After successful completion of the course, students are able to use the basic techniques and notions used commonly in the theory of geometric flows of curves. Some of these techniques and notions are the Maximum principle, Monotonicty/Entropy formulas, Harnack estimates, ancient solutions, and singularities models.

In this course we study a beautiful geometric heat equation known as the curve shortening flow (CSF). Given a simple closed curve in the Euclidean plane. This time dependent process manipulates the curve by moving its points perpendicularly to the curve with a speed proportional to the curvature; convex points move inwards, and concave points move outwards. A celebrated theorem of Grayson states that CSF shrinks any simple closed curve to a "round" point. That is, the rescaled solution converges to a circle as we approach the maximal time of existence. In this course, I provide a rigorous proof of this theorem and along the way you learn about some important concepts in the theory of geometric flows such as the maximum principle, singularity, Harnack estimate as well as entropy and monotonicity estimates.

**The first lecture and organizational meeting: **

Time: Mar 7, 2023 11:00 AM Vienna

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https://tuwien.zoom.us/j/67954671874