I will introduce Painleve equations which are first discovered by Painleve and Gambier in classifying degree 1 2nd order ODEs that doesn’t have movable branch points. I’ll show how Painleve equations and 4d rank one N=2 theories are related. In particular, the realization of Painleve equation as isomonodromic deformations of system of linear ODEs in isospectral deformations limit can be related to the oper limit of flat connection of Hitchin systems. The relation to 4d gauge theory provides a way to construct general solutions to Painleve equations in terms of 4d gauge theory quantities like instanton partition function.

I will introduce Painleve equations which are first discovered by Painleve and Gambier in classifying degree 1 2nd order ODEs that doesn’t have movable branch points. I’ll show how Painleve equations and 4d rank one N=2 theories are related. In particular, the realization of Painleve equation as isomonodromic deformations of system of linear ODEs in isospectral deformations limit can be related to the oper limit of flat connection of Hitchin systems. The relation to 4d gauge theory provides a way to construct general solutions to Painleve equations in terms of 4d gauge theory quantities like instanton partition function.

Equations of hypertree divisors on the Grothendieck-Knudsen moduli space of stable rational curves, introduced by Castravet and Tevelev, appear as numerators of scattering amplitude forms for n massless particles in N=4 Yang-Mills theory in the work of Arkani-Hamed, Bourjaily, Cachazo, Postnikov and Trnka. We re-interpret and generalize leading singularities of MHV scattering amplitude forms as probabilistic Brill-Noether theory: the study of statistics of images of n marked points on a Riemann surface under a random meromorphic function. This leads to a beautiful physics-inspired geometry for various classes of algebraic curves: smooth, stable, hyperelliptic, real algebraic, etc.