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."
Joshua Zelinsky writes: "A man in New York City was arrested for photographing subway trains. A police officer confronted the man and when the man attempted to explain to the officer that what he was doing was legal the officer arrested him. The man was charged with three separate crimes including one count of disorderly conduct for speaking to the officer in an "unreasonable voice." So far, two of the three charges have been dropped but the third is pending."
Joshua Zelinsky writes: "Wikipedia is being . censored by a variety of ISPs in Great Britian. The activity was first noticed when it appeared that almost all edits from Great Britain were coming through only a handful of IP addresses. This created serious problems in preventing vandalism from Great Britain. It then emerged that this was due to the system being used by ISPs to censor Wikipedia. At present it is not known in full which pages are being censored. However, at least one page which is being censored is a page is Virgin Killer about an album from a German heavy metal band which has a naked young girl on the cover. Individuals in Great Britain attempting to access the page report that they are simply getting error messages and not even being told that the page is being censored. Some commentators have already started referring to the Great Firewall of Britain."
Joshua Zelinsky writes: "Researchers estimated the successful response rates to spam by sending out millions of spam messages using the Storm botnet. They received 28 responses out of 350 million messages sent out and estimate that approximately one of every 12,500,000 spam emails generates a sale. Most disturbingly even with this low response rate the researchers estimate that a spammer will still make an easy profit."
Joshua Zelinsky writes: "A Polish company, 4fun.tv, has plagiarized from the Polish Wikipedia by using content without attribution. When Wikinews inquired about the apparent plagiarism, the company first claimed that they had taken the text from an old encyclopedia that was in the public domain that Wikipedia must have taken it from also. When this was shown to be false, the company's representative threatened the reporter saying that "he found the reporter's picture on the Internet and asked how the person would feel if this picture was to appear on billboards or press releases that advertised the TV station""