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### Course Information

**Lecturer:**

Mariel Saez, Associate Professor, Pontifica Universidad Catolica de Chile

**Course Synopsis:**

The study of geometric flows has gained a lot of attention in mathematics over the last few decades, both as a powerful tool in addressing problems in several branches of the discipline, as well as for the interest in its own. These flows can often be understood as non-linear quasi-parabolic equations with a geometrical meaning.

An important example is the extrinsic flow, known as the “Mean curvature flow”, that has been extensively studied in the smooth setting (see for instance [10] and references there in). Particularly, when the evolution of curves is considered, the flow is known as “Curve Shortening Flow” and it is fully understood for closed embedded curves. Nonetheless, an early motivations for curve shortening flow comes from the work of Mullins [7] in the 50’s, where the evolving curves are interpreted as evolving interfaces that appear naturally in material sciences. However, in that context it is expected that certain natural “built-in” singularities (which are not present in the smooth case) appear through the evolution. Such behavior is mathematically modelled by the evolution of networks under curve shortening flow, which has proven to be more challenging to study than its smooth counterpart.

In these lectures I will give an overview of the classical theory for smooth curves and compare it to its counterpart in the network case. More precisely, I will discuss the results of Gage-Hamilton and Grayson and more recent proofs of these theorems in [1, 2, 5], then I will define the flow of networks and possible different approaches to to study this flow. I will describe existence results and long time behavior for networks with no loops (c.f. [9]) and some results in the case with loops (c.f. [8]).

**References
**

- B. Andrews and P. Bryan A comparison theorem for the isoperimetric profile under curve-shortening

flow*Comm. Anal. Geom*.,**19**, (3) 503-539, 2011. - K. Chou and X. Zhu The curve shortening problem,
*Chapman & Hall/CRC*,*Boca Raton, FL*,2001. - M. Gage and R. Hamilton. The heat equation shrinking convex plane curves.
*J. Differential Geom*.**23**(1):69-96,

1986 - M . Grayson. The heat equation shrinks embedded plane curves to round points.
*J. Dierential Geom*.**26**

(2):285-314, 1987. - G. Huisken. A distance comparison principle for evolving curves.
*Asian J. Math*.**2**(1):127-133, 1998. - R. Mazzeo and M. Saez Self-similar expanding solutions for the planar network flow.
*Seminaires & Congres***19**,

2008, p. 159-173. - W. Mullins Two dimensional motion of idealized grain boundaries,
*J. Appl. Phys*.,**27**(1956), 900-904. - O. Schnurer, A. Azouani, M. Georgi, J. Hell, N. Jangle, A. Koeller, T. Marxen, S. Ritthaler, M. Saez, F.

Schulze, B. Smith. Evolution of convex shaped lenses networks under curve shortening flow.*Trans. Amer. Math.*

*Soc.***363**(2011), 2265-2294. - C. Mategazza, M. Novaga, A. Pluda and F. Schulze. Evolution of networks with multiple junctions

http://cvgmt.sns.it/media/doc/paper/3172/network-survey.pdf - C. Mantegazza Lecture notes on mean curvature flow,
*Progress in Mathematics*,**290**Birkhauser/Springer Basel

AG, Basel, 2011. - C. Mategazza, M. Novaga and V. Tortorelli Motion by curvature of planar networks,
*Ann. Sc. Norm. Super.*

*Pisa Cl. Sci.***(5)**, 3, 2004.

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### Lecturer Biography

**Associate Professor Mariel Saez, Pontifica Universidad Catolica de Chile**

Mariel Saez is currently an associate professor at the mathematics department of Ponticia Universidad Catolica de Chile. Her research primarily focuses in geometric analysis, with emphasis on geometric flows.

Professor Saez obtain her Ph.D. from Stanford University in 2005 and afterwards held a postdoctoral position at the Max Planck institute for Gravitational Physics in Potsdam, Germany. She has also held visiting positions at several institutions such as Free University in Berlin, the Mathematical Sciences Research Institute (MSRI) in Berkeley, California, Monash University in Melbourne and University College London.