What's Happening in Mathematics?
page 7 (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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The Ramsey number R(ℓ, k), proved to exist by F.P. Ramsey in 1930, is the minimum n such that every red/blue colouring of the edges of the complete graph K_n contains either a blue K_ℓ or a red K_k. The first reasonable upperbounds were given by P. Erdös and G. Szekeres in 1935. Now Marcelo Campos, Matthew Jenssen, Marcus Michelen, and Julian Sahasrabudhe have found an improved lower bound for R(3,k), within a factor of 3 of the best known upper bound.
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In 1977 B. Halpern and C. Weaver conjectured that any smooth embedded paper Möbius
band must have aspect ratio greater than √3. Richard Evan Schwartz proved this in a preprint in 2023, now published in the Annals of Mathematics. He also proved that any sequence of smooth embedded paper Möbius bands whose aspect ratio converges to √3 must converge, up to isometry, to an equilateral triangle of semi-perimeter √3. |
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When can a manifold have Kervaire invariant 1? Such a shape must be very twisted and weird. In 1969, William Browder proved that dimensions of the form 2^n − 2 are the only ones possible. Later it was proved that such shapes cannot exist in dimensions 254 and higher, and soon only dimension 126 remained open. In a paper posted online, Weinan Lin , Guozhen Wang , and Zhouli Xu have proved that there is such a shape in 126 dimensions, completing the analysis of this problem.
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In 1850 Sir Frederick Pollock conjectured that every positive integer is a sum of at most 11 centred enneagonal (often called 'nonagonal' but that's half-Latin, half-Greek) numbers —discrete analogues of the regular 9-sided polygon. Miroslav Kureš has now proved this with a 5-line proof.
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Curves of constant width in two dimensions are well known. The Reuleaux triangle has miminal area for a given width. The analogous problem in higher dimensions has been open. Now A. Arman and coworkers have constructed a constant-width solid in n dimensions with constant width 2 and volume less than (0.9)^n times the volume of a ball of unit radius. The proof occupies 8 pages.
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The moving sofa problem asks for the shape of smallest area that can be moved round a right-angled corner in a corridor of fixed width. Leo Moser posed it in 1966 and Joseph Gerver proposed a shape nmade from 18 analytic cuves. Now Jineon Baek has proved that Gerver's solution is correct.
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Robert Langlands
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After thirty years of effort, nine mathematicians have collectively proved a key component of the Langlands Programme, which aims to unify three major areas of mathematics: number theory, geometry and function fields. The 800-page proof is spread over five papers. The team was led by Dennis Gaitsgory (Max Planck Institute for Mathematics, Bonn) and Sam Raskin (Yale University).
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The bunkbed conjecture, stated by Pieter Kasteleyn in 1985, is about a pair of graphs stacked on top of each other with vertical links between them. Nikita Gladkov, Igor Pak, Aleksandr Zimin have announced that this highly intuitive conjecture is false. Their counterexample has 7222 vertices.
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The Kakeya set conjecture asserts that a subset of n-dimensional space that contains a unit line segment in every direction, must have Minkowski and Hausdorff dimension equal to n. Hong Wang and Joshua Zahl have released a preprint that resolves the three-dimensional case.
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In 1978 Roger Apéry announced a proof that zeta(3) is irrational. At first it looked crazy, but he was right. The proof seemed to rely on strange coincidences, with no general method. However, Frank Calegari , Vesselin Dimitrov, and Yunqing Tang have now generalised Apéry's method, with results including irrationality proofs for an infinite set of zeta-like values.
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Gregorio Ricci-Curbastro |
In 1968 John Milnor conjectured that any geodesically complete manifold with nonnegative Ricci curvature has finitely generated fundamental group (roughly, finitely many holes). Elia Brué, Aaron Naber, and Daneiel Semola have found a counterexample in 7 dimensions, improved to 6. Dimensions 4 and 5 remain open.
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Scientists at Barcelona Zoo have discovered that giraffes prefer carrots to courgettes (zucchini). Big deal? Well, if these vegetables are mixed in different proportions the giraffes make their choices to maximise their preferred food. So they are making decisions based on statisticalinformation.
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The collective behavior of active agents is a hot area in biological and soft matter physics. Christina Hueschen, Alexander Dunn, and Rob Phillips apply Toner-Tu flocking theory to describe the movement of a herd of wildebeest in hilly terrain.
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The recently discovered “hat” aperiodic monotile mixes unreflected and reflected
tiles in every tiling it admits. David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss show that a close relative of the hat admits only non-periodic tilings if we forbid reflections. Modifying this polygon’s edges gives a family of shapes called spectres that are admit only chiral non-periodic tilings. |
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Sets of tiles that can tile the plane, but not in any periodic lattice pattern, have long been known. Roger Penrose found such a set with just two tiles, but it has been open whether a single tile can do the trick. David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss have now found such a tile.
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In how many ways can n queens be placed on an nxn chessboard so that they do not attack each other? there are 92 different configurations when n = 8, the usual chessboard. Now Michael Simkin has proved that when n is large there are approximately (0.143n)^n configurations. The proof sandwiches the exact number between two bounds, both close together, using sophisticated methods.
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Squaring the circle using ruler and compasses is known to be impossible. In 1990 Miklós Laczkovich showed that a circle can be dissected into finitely many pieces that reassemnble to form a square. (The pieces are complicated and disconnected, unlike a conventional jigsaw.) Andras Máthé, Oleg Pikhurko, and Jonathan Noel have improved his construction. They need about 10^200 pieces, but they're simpler. Conjecturally, 22 will do the job.
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Joshua Evan Greene and Andrew Lobb have proved that for every smooth Jordan curve K and rectangle R in the Euclidean plane, there exists a rectangle similar to R whose vertices lie on K. The proof relies on Shevchishin's theorem that the Klein bottle does not admit a smooth Lagrangian embedding in ℂ^2.
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