"The baroque shape has a surprising property that disproves a long-standing geometrical conjecture: no matter how you shift or rotate it, one noperthedron can't fall through a straight hole in an identical noperthedron. Prime Number Patterns Prime numbers, divisible only by themselves and 1, have long fascinated mathematicians. Discovering new ones is difficult as you get to larger and larger numbers. But this year mathematicians have found a set of probabilistic patterns that govern how the primes are distributed."
"The patterns involve random chaotic behavior and fractals. A Grand Unified Theory A gargantuan effort involving nine mathematicians and five papers spanning almost 1,000 pages recently proved the geometric Langlands conjecture. The conjecture connects the properties of different Riemann surfaces, which are structures with coordinates that have real and imaginary parts. It is part of a broader set of problems called the Langlands program, which, if fully proven, could provide a grand unified theory of mathematics."
Researchers achieved major advances across geometry, topology, and number theory. A newly identified noperthedron with 90 vertices, 240 edges and 152 faces disproves a long-standing geometrical conjecture by never falling through a straight hole in an identical copy. Mathematicians discovered probabilistic patterns governing prime distribution that involve random chaotic behavior and fractals. A collaborative effort of nine mathematicians produced five papers totaling almost 1,000 pages proving the geometric Langlands conjecture, linking properties of different Riemann surfaces and advancing the Langlands program toward a unifying mathematical framework. Work also engaged long-standing questions about knot complexity.
Read at www.scientificamerican.com
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