The first of these sections provides an alphabetized set of articles on 99 specific mathematical concepts such as the axiom of choice, expander graphs, and Hilbert space. The second core section includes long surveys of 26 branches of research mathematics such as algebraic geometry and combinatorial group theory. The third describes 38 important mathematical problems and theorems such as the four color theorem, the Birch and Swinnerton-Dyer conjecture, and the Halting problem.[2][3][4][5][6][7] A collection of biographies of nearly 100 famous deceased mathematicians, arranged chronologically,[2][3][4][5][7][9] also including a history of Nicolas Bourbaki's pseudonymous collaboration.[6][9] Essays describing the influences and applications of mathematics in the sciences, technology, business, medicine, and the fine arts.[2][3][4][5] A section of perspectives on the future of mathematics, problem solving techniques, the ubiquity of mathematics, and advice to young mathematicians.[4][5]