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Geometric flows have been an active topic of research for the last seven years or so,
ever since Perelman's groundbreaking work appeared. In this talk, we begin with a
brief overview of the most well--known flow, namely, Ricci flow. We then move on to our work
on Ricci flow of unwarped and warped product manifolds through a study of generic examples.
In particular, we look at features such as singularity formation, isotropisation at
specific values of the flow parameter and evolution characteristics. Subsequently, we move on to geometric flows with higher order terms such as those involving $(Riemann)^2$. We discuss second order (in Riemann curvature) geometric flows (un-normalised) on locally homogeneous
three manifolds of various types. Novelties specifically associated with the presence of the higher order terms are pointed out. Finally, we discuss a new geometric flow governed by the Bach tensor, which involves higher derivatives of the metric, as well as higher orders.
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