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Professor Michael Magee's Outputs (5)

On Selberg's Eigenvalue Conjecture for moduli spaces of abelian differentials (2019)
Journal Article
Magee, M. (2019). On Selberg's Eigenvalue Conjecture for moduli spaces of abelian differentials. Compositio Mathematica, 155(12), 2354-2398. https://doi.org/10.1112/s0010437x1900767x

J.-C. Yoccoz proposed a natural extension of Selberg’s eigenvalue conjecture to moduli spaces of abelian differentials. We prove an approximation to this conjecture. This gives a qualitative generalization of Selberg’s theorem to moduli spaces of abe... Read More about On Selberg's Eigenvalue Conjecture for moduli spaces of abelian differentials.

An asymptotic formula for integer points on Markoff-Hurwitz varieties (2019)
Journal Article
Gamburd, A., Magee, M., & Ronan, R. (2019). An asymptotic formula for integer points on Markoff-Hurwitz varieties. Annals of Mathematics, 190(3), 751-809. https://doi.org/10.4007/annals.2019.190.3.2

We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation x21+x22+⋯+x2n=ax1x2⋯xn+k. When n≥4, the previous best result is by Baragar (1998) that gives an exponential rate of growth with exponent β that is... Read More about An asymptotic formula for integer points on Markoff-Hurwitz varieties.

The cycle structure of a Markoff automorphism over finite fields (2019)
Journal Article
Cerbu, A., Gunther, E., Magee, M., & Peilen, L. (2020). The cycle structure of a Markoff automorphism over finite fields. Journal of Number Theory, 211, 1-27. https://doi.org/10.1016/j.jnt.2019.09.022

We begin an investigation of the action of pseudo-Anosov elements of Out(F2) on the Marko-type varieties X : x2 + y2 + z2 = xyz + 2 + over nite elds Fp with p prime. We rst make a precise conjecture about the permutation group generated by Out(F2) on... Read More about The cycle structure of a Markoff automorphism over finite fields.

Matrix group integrals, surfaces, and mapping class groups I: U(n) (2019)
Journal Article
Magee, M., & Puder, D. (2019). Matrix group integrals, surfaces, and mapping class groups I: U(n). Inventiones Mathematicae, 218(2), 341-411. https://doi.org/10.1007/s00222-019-00891-4

Since the 1970’s, physicists and mathematicians who study random matrices in the GUE or GOE models are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces.We establish a new aspect of this t... Read More about Matrix group integrals, surfaces, and mapping class groups I: U(n).