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Associating Finite Groups with Dessins D'enfants

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Associating Finite Groups with Dessins D'enfants Introduction and Definitions Dessins d'Enfants Past Work Current Progress Associating Finite Groups with Dessins d'Enfants Luis Baeza Edwin Baeza Conner Lawrence Chenkai Wang Edray Herber Goins, Research Mentor Kevin Mugo, Graduate Assistant Purdue Research in Mathematics Experience Department of Mathematics Purdue University August 8, 2014 Purdue Research in Mathematics Experience Associating Finite Groups with Dessins d'Enfants Introduction and Definitions Dessins d'Enfants Past Work Current Progress Outline of Talk 1 Introduction and Definitions Utilities Problem Riemann Surfaces Finite Automorphism Groups of The Sphere Bely˘ı’sTheorem 2 Dessins d'Enfants Esquisse d'un Programme Passports Riemann's Existence Theorem Platonic Solids 3 Past Work Work of Magot and Zvonkin Archimedean and Catalan Solids Johnson Solids 2013 PRiME 4 Current Progress Approach Johnson Solids with Cyclic Symmetry Johnson Solids with Dihedral Symmetry Future Work Purdue Research in Mathematics Experience Associating Finite Groups with Dessins d'Enfants Introduction and Definitions Utilities Problem Dessins d'Enfants Riemann Surfaces Past Work Finite Automorphism Groups of The Sphere Current Progress Bely˘ı’sTheorem Utilities Problem Question Suppose there are three cottages, and each needs to be connected to the gas, water, and electric companies. Using a third dimension or sending any of the connections through another company or cottage are disallowed. Is there a way to make all nine connections without any of the lines crossing each other? Purdue Research in Mathematics Experience Associating Finite Groups with Dessins d'Enfants Introduction and Definitions Utilities Problem Dessins d'Enfants Riemann Surfaces Past Work Finite Automorphism Groups of The Sphere Current Progress Bely˘ı’sTheorem Graphs A (finite) graph is an ordered pair V ; E consisting of vertices V and edges E. v = jV j = the number of vertices e = jEj = the number of edges f = jF j = the number of faces A connected graph is a graph where, given any pair of vertices z1 and z2, one can traverse a path of edges from one to the other. Two vertices z1 and z2 are adjacent if there is an edge connecting them. A bipartite graph is a graph where the vertices V can be partitioned into two disjoint sets B and W such that no two edges z1; z2 2 B (respectively, z1; z2 2 W ) are adjacent. A planar graph is a graph that can be drawn such that the edges only intersect at the vertices. Purdue Research in Mathematics Experience Associating Finite Groups with Dessins d'Enfants Introduction and Definitions Utilities Problem Dessins d'Enfants Riemann Surfaces Past Work Finite Automorphism Groups of The Sphere Current Progress Bely˘ı’sTheorem Which Graphs are Planar? Theorem (Leonhard Euler, 1750) A finite connected graph is planar if and only if v − e + f = 2. Theorem (Kazimierz Kuratowski, 1930; Klaus Wagner, 1937) A finite graph is planar if and only if it does not have K5 or K3;3 as a minor. Solution Suppose there are three cottages, and each needs to be connected to the gas, water, and electric companies. There is no way to make all nine connections without any of the lines crossing each other because the Utility Graph has v = 6, e = 9, and f = 3. Purdue Research in Mathematics Experience Associating Finite Groups with Dessins d'Enfants Introduction and Definitions Utilities Problem Dessins d'Enfants Riemann Surfaces Past Work Finite Automorphism Groups of The Sphere Current Progress Bely˘ı’sTheorem Can We Draw Graphs on Other Objects? Purdue Research in Mathematics Experience Associating Finite Groups with Dessins d'Enfants Introduction and Definitions Utilities Problem Dessins d'Enfants Riemann Surfaces Past Work Finite Automorphism Groups of The Sphere Current Progress Bely˘ı’sTheorem Riemann Surfaces A Riemann Surface is a triple (X ; fUαg; fµαg) satisfying: Coordinate Charts and Maps: For some countable indexing set I , [ X = Uα and µα : Uα ,! C: α2I Locally Euclidean: Each µα(Uα) is a connected, open subset of C; and the −1 composition µβ ◦ µα is a smooth function. C X C −1 µα µβ ⊆ −−−−−! ⊆ −−−−−! ⊆ µα (Uα \ Uβ ) Uα \ Uβ µβ (Uα \ Uβ ) Hausdorff: For distinct z 2 Uα and w 2 Uβ there exist open subsets µα(z) 2 Uα ⊆ µα(Uα) −1 −1 such that µα (Uα) \ µβ (Uβ ) = ;: µβ (w) 2 Uβ ⊆ µβ (Uβ ) 1 2 3 P (C) = C [ f1g ' S (R) is an example. We always embed X ,! R . Purdue Research in Mathematics Experience Associating Finite Groups with Dessins d'Enfants Introduction and Definitions Utilities Problem Dessins d'Enfants Riemann Surfaces Past Work Finite Automorphism Groups of The Sphere Current Progress Bely˘ı’sTheorem The Sphere as The Extended Complex Plane Through stereographic projection, we can establish a bijection between the 2 unit sphere S (R) and the extended 1 complex plane P (C) = C [ f1g. Definition Define stereographic projection as that map from the unit sphere to the complex plane. 