82262053
submission
JoshuaZ writes:

*How many spheres of the same size can one pack in a given region? Kepler conjectured that the optimal packing in 3 dimensions was the packing we're used to seeing groceries use to pack oranges. That problem was solved in the 1990s in one of the first computer-aided proofs https://en.wikipedia.org/wiki/Kepler_conjecture However, the version of the question for dimensions other than 2 or 3 has remained open. This is an important problem to solve since dense sphere packings in high dimensions give rise to more efficient error-correcting codes which are important for many practical applications such as communication technology and storage.
For a long time, the problem of proving the optimal packing for 8 and 24 dimensions were open. The optimal packings were believed to be specific packings which arise from the E_8 and the Leech lattice, a special lattice which exists in 24 dimensions https://en.wikipedia.org/wiki/Leech_lattice. Now, it appears that mathematicians have finally proven that the expected packings in 8 and 24 dimensions are actually ideal. Maryna Viazovska proved that the 8-dimensional lattice is ideal http://arxiv.org/abs/1603.04246 and followup work by Viazovska and four other authors http://arxiv.org/abs/1603.06518 adapts the strategy for 8 dimensions to prove that in 24 dimensions the conjectured solution really is the most efficient.*
78303561
submission
JoshuaZ writes:

*November 2nd marked fifteen years of continuous habitation by humans in the International Space Station. Expedition 1 arrived on November 2nd, 2000 and there has been a continual presence through the current Expedition 45. Over that time, the station has grown with the addition of many new modules such as the Destiny module https://en.wikipedia.org/wiki/Destiny_(ISS_module), added in 2001, as well Columbus, was added in 2007 https://en.wikipedia.org/wiki/Columbus_(ISS_module), and Kibo, added also in 2007, https://en.wikipedia.org/wiki/Kibo_(ISS_module), among others. Current plans call for the station to be operational through 2020 but what will happen after that is uncertain. http://www.cbsnews.com/news/nasa-looks-to-post-2020-international-space-station-operations/*
78297137
submission
JoshuaZ writes:

*A major open problem in graph theory is how efficiently one can tell given two graphs whether or not they are isomorphic, that is, the same graph with just the labels changed. This problem is famous along with factoring integers as a problem that is potentially in between P and NP in difficulty. Now, Laszlo Babai has reported that he has a quasipolynomial time algorithm which he sketched out at a set of talks at the University of Chicago http://www.math.uchicago.edu/calendar?calendar=Combinatorics%20and%20Theoretical%20Computer%20Science. Scott Aaronson was one of the first to break the news and his latest blog entry and its comments contains further discussion of the result http://www.scottaaronson.com/blog/?p=2521. The new algorithm places the problem of graph isomorphism as at most just barely above P. Babai's result depends on the classification of finite simple groups https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups a deep result in algebra whose proof consists of thousands of pages over hundreds of distinct papers. Unlike the problem of factoring integers, improvements in this algorithm are unlikely to impact cryptography in any direct way since no cryptographic systems depend on the difficulty of determining when groups are isomorphic.*
74857101
submission
JoshuaZ writes:

*"Researchers from Brown University have tentatively identified an alloy of hafnium, nitrogen and carbon as having an expected melting point of about 7,460 degrees Fahrenheit (4120 Celsius). This exceeds the previous record breaker tantalum hafnium carbide which melts at 7,128 F (3942 C) and had stood as the record holder for almost a century. However, at this point, the record setter is still hypothetical, based on simulations. The new record has not yet been confirmed by experiment. http://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.020104 is the actual article while http://www.washingtonpost.com/news/speaking-of-science/wp/2015/07/28/behold-a-new-record-for-the-worlds-highest-melting-point/ is a lay summary. If the simulations turn out to be correct, the new alloy may be useful in parts like jet engines, and the door will be opened to using similar simulations to search for substances with even higher melting points or with other exotic properties.*