Felipe A. Ramirez

Different Speeds Suffice for Rendezvous of Two Agents on Arbitrary Graphs

We consider the rendezvous problem for two robots on an arbitrary connected graph with n vertices and all its edges of length one. Two robots are initially located on two different vertices of the graph and can traverse its edges with different but constant speeds. The robots do not know their own speed. During their movement they are allowed to meet on either vertices or edges of the graph. Depending on certain conditions reflecting the knowledge of the robots we show that a rendezvous algorithm is always possible on a general connected graph.

Higher cohomology for Anosov actions on certain homogeneous spaces

We study the smooth untwisted cohomology with real coefficients for the action on [SL(2,ℝ)×…×SL(2,ℝ)]/Γ by the subgroup of diagonal matrices, where Γ is an irreducible lattice. We show that in the top degree, the obstructions to solving the coboundary equation come from distributions that are invariant under the action. We also show that in intermediate degrees, the cohomology trivializes. It has been conjectured by A. Katok and S. Katok that, analogously to Livšic’s theorem for Anosov flows for a standard partially hyperbolic ℝd - or ℤd - action, the obstructions to solving the top-degree coboundary equation are given by periodic orbits, and also that the intermediate cohomology trivializes, as it is known to do in the first degree by work of Katok and Spatzier. Katok and Katok proved their conjecture for abelian groups of toral automorphisms. Our results verify the “intermediate cohomology” part of the conjecture for diagonal subgroup actions on SL(2,ℝ)d /Γ and are a step in the direction of the “top-degree cohomology” part.

Limit theorems for rank-one Lie groups

We investigate asymptotic behaviour of averaging operators for actions of simple rank-one Lie groups. It was previously known that these averaging operators converge almost everywhere, and we establish a more precise asymptotic formula that describes their deviations from the limit.

Invariant distributions and cohomology for geodesic flows and higher cohomology of higher-rank Anosov actions

Abstract We are motivated by a conjecture of A. and S. Katok to study the smooth cohomologies of a family of Weyl chamber flows. The conjecture is a natural generalization of the Livsic Theorem to Anosov actions by higher-rank abelian groups; it involves a description of top-degree cohomology and a vanishing statement for lower degrees. Our main result, proved in Part II, verifies the conjecture in lower degrees for our systems, and steps in the “correct” direction in top degree. In Part I we study our “base case”: geodesic flows of finite-volume hyperbolic manifolds. We describe obstructions (invariant distributions) to solving the coboundary equation in unitary representations of the group of orientation-preserving isometries of hyperbolic N -space, and we study Sobolev regularity of solutions. (One byproduct is a smooth Livsic Theorem for geodesic flows of hyperbolic manifolds with cusps.) Part I provides the tools needed in Part II for the main theorem.

Higher cohomology of parabolic actions on certain homogeneous spaces

We show that for a parabolic Rd-action on PSL(2,R)d/Γ, the cohomologies in degrees 1 through d − 1 trivialize, and we give the obstructions to solving the degree-d coboundary equation, along with bounds on Sobolev norms of primitives. In previous papers, we have established these results for certain Anosov systems. This work extends the methods of those papers to systems that are not Anosov. The main new idea is defining special elements of representation spaces that allow us to modify the arguments from the previous papers. We discuss how to generalize this strategy to Rd-systems coming from a product of Lie groups, as in the systems analyzed here.