Georgia Institute of Technology College of Sciences

This talk evolves around Markov functions, i.e., when a function of a Markov chain results in another Markov chain. We focus on two examples where this concept yields new results and insights: (1) the evolution of reflected stochastic processes in the study of stochastic networks, and (2) spectral analysis for a special high-dimensional Markov chain.

(Joint with: M. Benedicks and D. Schnellmann) Many interesting dynamical systems possess a unique SRB ("physical")measure, which behaves well with respect to Lebesgue measure. Given a smooth one-parameter family of dynamical systems f_t, is natural to ask whether the SRB measure depends smoothly on the parameter t. If the f_t are smooth hyperbolic diffeomorphisms (which are structurally stable), the SRB measure depends differentiably on the parameter t, and its derivative is given by a "linear response" formula (Ruelle, 1997). When bifurcations are present and structural stability does not hold, linear response may break down. This was first observed for piecewise expanding interval maps, where linear response holds for tangential families, but where a modulus of continuity t log t may be attained for transversal families (Baladi-Smania, 2008). The case of smooth unimodal maps is much more delicate. Ruelle (Misiurewicz case, 2009) and Baladi-Smania (slow recurrence case, 2012) obtained linear response for fully tangential families (confined within a topological class). The talk will be nontechnical and most of it will be devoted to motivation and history. We also aim to present our new results on the transversal smooth unimodal case (including the quadratic family), where we obtain Holder upper and lower bounds (in the sense of Whitney, along suitable classes of parameters).

Abstract: Four dimensions is unique in many ways. For example, n-dimensional Euclidean space has a unique smooth structure if and only if n is not equal to four. In other words, there is only one way to understand smooth functions on R^n if and only if n is not 4. There are many other ways that smooth structures on 4-dimensional manifolds behave in surprising ways. In this talk I will discuss this and I will sketch the beautiful interplay of ideas (you got algebra, analysis and topology, a little something for everyone!) that go into proving R^4 has more that one smooth structure (actually it has uncountably many different smooth structures but that that would take longer to explain).

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).