1,677 research outputs found
Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz Numbers
This article introduces mixed double Hurwitz numbers, which interpolate
combinatorially between the classical double Hurwitz numbers studied by
Okounkov and the monotone double Hurwitz numbers introduced recently by
Goulden, Guay-Paquet and Novak. Generalizing a result of Okounkov, we prove
that a certain generating series for the mixed double Hurwitz numbers solves
the 2-Toda hierarchy of partial differential equations. We also prove that the
mixed double Hurwitz numbers are piecewise polynomial, thereby generalizing a
result of Goulden, Jackson and Vakil
Independences and Partial -Transforms in Bi-Free Probability
In this paper, we examine how various notions of independence in
non-commutative probability theory arise in bi-free probability. We exhibit how
Boolean and monotone independence occur from bi-free pairs of faces and
establish a Kac/Loeve Theorem for bi-free independence. In addition, we prove
that bi-freeness is preserved under tensoring with matrices. Finally, via
combinatorial arguments, we construct partial -transforms in two settings
relating the moments and cumulants of a left-right pair of operators
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
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