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The deep and fruitful interactions between probability and geometry are well-established, including the powerful use of probabilistic constructions to prove existence of important objects (e.g., Dvoretzky's theorem and numerous other applications of the probabilistic method to prove existence statements), and the study of the behavior of a variety of important stochastic processes in a geometric setting. In this talk we will describe a useful paradigm in metric geometry (originating from the work of Keith Ball) that allows for a probabilistic interpretation of certain geometric questions whose statement does not have any a priori connection to probability. In particular, we will address the following topics.
  1. Using Markov chains to show that it is possible to extend Lipschitz functions between certain metric spaces.
  2. Using Markov chains to prove impossibility results for Lipschitz extension problems.
  3. Using Markov chains to prove that for certain pairs of metric spaces X, Y, any embedding of X into Y must significantly distort distances.
  4. Using Markov chains to show that good embeddings do exist between certain metric spaces.
  5. Using Markov chains in metric Ramsey theory.
  6. Markov chains as an invariant for Lipschitz quotients and a tool to understand isomorphic uniform convexity.
  7. Markov chains as a tool to prove nonlinear spectral calculus inequalities.
All of these applications involve geometric insights that introduce an unexpected link with probability theory, and then proving new probabilistic results that complete the solution of the problem at hand.

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