## Geometry and Analysis of Dirichlet forms.

#### Yuan ZhouBeijing University of Aeronautics and Astronautics (Beihang University)

First, for a given regular strongly local Dirichlet form $\mathscr E$, under the assumptions of the lower bound of Bakry-Emery's Ricci curvature, local doubling and local Poincar\'e, we obtain the coincidence of the intrinsic differential and distance structures induced by $\mathscr E$. Second, for the Dirichlet form $E_A$ generated by the operator $div (A\nabla)$, we establish some critical relations between the diffusion matrix $A$, and the coincidence of the intrinsic differential and distance structures induced by $E_A$. Finally, we discuss some applications. (Joint work with P.Koskela and N. Shanmugalingam)

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