Georgia Institute of Technology College of Sciences

Fix a complete non-Archimedean valued field K. Any subscheme X of (K^*)^n can be "tropicalized" by taking the (closure) of the coordinate-wise valuation. This process is highly sensitive to coordinate changes. When restricted to group homomorphisms between the ambient tori, the image changes by the corresponding linear map. This was the foundational setup of tropical geometry. In recent years the picture has been completed to a commutative diagram including the analytification of X in the sense of Berkovich. The corresponding tropicalization map is continuous and surjective and is also coordinate-dependent. Work of Payne shows that the Berkovich space X^an is homeomorphic to the projective limit of all tropicalizations. A natural question arises: given a concrete X, can we find a split torus containing it and a continuous section to the tropicalization map? If the answer is yes, we say that the tropicalization is faithful. The curve case was worked out by Baker, Payne and Rabinoff. The underlying space of an analytic curve can be endowed with a polyhedral structure locally modeled on an R-tree with a canonical metric on the complement of its set of leaves. In this case, the tropicalization map is piecewise linear on the skeleton of the curve (modeled on a semistable model of the algebraic curve). In higher dimensions, no such structures are available in general, so the question of faithful tropicalization becomes more challenging. In this talk, we show that the tropical projective Grassmannian of planes is homeomorphic to a closed subset of the analytic Grassmannian in Berkovich sense. Our proof is constructive and it relies on the combinatorial description of the tropical Grassmannian (inside the split torus) as a space of phylogenetic trees by Speyer-Sturmfels. We also show that both sets have piecewiselinear structures that are compatible with our homeomorphism and characterize the fibers of the tropicalization map as affinoid domains with a unique Shilov boundary point. Time permitted, we will discuss the combinatorics of the aforementioned space of trees inside tropical projective space. This is joint work with M. Haebich and A. Werner (arXiv:1309.0450).

The Alpert multiwavelets are an extension of the Haar wavelet to higher degree piecewise polynomials thereby giving higher approximation order. This system has uses in numerical analysis in problems where shocks develop. An orthogonal basis of scaling functions for this system are the Legendre polynomials and we will examine the consequence of this. In particular we will show that the coefficients in the refinement equation can be written in terms of Jacobi polynomials with varying parameters. Difference equationssatisfied by these coefficients will be exhibited that give rise to generalized eigenvalue problems. Furthermore an orthogonal basis of wavelet functions will be discussed that have explicit formulas as hypergeometric polynomials.

Incompressible Euler equation is known to be the geodesic flow on the manifold of volume preserving maps. In this informal seminar, we will discuss how this geometric and Lagrangian point of view may help us understand certain analytic and dynamic aspects of this PDE.

I will discuss the variational approach to determining the stability of pendant liquid drops. The outline will include some theoretical aspects and questions which currently can only be answered numerically.