Let g,n be an orientable surface of genus g with n punctures. We study actions of the
mapping class group Mod_g,n of g,n via Hodge-theoretic and arithmetic techniques. We …

We show that the minimum rank of a non-isotrivial local system of geometric origin on a
suitably general $ n $-pointed curve of genus $ g $ is at least $2\sqrt {g+ 1} $. We apply this …

The solutions of the (nonlinear) Painlevé VI differential equation having icosahedral linear
monodromy group will be classified up to equivalence under Okamoto's affine F 4 Weyl …

We survey some recent developments at the interface of algebraic geometry, surface
topology, and the theory of ordinary differential equations. Motivated by" non-abelian" …

The behavior and the location of singular points of a solution to Painlevé VI equation could
encode important geometric properties. For example, Hitchin's formula indicates that …

We classify the finite orbits of the mapping class group action on the character variety of
Deroin--Tholozan representations of punctured spheres. In particular, we prove that the …

Motivated in part by the modular properties of enumerative invariants of K3-fibered Calabi-
Yau threefolds, we introduce a family of 39 Calabi-Yau mirror pairs $(X, Y) $ with $ h_ {1 …

Grothendieck's dessins d'enfants are applied to the theory of the sixth Painleve and Gauss
hypergeometric functions, two classical special functions of isomonodromy type. It is shown …

Using an expansion method in the variables xi that appear in (n− 1)-dimensional integrals
representing the n-particle contribution to the Ising square lattice model susceptibility χ, we …

This is basically the text of a survey talk (entitled'Painleve, Klein and the icosahedron') given
at Hitchin's 60th birthday conference. It discusses the search for and construction of …