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.

The space of totally nonnegative real matrices, namely the real n by n matrices with all minors nonnegative, intersected with the ``unipotent radical'' of upper triangular matrices with 1's on the diagonal carries important information related to Lusztig's theory of canonical bases in representation theory.   This space of matrices (and generalizations of it beyond type A) is naturally stratified according to which minors are positive and which are 0, with the resulting stratified space described combinatorially by a well known partially ordered set called the Bruhat order.   I will tell the story of these spaces and in particular of a map from a simplex to these spaces that has recently been used to better understand them.  The fibers of this map encode exactly the nonnegative real relations amongst exponentiated Chevalley generators of a Lie algebra.   This talk will especially focus on recent joint work with Jim Davis and Ezra Miller uncovering overall combinatorial and topological structure governing these fibers.  Plenty of background, examples, and pictures will be provided along the way.