power." However, the program faced heavy criticism from the mathematics community for its omission of topics such as the mean value theorem, and for its perceived lack of mathematical rigor. Cryptanalysis work During World War II Gleason was part of OP-20-G, the U.S. Navy's signals intelligence and cryptanalysis group. One task of this group, in collaboration with British cryptographers at Bletchley Park such as Alan Turing, was to penetrate German Enigma machine communications networks. The British had great success with two of these networks, but the third, used for German-Japanese naval coordination, remained unbroken because of a faulty assumption that it employed a simplified version of Enigma. After OP-20-G's Marshall Hall observed that certain metadata in Berlin-to-Tokyo transmissions used letter sets disjoint from those used in Tokyo-to-Berlin metadata, Gleason hypothesized that the corresponding unencrypted letters sets were A-M (in one direction) and N-Z (in the other), then devised novel statistical tests by which he confirmed this hypothesis. The result was routine decryption of this third network by 1944. (This work also involved deeper related to permutation groups and the graph isomorphism problem.) OP-20-G then turned to the Japanese navy's "Coral" cipher. A key tool for the attack on Coral was the "Gleason crutch", a form of Chernoff bound on tail distributions of sums of independent random variables. Gleason's classified work on this bound predated Chernoff's work by a decade. Toward the end of the war he concentrated on documenting the work of OP-20-G and developing systems for training new cryptographers. In 1950 Gleason returned to active duty for the Korean War, serving as a Lieutenant Commander in the Nebraska Avenue Complex (which much later became the home of the DHS Cyber Security Division). His cryptographic work from this period remains classified, but it is known that he recruited mathematicians and taught them cryptanalysis. He served on the advisory boards for the National Security Agency and the Institute for Defense Analyses, and he continued to recruit, and to advise the military on cryptanalysis, almost to the end of his life. Mathematics research Gleason made fundamental contributions to widely varied areas of mathematics, including the theory of Lie groups, quantum mechanics,<ref name="ag-qm">. </ref> and combinatorics. According to Freeman Dyson's famous classification of mathematicians as being either birds or frogs, Gleason was a frog: he worked as a problem solver rather than a visionary formulating grand theories.<ref name=50years>.</ref> Hilbert's fifth problem In 1900