A classical theorem of Jordan asserts that each finite subgroup of the complex general linear group GL(n) 

is ``almost commutative": it contains a commutative normal subgroup 

with index bounded by an universal constant that depends only on n.

We discuss an analogue of this property for the groups of birational (and biregular)

 automorphisms of complex algebraic varieties

and the groups of diffeomorphisms of real manifolds. 

In water-limited regions, competition for water resources results in the formation of vegetation patterns; on sloped terrain, one finds that the vegetation typically aligns in stripes or arcs. The dynamics of these patterns can be modeled by reaction-diffusion PDEs describing the interplay of vegetation and water resources, where sloped terrain is modeled through advection terms representing the downhill flow of water. We focus on one such model in the 'large-advection' limit, and we prove the existence of traveling planar stripe patterns using analytical and geometric techniques. We also discuss implications for the stability of the resulting patterns, as well as the appearance of curved stripe solutions.

This is a joint Nonlinear PDEs seminar with Analysis seminar

A neighborhood of the zero section of the normal bundle of an embedded complex manifold can be seen as a first-order approximation of a neighborhood of the embedded manifold. One would like to know if these two neighborhoods are biholomorphically equivalent. This can be realized as a linearization problem. There are formal obstructions

to the linearization. The Grauert's formal principle is to determine whether the two neighborhoods are holomorphically equivalent when formal obstructions vanish. We will present convergence results under small divisors conditions similar to those in local complex dynamical systems, but in the form represented via cohomology groups in connection with tangent and normal bundles of the embedded manifold. This is joint work with Laurent Stolovich.