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[PAST EVENT] Mathematics Colloquium
November 22, 2013
2pm - 3pm
Abstract: In 1884, Felix Klein wrote his influential book, ``Lectures on the Icosahedron,'' where he explained how to express the roots of any quintic polynomial in terms of elliptic modular functions. His idea was to relate rotations of the icosahedron with the automorphism group of 5-torsion points on a suitable elliptic curve. In fact, he created a theory which related rotations of each of the five regular solids (the tetrahedron, cube, octahedron, icosahedron, and dodecahedron) with the automorphism groups of 3-, 4-, and 5-torsion points.
Using modern language, the functions which relate the rotations with elliptic curves are BelyI maps. In 1984, Alexander Grothendieck introduced the concept of a Dessin d'Enfant in order to understand Galois groups via such maps. We will complete a circle of ideas by reviewing Klein's theory with an emphasis on the octahedron; explaining how to realize the five regular solids (the Platonic solids) as well as the thirteen semi-regular solids (the Archimedean solids) as Dessins d'Enfant; and discussing how the corresponding BelyI maps relate to moduli spaces of elliptic curves.
Using modern language, the functions which relate the rotations with elliptic curves are BelyI maps. In 1984, Alexander Grothendieck introduced the concept of a Dessin d'Enfant in order to understand Galois groups via such maps. We will complete a circle of ideas by reviewing Klein's theory with an emphasis on the octahedron; explaining how to realize the five regular solids (the Platonic solids) as well as the thirteen semi-regular solids (the Archimedean solids) as Dessins d'Enfant; and discussing how the corresponding BelyI maps relate to moduli spaces of elliptic curves.
Contact
Chi-Kwong Li