“Enumeration of Grothendieck’s Dessins and KP Hierarchy” by Peter Zograf

Peter Zograf has a new paper on the ArXiv entitled “Enumeration of Grothendieck’s Dessins and KP Hierarchy”.

Abstract.Branched covers of the complex projective line ramified over 0, 1 and \infty (Grothendieck’s Dessins d’Enfant) of fixed genus and degree are effectively enumerated. More precisely, branched covers of a given ramification profile over \infty and given numbers of preimages of 0 and 1 are considered. The generating function for the numbers of such covers is shown to satisfy a PDE that determines it uniquely modulo a simple initial condition. Moreover, this generating function satisfies an infinite system of PDE’s called the KP (Kadomtsev-Petviashvili) hierarchy. A specification of this generating function for certain values of parameters generates the numbers of Dessins of given genus and degree, thus providing a fast algorithm for computing these numbers.

You can download the paper here.


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.
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