2 ∼ 1 S (R) −! P (C) 2 2 u + i v u; v; w = 2x ; 2y ; x +y −1 7! x + i y = x2+y 2+1 x2+y 2+1 x2+y 2+1 1 − w Purdue Research in Mathematics Experience Associating Finite Groups with Dessins d'Enfants Introduction and Definitions Utilities Problem Dessins d'Enfants Riemann Surfaces Past Work Finite Automorphism Groups of The Sphere Current Progress Bely˘ı’sTheorem M¨obiusTransformations Definition a z + b Rational functions of the form f (z) = where c z + d a b 2 GL2(C) = γ 2 Mat2×2(C) a d − b c 6= 0 c d are called M¨obiusTransformations. We denote the collection of M¨obius 1 Transformations by Aut P (C) . For example, the function f (z) = 1=z is also a M¨obiustransformation. Geometrically, this function represents a flip along the y-axis. Purdue Research in Mathematics Experience Associating Finite Groups with Dessins d'Enfants Introduction and Definitions Utilities Problem Dessins d'Enfants Riemann Surfaces Past Work Finite Automorphism Groups of The Sphere Current Progress Bely˘ı’sTheorem Finite Automorphism Groups of The Sphere Proposition (Felix Klein) 1 The following groups are finite subgroups of Aut P (C) : n Zn = hrjr = 1i : r(z) = ζnz 1 D = hr; sjs2 = r n = (sr)2 = 1i : r(z) = ζ z s(z) = n n z 2 3 3 z + 2 ζ3 z + 2 A4 = hr; sjs = r = (sr) = 1i : r(z) = s(z) = z − ζ3 z − 1 2 3 4 z + ζ4 z + 1 S4 = hr; sjs = r = (sr) = 1i : r(z) = s(z) = z − ζ4 z − 1 3 2 3 5 ' − ζ5 z ' − z A5 = hr; sjs = r = (sr) = 1i : r(z) = 3 s(z) = ' ζ5 z + 1 ' z + 1 p 2πi=n 1+ 5 where ζn = e is a root of unity, and ' = 2 is the golden ratio. 1 Conversely, if G is a finite subgroup of Aut P (C) , then G is isomorphic to one of the five types of groups above. Purdue Research in Mathematics Experience Associating Finite Groups with Dessins d'Enfants Introduction and Definitions Utilities Problem Dessins d'Enfants Riemann Surfaces Past Work Finite Automorphism Groups of The Sphere Current Progress Bely˘ı’sTheorem Bely˘ı’sTheorem Theorem (Andr´eWeil, 1956; Gennadi˘ıVladimirovich Bely˘ı,1979) Let X be a compact, connected Riemann surface. X is a smooth, irreducible, projective variety of dimension 1. In particular, X is an algebraic variety; that is, it can be defined by polynomial equations. P i j If X can be defined by a polynomial equation i;j aij z w = 0 where the coefficients aij are not transcendental, then there exists a rational function 1 β : X ! P (C) which has at most three critical values. 1 Conversely, if there exists rational function β : X ! P (C) which has at most three critical values, then X can be defined by a polynomial equation P i j i;j aij z w = 0 where the coefficients aij are not transcendental. Purdue Research in Mathematics Experience Associating Finite Groups with Dessins d'Enfants Introduction and Definitions Utilities Problem Dessins d'Enfants Riemann Surfaces Past Work Finite Automorphism Groups of The Sphere Current Progress Bely˘ı’sTheorem Bely˘ıMaps 3 Denote X as a Riemann Surface. We always embed X ,! R . 1 A rational function β : X ! P (C) is a map which is a ratio β(z) = p(z)=q(z) in terms of relatively prime polynomials p; q 2 C[X ]; define its degree as deg β = max deg p; deg q . Theorem (Fundamental Theorem of Algebra) 1 −1 For w 2 P (C), denote β (w) = z 2 X p(z) − w q(z) = 0 . Then −1 β (w) ≤ deg β. 1 −1 w 2 P (C) is said to be a critical value if β (w) 6= deg β. A Bely˘ımap is a rational function β such that its collection of critical 1 values w is contained within the set f0; 1; 1g ⊆ P (C). Purdue Research in Mathematics Experience Associating Finite Groups with Dessins d'Enfants Introduction and Definitions Utilities Problem Dessins d'Enfants Riemann Surfaces Past Work Finite Automorphism Groups of The Sphere Current Progress Bely˘ı’sTheorem Examples 1 2 Denote X = P (C) = C [ f1g ' S (R) as the Riemann Sphere. deg β = n : β(z) = z n 4 z n deg β = 2 n : β(z) = z n + 12 p z 4 + 2 2 z3 deg β = 12 : β(z) = p 2 2 z 3 − 13 8 4 3 1 z + 14 z + 1 deg β = 24 : β(z) = 108 z 4 z 4 − 14 20 15 10 5 3 1 z + 228 z + 494 z − 228 z + 1 deg β = 60 : β(z) = 1728 z 5 z 10 − 11 z 5 − 15 Purdue Research in Mathematics Experience Associating Finite Groups with Dessins d'Enfants Introduction and Definitions Esquisse d'un Programme Dessins d'Enfants Passports Past Work Riemann's Existence Theorem Current Progress Platonic Solids Alexander Grothendieck (March 28, 1928 { ??) http://en.wikipedia.org/wiki/Alexander_Grothendieck Purdue Research in Mathematics Experience Associating Finite Groups with Dessins d'Enfants Introduction and Definitions Esquisse d'un Programme Dessins d'Enfants Passports Past Work Riemann's Existence Theorem Current Progress Platonic Solids Dessins d'Enfant 1 Fix a Bely˘ımap β : X ! P (C).
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