The cylindrical singularities are prevalent but complicated in geometric flows. We discuss one of the simplest extrinsic flows, the mean curvature flow, and illustrate how the local dynamics of the singularities influence the singular set itself, and the geometry and topology of the flow. This talk is based on a series of joint work with Zhihan Wang (Cornell) and Jinxin Xue (Tsinghua).

In joint work with Lev Rozansky we construct a coherent sheaf realization of the tensor powers of $U_q(gl(1|1))$ vector representation $\mathbb{C}^{1|1}$. The construction reveals the non-semisimple nature of the representation of $U_q(gl(1|1))$. I plan to explain how this
construction extends to а wider class of the representations. The origin of the construction is a boson/fremion correspondence in physics context and Legandre transformation in the mathematical context.