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Description
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.
- Using Markov chains to show that it is possible to extend Lipschitz functions between certain metric spaces.
- Using Markov chains to prove impossibility results for Lipschitz extension problems.
- Using Markov chains to prove that for certain pairs of metric spaces X, Y, any embedding of X into Y must significantly distort distances.
- Using Markov chains to show that good embeddings do exist between certain metric spaces.
- Using Markov chains in metric Ramsey theory.
- Markov chains as an invariant for Lipschitz quotients and a tool to understand isomorphic uniform convexity.
- Markov chains as a tool to prove nonlinear spectral calculus inequalities.