During June 4-8, 2012, I lead a research group on the subject at the Research Experiences for Undergraduate Faculty (REUF) at ICERM in Providence, Rhode Island. Here is a summary of the main results we found during the 2012 summer program.

- Every Belyĭ map of degree is in the form where .

- Up to fractional linear transformation, every Belyĭ map of degree is in the form where .

- Consider four distinct complex numbers , , , and with cross-ratio and define the rational function . Then is a Belyĭ map whose associated Dessin d’Enfant has vertices and .

- Every planar complete bipartite graph be realized as the Dessin d’Enfant of some Belyĭ map, namely either or .

- Every path graph be realized as the Dessin d’Enfant of some Belyĭ map, namely .

- Every bipartite cycle graph be realized as the Dessin d’Enfant of some Belyĭ map, namely .

- The Möbius Transformations and generate a subgroup of isomorphic to .

- Let be a rational function. The composition is a Belyĭ map for every Belyĭ map if and only if is a Belyĭ map which maps the set to itself.

- Let be the Dessin d’Enfant associated to a Belyĭ map . For each , let be the Dessin d’Enfant associated to the composition . That is, , , and .

- If , then is the original Dessin d’Enfant.

- If , then can be obtained from by interchanging the white vertices with the midpoints of the faces .

- If , then can be obtained from by interchanging the black vertices with the white vertices . In other words, is the dual graph to .

- If , then can be obtained from by interchanging the black vertices with the midpoints of the faces .

- If , then can be obtained from by cyclically rotating the black vertices to the midpoints of the faces to the white vertices .

- If , then can be obtained from by cyclically rotating the black vertices to the white vertices to the midpoints of the faces .

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## About Edray Herber Goins, Ph.D.

Edray Herber Goins grew up in South Los Angeles, California. A product of the Los Angeles Unified (LAUSD) public school system, Dr. Goins attended the California Institute of Technology, where he majored in mathematics and physics, and earned his doctorate in mathematics from Stanford University. Dr. Goins is currently an Associate Professor of Mathematics at Purdue University in West Lafayette, Indiana. He works in the field of number theory, as it pertains to the intersection of representation theory and algebraic geometry.