton, noise is a huge problem. Template searches find what you're looking for — even if it's not there. Everyone expects gravitational waves to exist... but did LIGO find any?
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On 12 September 2015 the Laser Interferometer Gravitational-Wave Observatory (LIGO) switched on its upgraded detectors. Two days later it made its first detection. The media reported one of the greatest scientific breakthroughs of the century and a Nobel prize followed. Now a group of physicists, who have done their own analysis of the data, are skeptical: “We believe that LIGO has failed to make a convincing case for the detection of any gravitational wave event.” New Scientist investigates...
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The 2017 Atlantic hurricane season caused $265 billion of damage. Christina Patricola and Michael Wehner use a supercomputer to analyse cyclones (aka hurricanes, typhoons) via high-accuracy mathematical climate models that include convection on small spatial scales. They conclude that man-made climate change increased the wind speed and total rainfall of hurricanes such as Katrina, Irma, and Maria by 15%-30%. Hurricanes as severe as Harvey will become twenty times more likely by 2100.
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After a ten-year search, Raimar Wulkenhaar and Erik Panzer have obtained an explicit solution to an equation in quantum field theory that everyone thought could be solved only numerically. The equation has applications to elementary particles.
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How it spreads, and how to test methods for defending against it
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Dorje Brody and David Meier model fake news as biased noise in communication channels. Their model predicts such things as the likelihood of flipping an election outcome if false stories are released with a given frequency. Meier says their work "highlights the importance of mathematical modelling in dealing with the challenges our society faces."
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The Councils of the IMA and the LMS have awarded the 2018 Christopher Zeeman Medal to Dr Hannah Fry of University College London for her contributions to the public understanding of the mathematical sciences. The Christopher Zeeman Medal was created and named in honour of Professor Sir Christopher Zeeman FRS, the first mathematician to deliver the Royal Institution Christmas Lectures.
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The Fields Medal is the most prestigious award in mathematics, despite its low monetary value. It is awarded every four years by the International Mathematical Union. The 2018 winners were Caucher Birkar, Alessio Figalli, Peter Scholze, and Akshay Venkatesh.
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Biologist and amateur mathematician Aubrey de Grey has made a big advance on the Hadwiger-Nelson problem, which dates from 1950. How many colours are needed to colour every point in the plane, with no two points unit distance apart having the same colour? Until recently it was known to be 4, 5, 6, or 7. De Grey has proved it's not 4.
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László Fejes Tóth's Zone Conjecture of 1973 states that if a unit sphere is covered by several zones, their total width is at least π. A zone of width w is the set of points withing spherical; distance w of a great circle. A proof was found in 2017 by Zilin Jiang and Alexandr Polyanskii.
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Discovery of the 50th Mersenne Prime
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On 3 January 2018 the GIMPS project found the largest known prime number, 2^(77,232,917)-1, with 23,249,425 decimal digits. Like most record primes, it's a Mersenne prime—a power of two, minus one. The mathematical significance of such discoveries is small, but record-breaking computations are a good way to test computers. To prove the point, the same GIMPS software earlier revealed a flaw in Intel's Skylake CPU.
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Claims it was 500 years earlier than we thought are now disputed
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According to a press release from the Bodleian Library, carbon dating shows that the ancient Indian Bakhshali manuscript contains the oldest known occurrence of the symbol 'zero', making its origins 500 years older than previously believed. Three samples were dated to 224-383 AD, 680-779 AD, and 885-993 AD; zero appears on the oldest. But Kim Plofker et al. [The Bakhshali Manuscript: A Response to the Bodleian Library's Radiocarbon Dating , History of Science in South Asia, 5 134-150] argue that the birch bark folios dated 224-383 AD and 680-779 AD were consecutive in the manuscript, written by the same hand, and contain consecutive parts of the same calculation. If so, zero is not 500 years older than we thought.
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Current cosmology maintains that the universe contains 4.9% ordinary matter, 26.8% dark matter and 68.3% dark energy. But dark matter has never been detected directly. Alternatives have been proposed (and largely ignored); this is a new one. Colin Rourke's A New Paradigm for the Universe formulates Mach's Principle mathematically and removes the need for dark matter. In the proposed theory there is no Big Bang and the universe is much older than current estimates. Available on the arXiv and Amazon.
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Guozhen Wang and Zhouli Xu have answered a long-standing question by proving that the 61-sphere has a unique smooth structure. The only other odd-dimensional spheres with this property have dimensions 1, 3, 5. Download preprint from arXiv or access published paper in the Annals of Mathematics (account needed).
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