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.
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.
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.
JoshuaZ writes: "Astronomers have found an unusual small star. SDSS J102915+172927 is a small faint star with very little of any elements other than hydrogen or helium. The star's composition is surprising since standard theories of star formation require heavier elements in small stars in order to allow the stars to be heavy enough to come together. Possibly the most unusual aspect of this star is the complete non-detection of lithium which would be expected in a star of this size. The only elements created shortly after the Big Bang were lithium, hydrogen and helium, and the star should have lithium levels much higher since they should correspond closely with the levels believed to have been formed shortly after the Big Bang. The actual paper can be found at http://www.eso.org/public/archives/releases/sciencepapers/eso1132/eso1132.pdf."