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Friday, February 5, 2016

**Abstract:** A special case of the Sobolev Embedding Theorem states that if $u\in W^{1,p}(\mathbb{R}^n)$ and, $p\in[1,n)$ the $\frac{np}{n-p}$-integral of $u$ is dominated by the $p$-integral of its gradient. In Euclidean spaces, one has control over the size of a Sobolev function by the size of its gradient. Now suppose we consider metric-measure spaces on which we have similar control locally. We say that a metric-measure space supports a Poincare inequality if the size of every measurable function is controlled by the size of its upper gradients, in the sense of averaged integrals over balls. Examples of such spaces include Euclidean spaces and compact Riemannian manifolds. In this talk, I will state properties that these metric-measure spaces share and show the typical techniques one employs when working in these spaces. Knowledge of Math 540 is assumed.